INTELLECTUAL A RTHM _ETIC OI, AN ANXLYSIS OF TIlE SCIENCE OF NUMIBERSS WITH ESPECIAL RE FERENCE TO MENTAI. TRAINING AND DEVELOPMENT BY CHARLES DAVIES, LL.D., LU'JTHOR OF A SERIES OF ARITHMETICS, ELEMENTARY ALGEBRt, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE GEOMETRY. SHADES, SHADOWS, AND PERSPECTIVE, ANALYTICAL GEOMETRY, AND DIFFERENTIAL AND INTEGRAL CALCULUS. 1 O3 NEW YORK: PUBLISHED BY A. S. BARNES & CO, tlTI JM(LRE: J. WV. BOND & 0.-CINCINNATI: 11. W. DERBY. —HICAGo(... OOK & CO.-sBT. LOUIS: L. & A. CARR.-NEW ORLEANS: W. F'LDEMM'ING.-MOBILE: RANDALL & WILLIAMS. SUGGESTIONS. Tugs work is designed both for primary and advanced classes The first part is adapted to beginners, while the latter part is peculiarly fitted to give to the more advanced student that tiho rough mental drilling, in the Analysis of NVumbers, which fur nishes the true basis of all mathematical knowledge.'It is suggested that classes in Higher Arithmetic, and even in Algebra, not familiar with works of this kind, will be greatly benefited by a thorough exercise in this most important branch of mathematical science. The Teacher should require the class to dispense with their books at the time of recitation, He should read each example, and then call upon some member of the class to solve it. The pupil should rise and repeat the example in the same language used by the teacher, and should then proceed to analyze it. The analysis will be found to consist of three parts; two pro positions and a conclusion; thus: What will 4 barrels of cider cost at 3 dollars a barrel? 1ST PROPOSITION: Four barrels will cost 4 times as much as 1 barrel. 2D PROPOSITION: If I barrel costs 3 dollars, 4 barrels will cost 4 times 3 dollars, which-are 12 dollars: CONCLUSION: Therefore, 4 barrels of cider at 3 dollars a barrel, will cost 12 dollars. The pupil should never be allowed to omit either of the steps; and he should be required alwavs to adhere strictly to a correct and uniform phraseology in the analysis. The forms of analysis are thought to be of great service bot!: to the teacher and pupil. It is also suggested, that the pupil be thoroughly drilled is Lessons III. and IV., Sect. VII., ls bhey afford very valuable mental exercises and a great t:iOeti if Arithmetical processes Entered according to Act of Congress, in the year One Thousand Eiglt Hundred and Fifty-four, BY CHARLES DAVIES. In the Clerk's Office of the District Court of the Southern District of New York. INTRODUCTION. EVERY book of instruction should have a specific object to which the entire work, both in matter and method, should strictly conform. It is the object of this book to train and develop the mind by means of the science of numbers. Numbers are the instruments here employed to, strengthen the mnemnory, to cultivate the faculty of abstraction and to give force and vigor to the reasoning powers. All our ideas of numbers are either of unity or of multiplicity —unity being the elementary idea from which all others are derived. A true analysis must conform to the nature of the subject analyzed. It must separate all the ideas and principles into their primary elements, and then explain and make manifest the laws by which these elements are connected with each other. Hence, the analysis of numbers must begin with the unit 1, —for this is the foundation, and the science is but the development of the various processes by which all other numbers are derived from 1, as a base, and a comparison of the base 1, with the numbers so derived. Every number has what we call a base: that is, "number being a collection of things* of the same kind," one of these things is the base of the number; and this thing, is called a unit. If we have the numl. iV INTRODUCTION. her 3 hundred, we may consider it in several points of view: I st. It is one hundred taken 3 times, and if we regard one hundred as the base, then, the base is taken 3 tinmes to make up the number; and 100 is the unit. 2nrd. We may consider the number as made up cf 30 tens, and if we regard 10 as the base, then the base is taken 30 times; and 10 is the unit. 3rd. We may also consider the number as made up of 300 ones, in which, the base is 1, and the unit of the nuinmber 1. Again, if we analyze the number, cwt. qr. lb. oz. dr. 13 2 20 12 4 We see, that lcwt. is the base of 13cwt.; qgr. the base of 2qr.; llb. the base of 2016.; loz. the base of 12oz.; and ldr. the base of 4dr4.; and all these bases may be referred to 1 dram as a primary base; hence, as in simple numbers, every base may be referred to the unit 1: therefore, in every entire number, 1 is the primnary base. Let us see if the same be true in fractional numbers. If we have the fraction I it denotes: 1st. That something regarded as a whole has been divided into 8 equal parts: and, 2nd. That 7 of these parts are taken. In this collection of 7 things, (each of which is -), i is the base of the fiactional number; butt it is not the primary base; for 8 implies, either 8 of 1 or - of solme collection of l's; if a collection of l's we call tnat collection unity, which may be referred to the primary base 1: hlence, every number, either integral or fractional, has the Ztnit 1 for a primary base. A fractional number, therefore, is merely a collee INTRODUCTION. V tion of the equal parts of unity, and to one of these parts we give the name of fractional unil. The unlit which is divided is called the unit of the friaction, and may be a collection of units, (as what is 2 of 40?) or it may be the unit 1. The term UNITY, in mathematical science, is applied -to any number or quantity regarded as a whole: the term unit, in arithmetic, to any number which is used as the base of a collection. Thus, 10 is a unit of the second order, being the base for the collection of 10's 100 is a unit of the third order, being the base for the collection of hundreds, and similarly for other bases. Thus, also, in the fraction 7, 1 is the fractional unit, being the fractional base, while the primary base is the unit 1. Every arithmetical process, therefore, has a direct reference to the unit I; and with this view of the subject before him, the pupil always has the means of making a correct analysis. Addition is the process of finding a number which shall contain as many units, and no more, as are found in all the numbers added. Multiplication is taking one number, called the multiplioand, as many times as there are units in another number, called the multiplier, and the number which shows the result of such taking, is called the product: and similarly foer all other arithmetical processes. A clear conception of elementary principles, by which we mean, those principles that result from a final analysis, lies at the foundation of all knowledge. It is not till we get such conceptions, and have learned the laws by which they are connected, that we have acquired any thing deserving the name of science. VI INTRODUCTION. To learn one thing at a time-to leain that thing thoroughly-and to learn its connections with other things are the golden steps that leadlo the temple of knowledge. It will be seen, in Lessons XVI. and XVII., Section VII., that UNITY has been em2loyed to denote any nlumber entering into an arithmetical question. This use of unity affords a powerful means of solving most questions which otherwise present great difficulties; uald is, it is believed, a link of closer connection between the subjects of arithmetic and algebra, than has before been used. It has been the author's aim, in the present work, to treat the subject of number in accordance with these principles, and to give to the whole a scientific form, ald logical development. That he might not fail in so difficult and delicate an undertaking, he has defined all tile termls, and given a full analysis of every process employed. The work is complete in itself. It is a mental analysis of the science of numbers, designed to be accessible to the youngest pupils because of its simple gradations, and useful to the advanced pupil because of its scientific arrangement, its logical connections and its higher analysis of the properties and relations of niumbers. In the preparation of this work, many valuable suggestions and methods have been furnished by practical teachers. They were cheerfully offered and thanklfully adopted. ELSHnKILL LANDING, Februaly, 1854. INTELLECTUAL ARITHMETIC. SECT-ION FIRST. LESSON I. Counting. One, * O........ Two,........ o o.. Three,..... Four,.. - Five,........... -;.. * * Six, Seven,...-.. Eight,..... -,... -..: Nine,....... -. Ten,............. * * * *- i Eleven,.... e * - -9-i -~- - ~ Twelve,.... Thirteen,.... e.... ~ 3 ~ ~ > ~ Fourteen,.... ~ ~. _.~ ~ ~ Fifteen, -..... -.- Sixteen,....... - - -,Seventeen,. - Eighteen,. E,. "' ~ 8 i. ~ 9. Nineteen,, " 01+, 4-: Twenty,q.. - ~ D,- D, X.:,,' SUGGESTIONs.-There is but one simple idea in Arithmetic-it is the idea of the unit ONE,. Any collection of units is a number. Hence, every number is derived from one, and consequently has one for a base. Countmg is merely naming numbers. In this lesson, the names of nulmbers are written opposite the collection. How many units in four a In six apples, whaet is the, unit t What is the unit in seven pears? How many units in twelve peaches? What is the unit that is counted in the lesson? Eow nany stars in the 4th line? How many in the 14th? Fronm what are all numbers derived? What is the base of ivery number I 8 INTELLECTUAL ARITHMETIC. [SECo. LESSON II. Pigures front One to Twenty..................... 7.............. 14....... a 13............. *14.......~..~, -,- - 15.r, s a i4..........@c- ~ c> ~:~ > W -~s ~ ~-~ — YY~. 17.... 20 g. -. Which figure stands for two? Which figure stands for four? Which figure stands for nine? Which stands for eight? What stands for ten? What stands for twelve? What stands for fourteen? What stands for sixteen? What stands for eighteen? What stands for twenty? What stands fr seven. teen? What stands for fifteen? What stands for nineteen? WThat stands for thirteen? SuGeEsTIoNs.-This lesson is intended to teach that numhber may be expressed by figures, as well as by words. The teacher Should explain to the pupil that the figure 2 and the word two, have the same meaning, and similarly for every fig-ire and its corresponding word. Either the figure or the word, denotes m many units as its name points out. LES. III] INTELLECTUAL ARITHMETIC. 9 LESSON III. Figures from One to One Hundred. Naught. 0 Thirty-four. 34 Sixty-eight. 68 One... 1 Thirty-five. 35 Sixty-nine. 69 Two.... 2 Thirty-six. 36 Seventy.. 70 Three... 3 Thirty-seven 37 Seventy-one. 71 Four... 4 Thirty-eight. 38 Seventy-two. 72 Five.. 5 Thirty-nine. 39 Seventy-three 73 Six.... 6 Forty... 40 Seventy-four 74 Seven... 7 Forty-one. 41 Seventy-five. 75 Eight.. 8 Forty-two. 42 Seventy-six. 76 Nine... 9 Forty-three. 43 Seventy-seven 77 Ten... 10 Forty-four. 44 Seventy-eight 78 Eleven.. 11 Forty-five. 45 Seventy-nine 79 Twelve.. 12 Forty-six.. 46 Eighty.. 80 Thirteen.. 13 Forty-seven. 47 Eighty-one. 81 Fourteen.. 14 Forty-eight. 48 Eighty-two. 82 Fifteen.. 15 Forty-nine. 49 Eighty-three 83 Sixteen.. 16 Fifty... 50 Eighty-four. 84 Seventeen. 17 Fifty-one.. 51 Eighty-five. 85 Eighteen..18 Fifty-two. 52 Eighty-six. 86 Nineteen.. 19 Fifty-three. 53 Eighty-seven 87 Twenty.. 20 Fifty-four. 54 Eighty-eight 88 Twenty-one. 21 Fifty-five. 55 Eighty-nine. 89 Twenty-two. 22 Fifty-six.. 56 Ninety.. 90 Twenty-three 23 Fifty-seven. 57 Ninety-one. 91 Twenty-four. 24 Fifty-eight. 58 Ninety-two. 92 Twenty-five. 25 Fifty-nine. 59 Ninety-three 93 Twenty-six. 26 Sixty... 60 Ninety-fiur. 94 Twenty-seven 27 Sixty-one. 61 Ninety-five. 95 Twenty-eight 28 Sixty-two. 62 Ninety-six. 96 Twenty-nine 29 Sixty-three. 63 Ninety-seven 97 Thirty.. 30 Sixty-four. 64 Ninety-eight 98 Thirty-one. 31 Sixty-five. 65 Ninety-nine. 99 Thirty-two. 32 Sixty-six. 6. 6 One hundred 100 Thirty-three. 33 Sixty-seven. 67 Two hundred200 SvUGGSTIom.-This lesson is intended to teach that the words nd. the figures are simply two forms of language. 10 INTELLECTUAL ARITHMETIC. [BEC. I. LESSON IV. Roman Table. I.. ~ ~ One XX.. Twenty II... Two XXI. Twenty-one II[... Three XXX. Thirty IV... Four XL.. Fortv V.. Five L.. Fifty VI.. Six LX.. Sixty VII... Seven LXX. Seventy VIII... Eight LXXX. Eighty IX... Nine XC. Ninety X.. Ten C.. One hundred XI...Eleven CC. Two hundred XII.. Twelve CCC.. Three hundred XIII... Thirteen CCCC. Four hundred XIV.. Fourteen D.. Five hnndred XV.. Fifteen DC S. ix hundred XVI.. Sixteen DCC. Seven hundred XVII. Seventeen DCCC. Eight hundred XVIII.. Eighteen DCCCC Nine hundred XIX.. Nineteen 1.. One thousand This table is read, one I, one; two I's, two; three I's, three; IV, four, &c. Whant stands for two? What stands for four? What stands for five? What stands for eight? What stands for ten? What stands for twenty? What stands f)r thirty What stands for forty? -What stands for fifty? What stands for sixty? What stands for seventy? What stands for eighty? What stands for ninety? What stands for one hun. dred? Whht stands for five hundred WThtt for one tho-usatnd? S, GGES'TON.-This- lesson is mntended to teach that nutmbers mlay lbe expressed by the Roman character's, as well as by worlde and by figures. Hence, there are thlee ways of explessing nulubers, viz,: by words, by figures, and by letters. rW vl.] INTELLECTUAL ARITHMETIC. i LESSON V. Additior Table from 1 to 3 inclusive 1. One and one, are how many* 2. 2. One and two, are how imanyy? 3. One and three, are how many? 4. One and four, are how many? 5. One and five, are how many? 6. One and six, are how many? 7. One and seven, are how many? 8. One and.eight, are how many? 9. One and nine, are how many? 10. One and ten, are how many? 11. Two and one, are how many? 12. Two and two, are how many? 13. Two and three, are how many? 14. Two and four, are how many? 15. Two and five, are how many? 16. Two and six, are how many? 17. Two and seven, are how many? 18. Two and eight, are how maily? 19. Two and nines are how many? 20.' Two and ten, are how many.? 21. Three and one, are how many? 22. Three and two, are how many? 23. Three and three, are how many? 24. Three and four, are how many? 25. Three and five, are how many? 26. Three and six, are how many? 27. Three and seven, are how many? 28. Three and eight, are how many? 29. Three and nine, are how many? 30. Three and ten, are how many?'e SUGGESTIONS. —The sum of two or more numbers contains ne many units as there are in the numbers added. Thus, 2 is the sumn of 1 and 1; 4 the sum of 2 and 2, or of I and 3. ADDIrION is the process of finding the sum of two or more numbers. The sign + (plus,) placed between two numbers signifies that they are to be added. 12 INTELLECTUAL ARITHMETIC. [SEC. I. QUESTIONS. 1. HIow many fingers have you on one hand, not counting the thumb? How many on both hands? 2. Countingg the thumb, how many have you on each hand 3 HIow many on both? 3. One and four are how many?. One and five? One and nine? One and tenll 4. James has one apple and buys five: how many will he then have? 5. John has six apples and buys one: how many will he then'have? 6. Charles has nine marbles and John gives him one: how many will he then have? 7. How many are 2 and 2. How many are 2 and 4? 2and- S3 2 and 6? 8. l;vw many are 2 and 7? How many are 2 and 8? How many are 2 and 9? 9. James has two tops and buys four: how many will he then have? Two and four are how many? 10. John has two apples and William gives him six: how many will he then have? 11. Bought two quills for two cents, and four quills for four. cents: how many quills did I buy? 12. James bought 2 apples for two cents and 8 inore for eight cents: how many did he buy in all Three and six are how many? 13. If you buy two peaches for two cents and 9 peaches for nine cents, how many peaches do you buy H? tow much do you pay for them? 14. How many are' 3 and 3? How many are 3 and 4? 15. John has three nuts in one halid and five in the other: how, many in both? 3 and 8 nre Ihow many!, 10. James has three pencils and John five hlow ntany have both? 3 and six are how 11 many m LoS. VI.1 INTELLECTUAL ARITHMETiC. 13 LESSON VI. Additiorn Table from 4 to 6 inclusti 1. Four and one are how many? 2. Four and two are how many? 3. Four and three are how many? 4. Four and four are how many? 5. Four and five are how many? 6. Four and six are how many? 7. Four and seven are how many? 8. Four and eight are how many? 9. Four and nine are how many? 10. Four and ten are how many? 11. Five and one are how many? 12. Five and two are how many? 13. Five and three and how many? 14. Five and four are how many? 15. Five and five are how many? 16. Five and six are how many? 17. Five and seven are how many? 18. Five and eight are how many? 19. Five and nine are how miiahy? 20. Five and ten are how many? 21. Six and one are how many'? 22. Six and two are how many'? 23.: Six and three are how many? 24. Six and four are how many? 25. Six -and five are how many? 26. Six and six are how many? 27. Six and seven are how many' 28. Six and eight are how many 2 29. Six and nine are how many? 30, Six and ten are how many? QUESTIONS. 1. JAiohrnhas four tops and Charles one: how umany ha -e both"? 2 14 INTELLECTUAL ARITHMETIC. [S0Ct I 2. William has four apples and James three: how many have both 2 3. How many are 4 and 4?1 How many are 4 ard 5? 4. John has four chestnuts in one hand and three r: the other: how many has he in both 2 5. Charles has four quills and John seven: how lnany have both 2 Four and 7 are how many? 6. John and James have each four tops: how niany have bot-h? Four and 9 are how many 2 7. William has four birds in one cage and seven in another: how many in both? 8. Jane has four pins in her cushion and puts in six more: how many will she then have 2 9. Mary has four needles and buys eight: how many will she then have?: 4 and 10 are how many I 10. John buys three pears for four cents and six pears for eight cents: how many pears does he buy? 11. How many are 5 and 1? How many are 6 and 3? 5 and 8? 5 and 9? 5 and 102 1'2. John has five marbles in one hand and eight in the other: how many in both? 13. Charles has five cents and his father gives him seven: how many: has he then? 14. John has five apples and Reuben gives him nine: how many has he then? 15. Isaac buys five sheets of paper for five cents, alld ten sheets more for ten cents: how many sheets does he buy 16. If I buy five oranges for five cents, and six oranges for six cents, how many do I blly 2 Five and 5 are how many? 17. How many are 6 and 1? How many are e and 2? G and 6? 6 and 8? 6 and 10 7 LaS. VII.] INTELLECTUAL ARITHMETIC. 15 18. I-ow many are 6 and 6? How many are 6 and S? 6 and 9? 6 and 2? 6 and 10? 19. William carries six apples to school in his basket and HIenry four: how many in both baskets? 20. John has six apples, and hi.; sister Jane gives him five: how many has he then? 21. Charles has six apples and wins eight from John: how many has he then? 22. ~Williamn buys three tops for six celnts and eight tops for tell cents: how many tops does he buy I 23. James buys six eggs for sx cents and eight eggs for nine cents: how many egg; does he buy? 24. Jane has 6 apples and Mary gives her 9: how many will she then have LES SON VII. Additionz Table from 7 to 9 inclusive. 1. Seven and one are how many? 2. Seven and two are how many? 3. Seven and three are how rnany? 4. Seven an-d four are how nianyy? 5. Seven and five are how many? 6. Seven and six are how many? 7. Seven and seven are how many? S. Seven and eight are how many? 9. Seven and nine are how many? 10. Seven and ten are how many? 11. Eight and one are how manv? 12. Eight and two are how many? 13. Eight and three are how many? 14. Eight and four are how many? 15. Eight and five are how many? 1G. Eight and six are how many? 16 INTELLECTUAL ARITEMETIC. SEC. L 17. Eight and seven are how many? 18. Eight and eight are how many? 19. Eight and nine are how many 3 20. Eight and ten are how many 3 21. Nine and one are how many? 22. Nine and two are how many? 23. Nine and three are how many? 24. Nine and four are how many? 25. Nine and five are how many? 26. Nine and six are how many 3 27. Nine and seven are how many? 28. Nine and eight are how many 3 29. Nine and nine are how many? 30. Nine and ten are how mdny. QUESTIONS. 1. Iow many are 7 and 1? How many are 7 and 2? 7 and 5? 7 and 4? 7 and 6? 2. James has seven oranges in one basket and six hi another: how many in both? 3. William has seven apples and John gives hinm nine: how many has he then? 4. A father has two sons and gives seven cents tv each: how many cents does he give to both? 5. If Henry buys seven apples, and lary gives him nine: how many will he then have 3 6. If George buys seven quills at one time and 8 at another: how many does he buy in all? 7. William has 5 marbles and Henry gives him 8: how many will he then have? 8. How many are 8 and 2? How many are 8 and 4? 8 and 6 8 and 5? 8 and 9? 9. A boy has eight marbles and gains five: how1 many has he then 8 and 10 are how many? 10. If he has eight and gains nine, how many wil he have? 6 and 4 and 5, are how many? aES. VIz.1 INTELLECTUAL ARITHMETIC. I 11. If George buys eight marbles for three cents and eight more for four cents, how many will he buy in all a 12. John has eight marbles and Charles gives him nine: how many has he then? 13. Eight and four are how many? Eight and seven how many? Eight and 6 are how many? 14. How many are 9 and 2? How many are 9'3 and 4 1 and 3 and 6, are how many 15. Charles has nine apples and buys five more: how many has he then? 9 and 9, are how Imany? 16. If he has nine and buys eight, how many, will he have? 4 and 5 and 6, are how many? 17. Nine and seven are how many? 8 and 7? 18. If James buys nine oranges for nine cents and eight more for 9 cents, how many will he buy in all? 9. Six sheets of paper cost nine cents and 2 pencils cost ore cent: what do the paper and pencils cost? 6 and 7 alnd 8, are how many? 20. How. many are 10 and 2? -How many arle 1I and 3 10and9? 8and 7and 6? 21. James has ten pencils and then buys eight: how many has he thenr? 9 and 4 and 3, are how many? 22. John gives ten chestnuts to Henry and nine to William: how many does he give to both? 23. James spends six cents for candy, four cents for gingerbread, and eight cents for nuts: how mnuch eioes he spend in all? I and 2 and 4 and 10, are hlo' Ianfy'F 24. Nancy has ten pins on her cushion, and sticks nine more there: how many will she then have? 25. Jane has ten needles and Lucy gives her seven: how many will she then have? 26. Oliver buys ten oranges for twelve cents and ten more for eight cents: how many does he buy? 2* 18 INTELLECTUAL ARITHMETIC. LSEC. L LESSON VIII. Numbers added with reference to Ten. 1, Five and'5 are how miany? 4 and 6? 3 and 7? 2 and 8 land 9 2. Ten and 10 are how many? 15 and 5 are how nany? 16 and4? 17 and 3? 18 and 2? 19-and? 14 and 6? 13 an 12 and7? 12? a8 nd9 3. Twenty and ten are how many? 25 and 5. 26 and 4? 27 and 3' 28 and 2? 29 and 1 24 ),nd 6? 23 and 7? 22 and 8? 21 and 9? 4. Thirty and 10 are how many? 35 and 5? 30 and4? 37 and3? 38 and2? 39 andl 1 34and 6? 33 and7? 3and 8 31 and 9 9? 5. Forty and 10 are how many 45 and 5? 46 and 4? 47 and3 3? 4Sand2? 9 and I 44 and 6? 43 and 7? 42 and 8 I 41 and 9? 6. Fifty and 10 ale how many? 55 and 5? 56 and4? 57 and3? 58 and 2? 593 and 1? 54 and 6? 53 and7?:52and 8? 51 andg9? 7. Sixty and 10 are how many?. 65 and 5? 60( and 4? 67 and 3? 68 and 2? 69 and 1? 64 and 6? 63 and 7? 62 and 8? 61 andl 9? 8. Seventy and 10 are how many? 75 and 5.? 76 a-ld 4? 77 and 3? 78 and 2? 79 andl 1? 9. Eighty and 10 are how many? 85 and 5?'. 81 and 1? 87 and 3 2 4 and 86? 2 and 88? 6 and 84? 7 and 83? 9 and 81? 8 and 82? 10. Ninety and ten are how.many? 99 a.d 11 92 and 8? 7 and 93? 3 and 97? 5 and 95? 6 and 94? 9 and'31? 98 and 2?'4 and t9 *SucGEsrrsoNs. —The first part of this lesson points out to tho pupil the mu.:thod' of adding, by considerinlg what nlunbels maoke exact tenm. This is very impolrtant, and should be inueh dws1e. -;,i: The secoud part shows that the first figure of a snun is Mllway derived firomi the units. The questions of the lesson should first be put in the orde2 iv which they %re.written, and then promiscuously, until: thoro' ty learned. LES. IX.] INTELLECTUAL ARITHMETIC. 1-1. Two and two are how many? 12 and'2?. 22 and 2? 32 and.2? 42 and 2? 52 and2? 62 and2? 72 and 2? 82 and2? 92 and 2? 94 and 2? 96 and 2? 98 and 2? 12. Three and 3 are how many? 13 and'-) 23 fand 3? 33 and 3?- 43 and3? 53 and 3? 63 and 3?' 73 and 3? 83 and 3? 93 and 3? 96 and 4?:13. Four and 4 are how rnmany? 4 and 14? 24 and 4? 34 and 4 44and4? 54and4? 64and 4? 74 and4? 84 and-4? 94 and4? 98 and2? 14. Five and 5 are how many? 15 and, 5? 25 and 5? 85 and 5? 45 and 5? 55 and 5. 65 and 5? 75 and 5? 85 and 5?. 95 and 5? 15. Six and 6 are how many? 16 and 6? 26 and 6?. 36 and 6? 46 and 6? 66 and 6? 76 and 6? 86 and 6? 96 and 6? 16. Seven and 7 are how many? 17 and 7? 27 and 7? 37 and7? 47 alnd 7? 57 and 7? 67 and 7?. 77 and 7? 87 and 7? 97 and 7? 17. Eight and8 are how many? 18 and 8? 28 ad 8? 38 and 8? 48 and 8? 58 and 8? 68 and eight? 78 and 8? 88 and 8? 98 and 8? 18. Nine and 9 are how many? 19 and 9? 29 and 9? 39 and9? 49and9? 59 and 9? 69 and 9 79 and 9. 89 and 9? 99 and 9? LESS ON IX Showing the formation, of Nruni2bers fromn 11 to 100. I. Eleven and 1 are hoNq many? Eleven and 219 Eleven and 3? Eleven and 4? Eleven and 5 lEleven and 6? Eleven ancd 7? Eleven and 8?. Elevei and 9 2 Eleven and 10 Eleven and 11." *M SutOc i:sIo.-This lesson indicntes hoTw all the numbers inay be.foi.trcd from 11 to include one hundred and nine. After the questionls have been put in the orderC in which they are writtel ihey should be put promiscuously. ARITHMETIICAL ARITHME'IC. L[SEC. L 2 Twenty-two and 1 are how many? Twentytwo and 2? Twenty-two and 3? Twenty-two and 4? Tw1venty-two and 5? Twenty-two and 6? Twentv-two and 7? Twenty-two and 8 2 Twenty-two and 9? Twenty-two and 10? Twenty-two and iI? 1. Thirty-three and 1 are how many? Thirty. three and 2? Thirty-three and 3 Thirty-three and 4? Thirty-three and 5 Thirty-three and 6? Thirty. three and 7? Thirty-three and 8? Thirty-three and 9? Thirty-three and 10? Thirty-three and 1 1. 4. Forty-four and 1 are how many? Forty-four and 2? Forty-four and 3? [Forty-four and 4? Forty-four and 52 Forty-four and 6? Forty-four and 7? Forty-four and 8? Forty-four and 9? Fortyfour and 10? Forty-four and 11? 5. Fifty-five and 1 are how many? Fifty-five and 2? Fifty-five and 3? Fifty-five and 4? Fifty-five and 5? Fifty-five and 6? Fifty-five and 7? Fiftylive and 8? Fifty-five and 9? Fifty-five and 107 Fifty-five and 11? 6. Sixty-six and 1 are how many? Sixty-six and 2? Sixty-six and 3? Sixty-six and 4? Sixty-six and 5? Sixty-six and 6? Sixty-six and 7? Sixtysix and 8? Sixty-six and 9? Sixty-six and 10? Sixty-six and 11 7. Seventy-seven and 1 are how many? Seventyseven and 2? Seventy-seven and 3? Seventy-seven and 4? Seventy-seven and 5? Seventy-seven and 6? Seventy-seven and 7? Seventy-seven and 8? Seventy-seve and 9? Seventy-seven and 10? Sev enty-seven and 11 8. Eighty-eight and 1 are how many? EigNtyeight and 2? Eighty-eight and 3? Eighty-eight and 4? Eighty-eight and 5? Eighty-eight and 6? Eighty-eight and 7? Eighty-eight and 8 Eighty LES X.J. INTELLECTUAL ARITHMETIC. 21 eight and 92 Eighty-eight,and. 10? Eighty-eigbt and 11? 9. Ninety-nine and I are how many? Ninetynine and 2? Ninety-nine and 3? Ninety-nine and 4? Ninety-nineand 5? Ninety-nine and 6? Ninety-nine and 7? Ninety-nine and 8? Ninety-nli and 9? Ninety-nine and 10? LESSON X. Practical Questions. 1. Let each of the following combinations be giveu as a separate example. HI-ow many are 10 and 20 and 4? 40 and 50 and 6? 10 and 30 and 9? 6 and 12 and 30? 10i and 40 and 6? 7 and 15 and 70? 10 and 50 anCd 3? 9 and 14 and 60? 20 and 30 and - 13 and 7 and 14? 15 and 20 ind 46? 19 and 11 and 16? 25 and? 15 and 4 21 and 9 and 13?. 35 alcl 12 and 3? 30 and 40 and 10? 40 and 60 and 9? 36 and 4 and 19. 8 and 20 and 10 38 and 12 and 16? 2, Jane has 13 pins in her cushion and Mary 27: how many pins have both? 3. John has a number of pears: he gives 8 et William, 12 to Charles, 9 to James atid has 1 lefthow Iany had he at first? 4. There are 4 bags of coffee: the first contains 16 pounds, the second 14 pounds, the third 7 pounds, an d the fourth 3 pounds: how manv pounds in all t.h-e cbags? 5. A farmner has 4 pastures containing sheep. In the first there are 3 sheep, in the second there are 6, in INTELLECTUAL ARITHMETIC. [$EO. L the shird there are 7, and in the fourth there are 8: hovw many are there in the four pastures? 6. James gave 18 cents for a squirrel, 82 cents foi a cage, and 115 cents for nuts: how many cents did he pay in all:7. A man bou a.eow for 25 dollars, a calf for 5 dollars, 3 lambs for- 8 dollars, and a pig for 2 dollars i what did he pay for all? S. ]-low many are 1 and 2 and 4 and 14: and 9? 9. How many:~ are:ga ~:andi 4 and 16 a:.niwd 5 atrd 4 and 5? 10. IHow many're 4 and- 14 and 16 and 6- and 7 and 8? 11. How many are- 15 and 13 and 12 and4 and 9?' 12. How many are 9 and 11 and 14 and 16 and 17? 13. How many are 1 and 2 and 4 and 3 and 6? 14. Hlow many are 2 and 2 and 4 and 3 and 5? 15. Howv many are 6 and 4 and 4 and 3 and 3? 16. How many are 6 and 4 and 3 annd 6 and 5? 17. How many are 7 and 7 and 4 and- 2 and 6? 18. How many are 9 and 2 and 8 and 7 and 5 nd 87 19. A lady bought some tape for 10 cents, some pins fcr 18 cents, a comb fbr 22 cents, and a pair of scissors for 30 cents: how much did she pay in ali f 20. A farmer has 15 sheep in one lot, 25 in another, and 30 in his barn-yard: how many has he in all? 21. A merchant buys 26 barrels of flour of one miller, 30 of another, 14 of another, and 36 of another: how many barrels does he buy in all? 22. A man bought a horse, saddle and bridle; for the horse he gave 75 dollars, for the saddle 25 dollars, and 7 dollars for the bridle: what did they all cast him? LES. X.] INTELLECTUAL ARITHMETIC. 2 23. A drover bought 12 sheep of one farnter, 30 of another, 18 of another, and 25 of another: how,inany did he buy in all? 24. James gave nine cents to a beggar woman, 11 CeC "s to a beggar man, and 8 cents to a beggar girl: ow much did he give in all? 25. If John has 14 cents in one pocket, 10 cents in another, 6 cents in his purse, and 8 cents in his hard, how mlany cents has he in all 2 26. Charles has 8 cents, William 18, Robert 4, and Samuel 9: how many cents have they all 2 27. If Lucius gives 36 cents for a pen-knife, 8 cents for paper, 6 cents for quills, and 7 cents for wafers, how rmuch does he pay in all? 28. James buys 9 sticks of white candy, 9 sticks of red candy, 2 of brown candy, and 8 of yellow candy: how many sticks does he buy in all 2 29. If James gave 18 cents for the white candy, 8.elnts for the red candy, 4 cents for the brlown candy, and 16 cents for the yellow candy, how much did he pay in all 2 30. Jane pays 18 cents for a slate, 12 cents for quills, 11 cents for paper, and 15 cents for pencils: what does she pay in all? 31. A grocer purchases 6 barrels of flour for 30 dollars, a load of hay for 12 dollars, and 10 bushels of oats for 15 dollars: how much did he pay in all? 32. A tailor paid 15 dollars for a piece of cloth, for a coat, 4 dollars for the lining, 2 dollars for the buttons, and charged 9 dollars for making: what was fie cost of the coat? 24 INTELLECTUAL ARITIIMETIC. [SEC. IL SECTION SECOND. LESSON I. Subtraction. 1. One and 1 are 2: if we take I from 2 wha.t remains*?. 2. One and 2 are 3: if we take 1 fromr 3, what remains? If we take 2 from 3, what remains? 3. One and 3 are 4: if we take 1 from 4, what remains? If we take 3 from 4, what remains? If we take 2 fiom 4, what remains? 4. One and 4 are 5: if we take 1 from 5, what remains? 5 less 4, are how many? 5 less 2, are how many? 5 less 3, are how many? 5. One and 5 are 6: 6 less 5, are how many? 6 less 1, are how many? 6 less 2, are how many? 6 less 3, are how many? 6 less 4, are how many? 6. Seven less 1 are how many? 7 less 2, are how many? 7 less.4, are how many? 7 less 5, are how many? 7 less 6, are how many? * SUGGETsoxs.-TThe pupil first gets the idea of more by addition. He adds two numbers together and finds that their sum is greater t-han either of them. He next sees that if one of them be taken away, the other will be left: hence, The d!ierence between two numbers is such a number as addled to the less will give the greater. SuBTRAcTIoN is the process of finding the difierence betweeu two numbers. Teach the pupil that every question in Subtraction requires him to find such a number as added to the less will give the greater. What is the difference between 6 and 2? VWhy is 4 the difference between 6 and 2? Because 4 added to 2 gives 6. — The sign - (minus), placed between two numbers siognifies that the number after rit is to be taken frouiit'l ibinmuber befoue it LES. I.] INTELLECTUAL ARITHIMETIC. 25 7. Eight less 1, are how many? 8 less 3, are how many 8 less 5, are how many?8 less 7, are how many? 8. Nine less 1, are how many? 9 less 3, are how many? 9 less 5, are how many? 9. Eight and 2, are how many? 10 less 2, are how many? 10 less 8, are how many? 10. Six and 4, are how many? 10 less 6, are how many? 11 less 5, are how many? 11. William has three apples and gives them all to James, how many has he left? 3 less 3, what remains? 12. William has six apples and gives three to James: how many has he left? 6 less 3, are hotw many? 13. 7 less 3, are how many? 14. 9 less 3, are how many? 15. 10 less 3, are how many 2 16. 14 less 3, are how many 2 17. 4 less 4; what remains? 18. 8 less 4, are how many? 19. Fourteen less 4, are how many? 20. Henry has five pears in a basket and gives them all to his sister: howm many has he left? 5 less 5 what remains? 21. 8 less 5, leaves how many? 22. 9 less 5, leaves how many? 23. 11 less 5, leaves how many? 24. James has six squirrels in a cage, and takes them all out: how many will, be left? 6 from 6i, leaves how many? 25. 9 less 6, leaves how many? 26. 8 less 6, are how many? 27. 16 less 6, are how man y? 28. WMhary has seven pins in her cushion and takes them all out: how many are left? 7 less 7, what remains 2(0 INTELLECTUAL ARITHMETIC. [SEC. IL 29. 12 less 7, are how many? 30. 15 less 7, are how many? 31. Reuben has eight plums and gives them all to John: how many has he left?2 8 less 8, what remains? 32. 10 less 8, are how many? 33. 14 less 8, are haw many? 34. 18 less 8, are how many? 35. There are nine chairs in a room, and Mary Cakes them all out: how many are left? 9 less 9, what remains? 36. 12 less 9, are how many? 37. 15 less 9, are how many? 38. 19 less 9, are how many 39. 16 less 6, are how many? 40. 15 less 10, are how many? 41. 14 less 4, are how many? 42. 25 less 5, are how many? 43. 36 less 16, are how Iany? 44. 25 less 12, are how many'? 15. 30 less 7, are how many? 46. 20 less 14, are how many? 47. 30 less 12, are how many? 48. 27 less 7, are how, many? 49. 29 less 10, are how many? 50. 39 less 20, are how many 1 51. 42 less 12, are how many 52. 19 less 7, are how many? 53. 50 less 20, are how many? 54. 14 less 8, are how many 55. 15 less 3, are how many? 56. 24 less 4, are how many? 5T7 16 less 5, are how many Q 58.: 17 less 8, are how many? 59. 19 less 7, are how many? 60. 19 less 9, are how many Q LE[S. I.] INTELLECTUAL ARITHMETIC 27 61. J29 less 8, are how many? 62. 34 less 3, are how many? 63. 35 less 6, are how many t 64. 50 less 8, are how many? 65. 57 less 6, are how many? 66. 59 less 5, are how many? 67.'53 less 7, are how many? 68. 60 less 20, are how mRany 69. 65 less 15, are how many? 70. 67 less 10, are how many? 71. 67 less 9, are how many? 72. 70 less 5, are how many? 73. 74 less 3, are how many? 74. 78 less 8, are how many? 75. 79 less 10, are how many? QUESTIONS. 1. There are nineteen peach-trees in an orchard, and six of them are blown down in a storm: how many are left standing? 2. Laura has twenty-five cents, and buys an arithmetic for eighteen cents: how much money will she have left 3. There are thirty-four pears in a basket, and nine of them are taken out: how many are left? 4. There are sixty-five pigeous in a flock, and John fires at them and kills nine: how many are left? 5. There are fifty-four sheep in a fold, and a wolf breaks in and kills seven: how'many are left? 6. There are forty-nine scholars in a school, and ten of them are girls: how many boys are there? 7. In another school there are twenty scholars, and iine are boys: how many girls are the ~-? 8. In Elizabeth's flower-bed there are thirty beauti fill lilies, and John breaks off seven of them: how many are unbroken? '28 INTELLECTUAL ARITHMETIC. [SEC. I. 9. James has thirty-seven cents: he spends six foi candy, eight for a pencil, and twelve for a pen-knife: how many has he left? 10. John has twenty-five cents, and spends six cents for a top, nine cents for a pencil, and two cents for a peach: how much has he left? 11. Wiliam has 37 cents, he buys a top for 10 cents, and 8 marbles for 2 cents: how much has he left? 12. A boy has 40 peaches: he gives 24 to Lucy and 9 to Elizabeth: how malny has he left? 13. James received a premium worth 56 cents; Jane received one worth 30 cents: what was the dif. ference of their values? 14. Charles has 49 cents, and buys a book which costs himr 29 cents: how much has he left? 15. A butcher buys 39 sheep, and kills 17: how many are left alive? 16. A grocer has a tub of butter containing 45 pounds; he sells 20 pounds to Mr. Wilson, and 15 pounds to Mr. Jones: how much is left? 17. IIow many are 55 less 17? 18. How many are 77 less 19? 19. Four men bought a horse for 56 dollars; the first paid 16 dollars, the second 20 dollars, and the third 14 dollars: what did the fourth pay? 20. Charles bought a penknife for 48 cents, and a top for 25 cents: how much more did he pay for the knife -than for the top 2 LESSON II. Questions involving Addttion and Substraction. 1. Six and 4 and 3 and 5, less 4, are how many? 2. Eight and 9 and 4 and 2, less 3, are how many? 3. Seven and 6 and 5 and 8, less 5, are how many? LES. II.] INTELLECTUAL ARITHMETIC. 29 4. Nine and 2 and 3 and 7, and 1, less 6, are how many? 5. Thirty and 5 and 8 and 15, less 10, are how many? 6. Twenty-one and 6 and 13, less 12, ale how many? 7. Forty-five and 15 and 12, less 9, are how many? 8. Sixty-nine and 11 and 5 and 2, less 8, are how many?. 9. Seventy-five and 5 and 6 and 12, less 8, are how many? 10. Forty-five and 8 and 4 and 3 and 6 and 7 and 9, less 12, are how many? 11. James has 4U cents, and pays 12 cents for a whistle and 25 cents for a knife: how much has he left 12. A man bought a calf for 5 dollars, a sheep for 4 dollars, and a pig for 2 dollars; also a cow feor 25 dollars: what did he pay in all, and how much more for the cow than for the other animals? 13. James has 26 nuts in one pocket, and 14 less in the other pocket: how many has he in both? 14. A man has 50 dollars; he pays 26 for a coat, 8 dollars for a pair of pantaloons, and 4 dollars for a vest: how much has he left? 15. A school-boy pays 56 cents for an Atlas, 30 cents for an Arithmetic, and 24 cents for a slate: what did he pay for all, and how much more for the Atlas than for the Arithmetic and slate? 16. Eighty-five and 5 and 9 and 3 and 12. less 8, less 3, are how many? 17. Fifty-nine and 5 and 9 and 8 anli 7 and 0j less 8, are how many? 18. Forty-seven and 9 and 6 and 4 and 5 anti 3, less 12, are how many? 3* 30 INTELLECTUAL ARITHMETIC. LSEC. n. 19. Seventy-two and 15 and 6, less 11, are how many? 20. Thirty eight and 4 and 9 and 7 and 6 and 2, less 12, are how many? 21. William had 16 marbles, James gave him 7, John gave him 8, and Reuben gave him enough to make his number, 40: how many marbles did Reuben give John? 22. A father gave seventeen cents to Lucy, 13 to Mary, and 4 to Jane, and then took back 9 from Lucy: how many had they left? 23. A man travelled 49 miles in three days; the first day he travelled 16 miles, and the second day 13 miles: how far did he travel the third day? 24. A tailor has a piece of cloth containing 393 yards; he sold 14 yards to one man, 13 to another, and made a coat which took two yards: how many yards were left? 25. A merchant bought some coffee, for which he paid 25 dollars, some sugar, for which he paid 12 dollars, and some tea, for which he paid 11 dollars; he sold the whole for 56 dollars: what did he gain? 4g5. A tailor bought a piece of cloth for 45 dollars; he made it into a coat and pantaloons; he paid 10 dollars for making, and then sold them for 60 dollars: did he make or lose, and how much? 27. James has 49 peaches; he gives 15 to Robert, 13 to John, and 9 to William: how many has he left? 28. Charles has some pears, and gives 12 to James, 11 to Henry, 13 to Reuben, and 8 to Elisha; when he finds that he has 9 left: how many had he at first? 29. Lucy has 12 pins on one cushion, and 15 on another: if she takes off 9, how many will she have left'? LES. 1I] INTELLECTUAL ARI'METC. 31 30. A man travelled 5 miles before breakfast, 19 miles between breakfast and dinner, and then travelled back 12 miles: how far was he from the place of starting 1 31. A cow has two calves, the first is worth 3 dollars, the second 4 dollars, and the cow is worth 25 dollars: how much more is the cow worth than the two calves, and what are they all worth? 32. A grocer buys some lemons for 15 dollar, some oranges for 25 dollars, and then sold the whole for 56 dollars: how much did he make? 33. Jane has 32 rose-buds on one bush, and 16 on another, and 38 only blossom: how many buds did not flower 2 34. A man owes 55 dollars; at one time he pays 24 dollars, at another 17 dolltrs, aed tien 14 dollars how much does he then owe? 35. William went after chestauts; he put 26 in one pocket, 15 in another, and 16 in a third: he lost 9 out of the first pocket, 3 from the secoad, atd 6 fiom the third: how many had he left! 36. Twenty-nine plus l plus 8 plus 4, less- 12 less 17 are how many~ 37. Mr. Jones owes his baker 17 do[lalrs, his gro.. cer 16 dollars, and his tailor 27 dollars: how much does he owe in all, and how much more to his baker and grocer than to his tailor 38. A farmer has 30 sheep in one lot, 25 in another, and 16 in a thbidi 9 shep3 escape from the first lot, 12 from nhe second, and 9 from the third: how many sIe'p had he in all, how atwty on t ped, and how maiy- were ft ina tihe falds R2 INTELLECTUAL ARITHMETIC. [SEC. IX. SECTION THIRD. LESSON I. Multipliers from 1 to 4 incuisive. Once 1 is how many?* Once 7 is how marny Once 2 is how many? Once 8 is how many?. Once 3 is how many? Once 9 is how many? Once 4 is how many? Once 10 is how many Once 5 is how many? Once 11 is how many? Once 6 is how many? Once 12 is how many? 2 times 1 are how many? 2 times 7 are how many 7 2 times 2 are how many? 2 times 8 are how many? 2 times 3 are how many? 2 times 9 are how many? 2 times 4 are how many? 2 times 10 are how many? 2 times 5 are how many? 2 times 11 are how many? 2 times 6 are how many? 2 times 12 are how many? 3 times 1 are how many? 3 times 7 are how many? 3 times 2 are how many? 3 times 8 are how many? 3 times 3 are how many? 3 times 9 are how'many' 3 times 4 are how many? 3 times 10 are how many i 3 times 5 are how many 2 3 times 11 are how many 1 3 times 6 are how many? 13 timnes 12 are how many? 4 times 1 are how many? 4 times 7 are how many? 4 times 2 are how many? 4 times 8 are how many? 4 times 3 are how many' 4 times 9 are how many 4 times 4 are how many? 4 times 10 are how many? 4 times 5 are how mlany?- 4 times 11 are how many? 4 times 6 are how many2 14 ti;-ries.12 are how many'? * MULTIPLICATION is a short process of takaz, one uumler as many timles as there are units in another. The number to be taken is called the muzltp4iNcand. The cumber denoting how many times the multiplicani is to be taken, is called the multiplier. The result, or answer, is called the product. The multiplicand and multiplier are called factors or prao hIwers of the product. LES, II.1 INTELLECTUAL ARITHMETIC~. 33 1. If James buys 2 oranges at 3 cents apiece, what do they cost?* 2. What will 4 peaches cost at 1. cent each? 3. What will 3 oranges cost at 3 cents apiece? 4. What will 4 lemons cost at 4 cents apiece? 5. What will 3 pounds of raisins cost at 12' cents pound? 6. What will be the cost of 4 tops at 12 cents apiece? 7. What will be the cost of 2 melons at 11 cents apiece? LESSON II. MAultipliers from 6 to 8 inclusive. 5 times. 1 are how many? 5 times 7 are how many? 5 times 2 are how many? 5 times 8 are how many? 5 tines 3 are how many? 5 titmes 9 are how many? 5 times 4 are howr many? 5 times 10 are how many? 5 times 5 are how many v 5 times 11 are how many? 5 times 1 are how m1any? 5 times 12 are how many? 6 tizes 1 are how many? 6 times 7 are how many? 6 times 2 are how many? 6 times 8 are how many? 6 times 3 are how many? 6 times 9 aire how many? 6 times 4 are how many? 6 times 10 are how many? 6 times 5 are how many? 6 times 11 are how many? 7 times 6 are how many' 6 tilnes 17 are how many? 7 times 1 are how many? 7 times 7 are how many 7 7 times 2 are how nlaDy?17 timne3 8 are how many. 7 times 3 are how many? 7 tirnes 9 are how many? 7 times 4 are how many? 7 times 10 are how many? 7 times 5 are how many? 7 times 11 are how many? 7 times 6 are how many? 7 times 12 are how ma.ny, * ANALYsrs.-Two orlanges will cost two times as much as one orange. Since 1 orange costs 3 cents 2 oranges will coust two timles 3 cents, which ave 6 cents: therlfore, 2 oranges at S scr apiece, -will cost 6 cents. 2 34 INTELLECTUAL ARITHMETIC. [SEC. III, 8 times 1 are how many? 8 times 7 are how many 1 8 times 2 are how many? 8 times 8 are how many. 8 times 3 are how many? 8 times 9 are how many? 8 times 4 are how many? 8 times 10 are how many? 8 times 5 are how many? 8 times 11 are how many? S times 6 are how many? 8 times 12 are how many? 1. If William buys 5 pine-apples at 4 cents each, what do they cost him? 2. What is the cost of 6 barrels of flour at 5 dol lars a barrel? 3. What is the cost of 7 yards of cloth at 9 dollars a yard? 4. What is the cost of 8 barrels of fish at 8 dollars a barrel? 5. What is the cost of 6 pounds of candles at 9 cents a pound? 6. What is the cost of 8 pounds of raisins at 11 cents a pound? 7. What is the cost of 7 pounds of sugar at 8 cents a pound? 8.,What is the cost of 5 pounds of beef at 11 cents a pound? LESSON III. Mlultipliers from 9 to 12 inclusive. 9 times 1 are how many? 9 times 7 are how many? 9 times 2 are how many? 9 times 8 are how many? 9 times 3 are how many? 9 times 9 are how many? 9 times 4 are how many? 9 times 10 are how many. 9 times 5 are how many? 9 times 11 are how many? 9 times 6 are how many. 9 times 12 are how many? NoT. —The sign of multiplication (X), placed between Iwo m lmore numbers, signifies that they are tc be multiplied Itether. LES. III.] INTELLECTUAL ARITIIHMETIC. 35 10 times 1 are how many? 10 times 7 are how many? 10 times 2 are how many 2 10 times 8 are how many. 10 times 3 are how many 10 times 9 are how many? 10-times 4 are how many? 10 times 10 are how many-? 10 times 5 are how many? 10 times l are how many? 10 times 6 are how many? 10 times 12 are how many 1 11 times 1 are howmany? 11 times 7 are how many? 11 times 2 are how many? 11 times 8 are how many? 11 times 3 are how many II times 9 are how many? 11 times 4 are how many? 11 times 10 are how many? 11 times 5 are how many?i 11 times 11 are how many?. 11 times 6 are how many?11 times 12 are how many? 12 times 1 are how many? 12 times 7 are how many? 12 times 2 are how many? 12 times 8 are how many? 12 times 3 are how many? 12 times 9 are how many? 12 timnes 4 are how many? 12 times 10 are how many? 12 times 5 are how many? 12 times 11 are how many? 12 times 6 are how many? 12 times 12 are how many' 1. If James buys 9 lemons at 2 cents each, what do they cost him? 2.\ What will 11 yards of calico cost at 11 cents a yard? 3. What will 12 dozen of apples cost at 12 cents a dozen? 4. What will 9 pine-apples cost at 11 cents apiece? 5. What will 12 yards of cloth cost at 9 dollars a yard? 6. What will 9 pumpkins cost at 12 cents apiece? 7. What will be the cost of 11 pairs of boots at 5 dollars a pair? 8. What will be the cost of 12 loaves of bread at 11 cents a loaf? 9. What will 9 slates cost at 11 cents apiece? 10. What will be the cost of 11 yards of brotad cloth at 12 dollars a yard? s8 INTELLECTUAL ARITIHMETIC. [SEC. IlL LESSON IV. QUESTIONS. 1. What is the product of 13 taken 2 timles * 2. What is the product of 14 taken 3 timnes3 3. What is the product of 15 taken 5 times 4. What is the product of 18 taken 6 times? 9 tinaes 7 times? 5. What is the product of 16 taken 4 tirnes 5 times? S times? 6. If Jamles reads 16 verses of the bible a day, how mnany veises will he read in a week? 7. ilow nmany are 7 titlles 15 Whilich the mulltiplicaund?'3 Which the mrultiplier 3 Which the product? 8. \Vhat will 9 pounds of butter cost at 19 cents a pm>lcnd 3 9. If a steamboat can go 16 miles an hour, how far can it go in 11 hours'? 10. HIow Imally are 2 times 20? 4 times 2i 0 i 5 times 20 3 7 times 20? 9 times 20? 1i1 timines 20? 12 timnes 20' 11. if 1 barrel of flour cost 9 dollars, what will 12;barrels cost? 1.2. It' 1 barrel of fish cost 7 dollars, what will be the cost of 9 barrels' 13. I' I yard of cloth cost 6 dollars, what will he the cost oft 8' yads? 14. If 1 yard of cloth costs 5 dollars, what will 15 yards cost? 15. If' a man on horseback can ride 8 miles in I 3hour,' how far can he ride in 12 hours? ANAiALYSS. —Thirteen is made up of 1 ten and 3 units. T hen I ten nmultiplied by 2 gives 2 tens, ori 20; and 3 units ilu!t1:pl;'.d ty o gives 6 units, whichl being added to 20 gives 26 f; f 3e psuduct. Let each question be analyzed in a hnilar mauonn ea. LES. Vo. INTmLLECOUAL ARIaTHM{ET[O. 37 16. How many are 2 times 18? 3 times 18{ s times? 17. How many are 5 times 17? 3 times 18 1 4 times? 18. If 1 cow cost 20 dollars, how much will 7'ows cost? How much will 9 cost? How much will 12 cost? 19. If a man travels 19 miles a day, how far will he travel in 9 days? In 10 days? In 1 days? Ir 12 days? 20. If a pound of raisins cost 15 cents, how much, will'5 poands cost? (6 pounds? 7 pounds9? I pounds' 21. if a man digs IS bushels of potatoes in I day, how tmany would he dig in 7 days!? n 10 days I in 12 da ys? 22. If' a mlan eat 15 ounces of bread ie a. day, how uarny ounces will he eat in a week? how tmany in days? In Il days. L E S OT v. A&ti dtion- Sub t ct io t.-futipJticateion. 1. W1That will be the cost of 5 oranges at 3 eents 2. CWhat will 7 books costs at 8 cents apiece? At 9 cents? At 15 cents? At 20 cents? 3. -What will e the cost of loaves of bread.at 9 cents a loaf? Of 8 loavest. Of 10 loaves? Of 1t oa-ves.? Of I2 loaves 4,: What will be the cost of 4 hats at 7 dollars rapice-e, and the cost of 2 coats at I2 dolt.as apieee t W 5. h att will be the cost of 1t chickens at 2. certt Un.,[ipce. Of 11? Of 9? Of 7?T Of 6 Of 5 t \5. W'bhIat will be the diffierente in. cost between: 2 ij;s at () cents apiece, and 3 tops at4 cents apiec hey aIL[ cost? R8 INTELLECTUAL ARITHMETIC. [SEc. Il. 7. What will be the cost of 8 yards of ribbon at 8 cents a yard? Of 5 yards? Of 6 yards? Of 9 yards? Of 12 yards? 8. What is the difference between the cost of 8 yards of cloth at 5 dollars a yard, and 12 yards at 2 lollars a yard? What would be the cost of the whole?, 9. James has 25 marbles and gives 9 to Willianm, then Charles gives him as many as he has left: how many more than he had at first will he then have? 10. What will be the whole cost of 8 apples at I cent apiece, and of 4 pears at 2 cents apiece? 11. What will be the whole cost of 6 oranges at 4 cents apiece, and of 2 lernons at 3 cents apiece? 12. What will be the whole cost of 6 lemons at 2 cents apiece, and of 2 apples at 1 cent apiece? 13. What will be the whole cost of 8 quills at 2 cents apiece, and of 12 sheets of paper at I cent each 1 14. What will be the whole cost of 6 spelling books at 8 cents apiece, and of 5 slates at 10 cents apiece? 15. What will be the difference of the cost of 6 yards of cloth at 5 dollars a yard, and of 4 yards of cloth at 6 dollars a yard 16. What will be the whole cost of ten sticks of candy at 2 cents apiece, and of 4 pounds of raisins at 11 cents a pound? 17. A farmer bought 9 sheep at three dollars apiece, and two calves at 4 dollars apiece: how much more did he pay for the sheep than for the calves? 18. A farmer bought 4 cows at 20 dollars apiece, and 12 sheep at 4 dollars apiece: what did the whole cost him 19. Fifteen plus 8 plus 9, less 5, are how manyi. 20. Forty plus 7, less 6, plus 8 plus 9s are i17 many? LES. V.o] INTELLECTUAL ARITIIMETIC. 39 21. Twenty-five plus 6 plus 8, less 3, plus 1, are how many? 22. Fifty plus 6 plus 9, less 8, less 3, less 4, are low many? 23. Sixty plus 8 plus 7 plus 1, less 6, are how aany? 24. Thirty less 7 less 6, plus 8, plus 9, iare how many? 25. Mary has 6 rose-bushes, and 9 buds on each; also 3 geraniums with 8 buds.-on each: how many buds are there in all? 26. A family consumes 12 pounds of meat in a day. Beef is 12 cents a pound and mutton 9: how much will they save by using mutton instead of beef?'/27. James bought 4 oranges at 3 cents apiece, and 2 quarts of chestnuts at 9 cents a quart: what did he pay in all? 28. James and John start from the same place, and run in opposite directions. James runs 30 rods and John 20: how far are they apart? How far would they be apart if they had run the same way? 29. Two men start from the same place and walk in opposite directions: the first walks 4 hours and goes 8 nmiles an hour, the second the same time and goes but 2 miles an hour: how far will they be apart? 30. Two men start from the same place and walk tilhe same way; the first wallrs 4 miles an hour, the second 3: how far will they be apart at the end of the first hour'. How far at the end of the second I The third? 31. A drover bought 3 sheep at 4 dollars apiece, iand 4 lambs at 2 dollars apiece: he gave in payment 2. calv.es at 5 dollars apiece, and the rest in cash. B:ow mruch money did he pay? 32. A jeweller bought a watch for 55 dollars, a 40 INTELLECTUAL ARITHMETIC. [SEC. In chain for 15 dollars, and a seal for 8 dollars; and then sold the whole for 80 dollars: did he make or lose, and how much? 33. Four boys bought a foot-ball for 75 cents. John paid 20 cents, James 33 cents, and William 18 cants: how much did Reuben pay? 34. A grocer bought a hogshead of sugar for 65 dollars, and a hogshead of molasses for 55 dollars; he sold them both for 125: did he make or lose, and how much? 35. A farmer bought 4 sheep for 15 dollars, and 6 lambs for 10 dollars; he sold the whole for 30 dok lars: did he make or lose, and how much? 36. A tailor has a piece of cloth containing 25 yards, and a second piece containing 35 yards; hp cut 8 yards from the first piece, and 12 from tho second: how many yards of cloth has he left? 37. A merchant has a piece of cloth, for which he paid 36 dollars; he wishes to sell it so asY to mlake a profit of 12 dollars: what must he ask for it? 38. A farmer has 150 sheep in three fields. In the first he has 45, in the second 55: how many has ha in the third? 39. There is an orchard with ten full rows of trees and 7 trees in each row; besides which there are 3 broken rows with 4 trees in each rov': how mlany trees are there in the orchard? I-low many more in the whole than in the broken rows? 40. An orchard contains 20 apple trees, 10 cherry trees, and 15 plum trees: how many more apple and cherry trees than plum trees? 41. James worked 4 days in a week for 12 cents a day, and William 2 days for 4 cents a day: how much more did James earn than Williamrrl 42. A butcher bought 16 sheep of one nman, 14 of another, and 25 of a third; he then killed 9, afterwards he killed 21: how many had he left? LES. vI.] INTELLECTUAL ARITHMETIC. 41 43. A man earned 16 dollars a month for 6 months and his son James 7 dollars a month ftr the samle tiime.: how much more did the father earn than the soli I LESSON VI Factorig nurbers. 1. Four are how many times 2? What are the factors of 4.' 2. Six are how many times 3? Ihow many times 2? What are the factors of 6? 3. Eight are how many times 4? How many times 2? Wh'at are the factors of 8? 4. Nine are how many ti~mes 3? What are the factors of 9? 5. Ten are how many times 5? How many times 2? WVhat are the factors of 10? 6. Twelve are how many times 6? I-Iow many times 2. IBow many tigmes 4? I-ow many times 3.? Wh t are the factors of 121? 7. Fouirteen are how many times 7? How many times 2'? WV ht are the flactors of 14?. lFifteen are how many times 5? Iow many times 32 What are the factors of 15? 9. Sixteen iare how many times 8-? How many times'27 I-Iow many times 4? What are, the f'a. tots of 16? 10. Eiohteen are how many times 9?I- oow many tinmes 2 How manly times 6? Iow many time3 3? IV hat are the factors of 18? SUouEwr;EsoN.- — In each of the following questions, two numbera ime u 0.'td, 11mnd it is required to fitd a thi cld which multip)lietl b1, thie svcetu!l, will give a product equal to tle first. T'he sec86,l Ud lhirid Il lll llbtfbl's ale called FACTORS of the filst. Let hie pupil poiut out the factors in every example. 4* 42 INTELLECTUAL ARITHMETIC. [SEC. Ill 11. Twenty are how many times 2? How many tines 10? How many times 5? How many times 4? What are the factors of 20? 12. Twenty-two are how many times 111 hIow many times 2? 13. Twenty-four are how many times 12? IHow rany times 2? How many times 8? How many times 3? 14. Twenty-six are how many times 2? How many times 13? What are the factors of 26? 15. Twenty-seven are how many times 9? How many times 3 2 What are the factors of 27? 16. Twenty-eight are how many times 14? How imany times 2? What are the factors of 28? 17. Thirty are how many times 15? How many times 2? How many times 10? How many times 3? What are the factors of 30? 18. Thirty-three are how many times 11? How many times 3?'19. Thirty-four are how many times 2? How mnal-yv times 171 20. Thirty-six are how many times 18? How many times 12? How many times 9? How many times 6? How manytimesmes 4? How many times 3? How many times 2? What are the factors of 36?21. Thirty-eight are how many times 19? How many times 2 2 What are the factors of 38? 22. Forty are how many times 20 How many times 102 How many times 8? How many times 52 How many times 4? I-ow many times 2? 23.:Forty-two are how many times 21? IHow many times 2? How many times 6? How many times 7? What are the factors of 42? 24. Forty-four are how many times 22? How many times 2?. How many times 11 IIHow many times 4. What are the fiactors of 44 9. LES. I.] INTELLECTUAL ARITHMETIC. 43 25. Forty-six are how many times 23? How niany times 2? What are the factors of 46? 26. Forty-eight are how many times 24? How many times 16? How many times 12? IHow many times 8? How many times 6?2 How many times 4? How many times 3? How many times 2?. What are the factors of 48? 27. Forty-nine are how many times 7. What are the factors of 49? 28. Fifty are how many times 25? How many times 10? How many times 5?2'hat are the factors of 50? 29. Fifty-four are how many times 9? Hlow many times 6? How many times 278 What al-e the fitctors of 54? 30. Fifty-six are how many times 7? How many times 8? How many times 28? What are the factors? 31. Sixty are how many times 10? How many times 12? How many times 5?. What are the factors? 32. Sixty-four are how many times 8? How many times 16? How many times 4? What are the ftactors? 33. Seventy are how many times 7? How many times 10? What are the factors? 34. Seventy-two are how many times 8? How many times 9? What are the factors? 35. Eighty-four are how many times 12? How many times 7? What are the factors 36. Ninety-six are how many times 12?. IIow many times 8? What are the factors I 44 INTELLECTUAL ARITHMETIC. [SEC. IV. SECTION FOURTH. LESSON I. In which Divisors are usedfrom 2 to 5.* 1. tiow many 2's are there in 2. 2 is contained in 2, how many times? 2. iHow mlany 2's are there in 4? 2 in 4, how many timles? 3. How many 2's are there in 6? 2 in 6, how n1mar.v times? 4. -Tow many 2's are there in 8? 2 in 8, how many times? 5. Ho(w many 2's are there in 10? 2 in 10, how mnany times? 6. HIow many 2's are there in 12? 2 in 2'how many times? 7. How many 3's are there in 32 3 is contained in 3, ho)w many timtes? S. lHow many 3's are there in 6? 3 in 6, how mrlany times? 9. I-low many 3's are there in 9? 3 in 9, how many vI 10. Tlow many 3's are there in 12? 3 in 12, how many? ii. flow many 3's are there in 15? 3 in 15, how many titmles? * DvIlSION is the process of finding how many times one nnmn her contains another.'he nuMnbel' to be divided is called the dividend. The ullmblhe by which we divide is called the dvi.!isor. The nmllber explressing how many t;imes the dividend ontainrs O0e divisor is called thle qluotient. Inl case tllt'ere is a num:ber left, it is called the remainder. rhe divisor and quotient are jixctors of the dividend. LEtS. I.] INTELLECTUAL ARITHMETIC. 45 12. IHow many 3's are there in 18 3 in 18, how many tires? 13. HI-ow many 4's are there in 4? 4 is contained hi 4, how many times? 14. Iow many 4's are there in 8 4 in 8, bow many times? 15. How many 4's are there in twelve? 4 in 12, how many times? 16. How many 4's are there in 161 4 in 16, how many tim-nes? 17. How many 4's are there in 20? 4 in 20, how many times? 18. Hsow many 4's are there in 24? 4 in 24, how many times? 19. low many 5's are there in 5? 5 is contained in 5, how marnly times? 20. How many 5's are there in 10? 5 in 10, how many ti mes? 21. HI-Iow many 5's are there in 15? 5 in 15, how many times? 22. How many 5's are therein220-1 5 in 20, how many tines? 23. How many 5's are there in 25? 5 in 25, how many:times? 24. IHow many 5's are there in 30? 5 in 30, how many times? QUESTIONS. 1. *"William has 8 apples, and divides them equally between two boys: how many does he give to each? g. Jafrles has 12 peaches, and divides them equally be.een his two sisters: how many does he give to enach?.~. ANALSIS.-Since 8 apples are to be divided equally be. ie.n 2. boys, one will have as many apples as 2 is contained timnes in 8, which are 4: therefore, if 8 apples be equally divided between 2 boys, each boy will have 4 apples. 46 INTELLECTUAL ARITHMETIC. [SEC. LV 3. Charles has a basket containing 20 pears, and divides them equally between his father and mother: how many does he grive to each? 4. A fBther bought 28 fish-hooks, and divided them between Johnl and Charles: how many had each? 5. A mother has a dozen needles and gives an equal numtuber to Jane and Mary: tow many will each have? 6. A lady having two parlors, bought 24 chairs, and put an equal number in each room: how many were there in each room? 7. There are 16 boys in a school-room, and but 2 benches: how many boys must sit on each seat? 8. * low many peaches, at 2 cents apiece, can you buy for 18 cents? 9. At 3 cents apiece, how many oranges can you buy for 9 cenlts? How many can yot -buy- for 12 cents? How many can you buy for 30 cents? How many can you buy for 24 cents? 10. A boy has 12 cents, and finds that he must give 3 cents apiece for tops: how many can lihe buy? If he has 21 cents, how many can he buy.? If he has 24 cents, how many can he buy? 11. If 5 barrels of flour cost 30 dollars, how much will 1 banlel cnst?t * ANALYSs.-Since one peach costs 2 cents, you can bny as many peaches for 1S cents as 2 is contained times in 1S: 2 is contained in 18, 9 timen ie therefore, you can buy 9 peaches at 2 c(enots alpiece for 18 cents. t Sincve ) barrelscf-flhoi)r cost 30 dollars, one barrel will cost as many doltars as I is contained tim'es in 0: 5 is contained in 30, 6 times; therefore, If5 barcels of flour cost 3f dollars, one barrel will cost 6 dollars. NoTE.-The follbowing rules result from the analysis of exampl ei, 1, 8;and 1 1: -E'x. 1. Divide the whole nslmber of. things by the number of parts,,to swhichL they re to be divided; the quotient will be theO number is? each part. Ex. 8. Divide the entire cost by the cost of a single tsingf; and hdc quotienst is the znumbzher of thingqs.. Ex. 11. Divided the entire cost by the numiber of things, and the quotient will be the cost of a single thinlg. LES. I.1 INTELLECTUAL.ARTTHMETIC'. 47 1.2. If' 4 yards of cloth cost 24 dollars, how much will 1 yard cost? I-How much will 2 yaids cost? 8? 4? 6? 13. If 3 yards of ribbon cost 36 cents, how mnuch will it cost a yard? How mllch will ii valds cost? 14. If 4 pounds of beef cost 48 cents, how much will 1 pound cost,? 2? 3? 5? 15. Twenty dollars are paid for 5 yards of cloth, how mnucthl is plid for 2 yalrds 4? 6? 10? 16. Firteen dollars are paid for 5 pairs of boots: how much must be paid fo)r 1 patir? 17. If 4 apples be equally divided between 4 boys, how many will each haive? 18. At 4 cents apiece, how maniy oranges can you buy for 8 cents Hflow many can you buy for 1I cenlts? 19. If it takes 4 sheets of paper for a book, how many books wrill 20 sheets make? H-ow manyl will 28 sheets malke? How many will 32 sheets mnjalke? How niany will 36 sheets make? I-Iow many will 40 sheets mak e? 20. There are 4 benches in a school-room, and 20 scholars: how manly must sit on each beinch? If there be 24 scholars, how many must sit on each bench? If there.are 32 scholars, how n:any must sit on each bench? If there be 36 scholars, how mally mlust sit on each bench? 21. If John pays 4 cents for one top, how many tops will he buy for 12 cents? Ilow many will he buy for 16 cents? HI-ow many will he'ouy for 20 cents Iow many will he buy for 28 cents? How many will he buy for 40 cents? 22. If Charles gives 4 cents a quart for chestnuts, how many' will he buy for 8 cents? I-ow many hir 16 cents' IHfow many for 36 cenits 23. In a school-house there are 5 benches and 20 48 INTELLECTUAL ARITHMETIC. [SEa. IV. scholars: blow many must sit on a bench? If there are 25 scholars, how many must sit oii a bench? If there. are 30, how manv would there sit on a bench? 24. If cloth is 5 doilars a yard, how many yards can be purchased fi:r 10 dollars? How much can be purchased for 20 dollars? For 30 dollars? For 40 dollars? For 50 dollars? 25. If flour is 5 dollars a barrel, how many barrels can be purchased for 15 dollars? How manly barrels can be purchased for 20 dollars? How many barrels for 30 dollars? 26. If tape is 5 cents a bunch, how many bunches can be bought for 20 cents? How many bunches can be bought for 50 cents? How many bunches can be-bought for 45 cents? 27. If 5 sheets of paper make a copy-book,,how many books will 20 sheets make? flow many books will 30 sheets make? LESSON II. In whichl Divisors are used from 6 to 10. 1. How nmary 6's are there in 6? 6 is contained in 6, how many times? 2. How many 6's are there in 12? 6 in 12, how many tin'es? 3. I How many 6's in 18? 6 in 18, how many times'! 4. I-low many 6's in 24? 6 in 24, how many times? 5. tIow many 6's in 30? 6 in 30, how many times? d6. How many 6's in 36? 6 in 36, how many times? LES. II.] INTELLECTUAL ARITHMETIC. 4"L 7. How many 7's are there in A? 7 is cantained in 7, how mariy times? S. I-How many 7's aile there in 14? 7 in 14, how many times? 9. How many 7's are there in 21? 7 in 21, holw many times? 10. How many 7's are there in 28? 7 in 28, how malny times. 11. How many 7's are there in 35 2 7 in 35, hoi many times 2 12.! How many 7's are there in 42? 7 in 42, how many times? 13. TIow many: 8's are there in 8? 8 is coti tained in 8, how manliy times? 14. How many 8's are there in 16.? 8 in 16, how many times? 15. How many S's are there in 24 2 8 in 24, how many times? 16. How malny 8's are there in 32? 8 in 32, hot. many times? ]7. How many\8's -are there in 40 2 8 in 40, hows many tim es? 18. How many 8's are there in 48 8 in 48, how many times? 19. I-ow many 9's are there in 92 9 in 9, how many times? 20. IIow many 9\s ar:e there in 18? 9 in 19, how many times? 21. How many 9's ae' there in 27? 9 in 27, how many times?'22. HIow many 9's are there in 36? 9 in 36, how many tines? 23. I-ow many 9's are there in 45? 9 in 45, how many times? 24. How many 9's are there in 54 2 9 in 54, how many times? 5 50 INTELLECTUAL ARITHMETIC. [SEC IV. 25. I-Tow many 10's are there in 30? 10 in 30, how many times? 26. Iowmallny 10's are there in 60? 10 in 60, how many times? 27. Iiow many 10's are there in 100. 10 in 100, Ihow many times? QUESTIONS. 1. If 6 sheets of paper make a copy-book, bow many boolos will 12 sheets make'? How many boolks will 24 sheets make? HIow many books will 30 sheets mnake? How many books will 48 sheets make. Itow many books will 60 sheets make? 2. If 1 yard of broadcloth costs 6 dollilrs, how many yards can be bought.for 30 dollars? I-low many yards can be bought for 36 dollars? flow many yards for 42 dollars? HIow many yards for 54 dollars? I-low many for 60 dollars? 3. If a man.travels 6 miles in 1 hour, how many hours will it take him to travel 12 miles fHow many hours will it take him to travel 24 miles? How long will it take him to travel 30 miles? IIow long to travel 54 miles? How long to travel 60 miles? 4. Forty-two apples are divided equally among 6 boys: how many does each one receive? 5. If 54 peaches be divided equally between 6 boys, how many will each receive? 6. If a yard of ribbon costs 6 cents, how many yards can be bought for 24 cents I-low many for 30 Cen ts? 7.. If you have 28 dollars, how many yards of cloth caOn you buy at 7 dollars a yard? HIow many yards for 35 dollars? HIow nmany for 70 dollars? 8. If you have'l6- apples to divide among 8 boys, how many do you give to each? If you have 32, LES. l1I.1 INTELLECTUAL ARITHMETIC. how many? If you have 64, how many? If you have 96, how marny? 9. A laborer engaged to work for 12 dollars a month; at the end of the time he received 96 dollars, hlow many months did he work' 10. James engaged'to wo rlde:for 9 cents a day, and at the end of the time received 72 cents: how many days did lie work? 11. If a man can do a piece of work in 56 hours, how many days will it take him to do it, working 7 hours a day? 12. If two boys are 36 yards apart, and the oln behind gains on the other 4 yard a minute, how many minutes before he will overtake him? 13. A man has 84 pounds of butter to be put in seven jars: how much must be put in each jar? 14. A lady paid 108 cents for 9 yards of ribbont: how much did she pay a yard? 15. If a milliner pays 121 cents for 11 yards of ribbon: how much does she pay for 1 yard? 16. HIow many dresses can be cut from 132 yards of silk, if each dress contains 12 yards? LESSON III. Addition, Subtraction, Multiplication, Division. 1. Two in 5, how many times, and what over? 2. Two in 7, how many times, and what over.* 3. Four in 15, how many times? 4. Five in 19, how many tinies? 5 Five in 36, how many times? 6. Seven in 42, how many times? 7. Nine in 60 how many times? w When there is a remainder, let it be simply mentioned 52 INTELLECTUAL ARITHMETIC. LSEC. IT. 8. Seven in 64, how many times? 9. Ten in 55, how many times? 10. Six in 70, how many times. 11. Nine in 100, how many times? 12 Five in 56, how many times? 13. Twelve are how many times 2? 14. Eighteen are how many times 6? 15. Nineteen plus 6, are how many times 5? 16. Twenty plus 8, are how many times 7? I-ow many times 4? 17. Thirty less 6, are how many times 6? How many times 4? How many times 3 18. Sixty less 5, are how many times 11? 19. Ninety plus 9, are how many times 11? 20. Eighty-seven plus 3, are how many times 10. 21. Forty-fi ve plus 4, are how many times 7 22. Sixty-nine plus 15, are how many times 12? How many times 6? How many times 7? 23. Six times 7 less 2, are how many times 10? 24. Forty plus 4 times 6, are how many times 8? 25.;Fifty plus 3 times 4, plus 2, are how many times 8? 26. Five times 6 plus 4 times 9, are how many times 1? 27. Seven times 8 plus 4, are how many times 6? 12? 10? 5?. 28. Eight times 5 plus 5, are how many times 9? 5? 29. Five times 11 less 5, are how many times 2? 10? 5?. 30.,Forty-six less 3 times 2, are how many times 10? "~? 5? 4? 2? 31. ~eyen times 9 plus 3 times 4, are how many tiMnesq.'5? 32. Six times'-4 less.3 times 4, are now many times 3? 4?.6? LES. IIn.] INTELLECTUAL ARITHMETIC. 53 QUESTIONS, 1. If 2 yards of cloth cost 6 dollars, what will 8 yards cost 2-* 2. If 3 oranges cost 12 cents, what will II cost? 3. If 4 boxes of raisins can be bought for 16 dollars, how much will 9 boxes cost? 4. If 7 pounds of sugar cost 56 cents, what will 12 pounds cost? 5. What will 6 lemons cost if 8 cost 24 cents? 6. Jam es has apples worth 2 cents apiece, howv mtaniy must he give for 6 oranges worth 3 cents apiece? 7. How many eggs at S cents a dozen must be giNven for 12 pounds of sugar \worth 6 cents a pound? 8. How nmany knives at 2 shillings apiece are worth 4 axes at 8 shillings apiece? 9. HIow much barley at 5 shillings a bushel must be given for 6 bushels of wheat at 10 shillings a bushel? 10. A farmer bought 4 yards of cloth at 3 dollars a yard and paidcl in labor at 2 dollars a day; how many days must he labor? 1li. John had oranges worth 4 cents a piece which he gave to Jamies for 2 quarts of cherries worth S cents a quart; how many oranges did he give for the cherries? NOTE.-If anv number be divided into two equal parts, one of the parts is called one half. If it be divided into three equal parts, one of the par'ts is called onte third, and two of tihe parts are called two ti tirdt. If it be divided into four equal parts, one of the parts is called one fourth, two of the parts, twzo fourths, three of them, three fourths, &., &c. * ANALYsI. —One y ard will cost one half as much as two yards. If two yards cost 6 dollars, one yard will cost on.half of six dollars, which are 3 dollars; 8 yards will cost 8 times as much as 1 yard: if 1 yard costs 3 dollars, 8 yardsw..vill cost 8 timles 3 dolulars which are 24 dollars'. therefore, if 2 yards of cloth cost 6 dollars, 8 yards -will cost 21 dollars. 54 INTELLECTUAL ARITIIMETIC. [SEC. ~V. 12. How many yards of cotton cloth, at 8 cents a yard, mlust be given for 6 pounds of butter worth 12 cents a pound? 1 3. A fairmer bought 4 yards of broadcloth wortb 5 dollars a1 yard, and paid for them in calves worth 4 doll;ars apiece. Ilow many mast he give? 14. I-ow many barrels of flour worth 6 dollars a 6arrel imust be given for 8 barrels of fish worth 3 dollars a barrel? 15. A inan bought S1 oranges at t;1e rate of O for 5 cents. 1-How imuch did they come to?i 16(. HIow many pears at'the rate of 2 for 3 cents can be 1bought for 24 cents? 17. Jaml-es bought 20 marbles at the rate of 5 for 0 cents. I-How much did they cost? 18. How nmuch cloth worth 2 dollars a yard, must be givel for 2 fiukins of butter worth 18 dollars a firkin? 1i9,. What will 8 pounds of beef cost, if 6 pounds cost 54 cents'? 20. A farmer has 6 dozen of eggs worth 12 cents a do~zenl, and wishes to exchange them for nutmlegs at 3 cents apiece: how many shall he receive? 21. A Ilran bought 4 barrels of flour at 7 dollars a barrel, and wishes to exchange them for' cloth at 2 dollars a yard: how nmany yards of cloth should he receive? 22. F1ourr men bought a boat for 60 dollars1: they paid 12 dollars fobr repairing her, and then sold her so that they made 5 dollars: what did they get for her. Hlow much more than they gave? * ANALYLIs.-SSince 3 oranges cost 5 cents, 18 oranges will cost as many times 5 cents as 3 is contained timles in 18, \wh)ich Ls 6 timles: therefore, 18 oranges will cost 6 times 5 cents which iare 30 ceLntu. LES. III.] INTELLECTUAL ARITHMETIC. 5D 23. A laborer engaged to work for a year at 144 dollars a monith, but was sent away at the end of 7 months: how muc.h should he receive? How mluch less than if he had staid the entire y ea.r 24. A drover has S calves for which lhe paid 40 dollars, and 9 sheep for which he paid 27 dollars,hat did he pay apiece fo(r the calves? WVhat for the sheep? How much for ail? 25. If 8 firkins of butter are worth 96 dollars, how Imany firkins must be given for 3 barrels of sugar worth 20 dollairs a barrel? 26. Six men agree to do a piece of work for 90 dollars: it tutrns out that each man malkes 5 dollars by the bargain: what was the cost of doing the work? 27. A mann has 40 eggs, fiom which he wishes to raise chickens: his eggs( are worth a cent apiece, and his chickens, when hatched, w-ill be worth 3 cents apiece. Now, if 15 of his eggs prove addle, how much Nwill he gain 2 28. How much honey at 16 cents a pound must be given for 6 pounds of coffee at 8 cents a pould? 29. -Iow many hatts at 4 dollars apiece, can a ma-n buy if he gives in payment seven handkerchiefs at I dollar' apiece, and 5 pair of boots at 6 dollars a pair? What will he have left? 30. A grocer purchased 4 barrels of sugar at 12 dollars a barrel, and 5 hogsheads of molasses at 20 dollars a -hogshead, and sold the whole for 150 dollars': dcld he make or lose, and how much? 31. Fiv!e men agree to do a piece of wo-rk for 60 dollars, each to receive an equal part. When the is half done, two of the men quit, and the other finish it: how much should each receive 2 ri'ee men made up a purse. The first put i, A.jtirs; the second twice as much as the first, and 56 INTELLECTUAL ARITHMETIC. [SEC. IV. the third twice as much as the second: how much was put in in all? What would each have put in had they contributed equally? 33. ilMr1. Johnson bought 14 yards of broadcloth at 4 dolla'rs a yal'd, and having cut off 9 yards, sold the remainder for one half of what he paid for the whole less 3 dollars: how much more was this for each vard than he gave? 4, If it6 en can do a piece of work in 9 days, how long will it take one man to do it?* 35. How many men in 9 days can perform as much labor as 12 men can in 6 days? 36. If a barrel of flour will last 5 men 25 days, how long will it last 9 men? 37. A maln has a piece of work which would employ 3 men fobr 8 days: how many men can do it in I d ajY In 4 days 4',8 If S men can do a piece of work in 5 days, how long will it take oiie man to do it? How many men will do it in one day? 39.,A piece of work requires 9 men for 10 days, how many men can do the same work in one day? In 3 days? In 6 days? 40. If 6 men can build a wall 12 rods long in 8 days, how long will it take one maln to build it? How many men can build it in 1 day l In 3 days? In 12 days? - ANALrs2s. —It will take 1 man 6 times as long to do the same orlk as it mwill 6 men: if it takes 6 men 9 days, it will take 1. man 6 times 9 days, which are 54 days: therefoi e, if it take 6 Ien 9 days to do a piece of work, it will take 1 man 64 days to do the same work. ANXALYSIs. —It will take 8 times as mniny men tio work in I day as it iwill,to do it in 8 days: if it takes 8 cdays, it will take 8 times)3 men, TwMch are A24 mni,;' work in one day: theref!9, if 3 men can do a piece mo.8 days, 24 men cand'o the samre work in 1 day. IES. IV.1 INTELLECTUAL ARITHMETIC. 57 LESSON IV. Combinations. The classes are to be drilled in the following tables until they are fully understo)od. The tables are to bo read across the page: thus, 1 and 1 are two; 1 foron 2 leaves one; once I is one; 1 in 1, once; and similarly fi)r the other lines.* 1 and 1; 1 fiom 2; once 1; 1 in 1 I and 2; 1 firom 3; once 2; 1 in 2 1 and 3; 1 fr,m 4; once 3; 1 in 3 1 ajnd 4; 1 friom 5; once 4; 1 in 4 r and 5; 1 fioom 6; once 5; 1 in 5 1 and 6; 1 friom 7; once 6; 1 in 6 1 and 7; 1 f'om S; once 7; 1 ii 7 1 and 8; 1 friom 9; once 8; 1 in 8 1 and 9; I from 10; once 9; 1 in 9 1 and 10; 1 firom 11; once 10; 1 in 10 2 and 1; 2 fiom 3; two timnes 1; 2 in 2 2' and 2; 2 fromn 4; two times 2; 2 in 4 2 and 3; 2 from 5; two times 3; 2 in 6 2 wand 4; 2 firom 6; two times 4; 2 in 8 2 and 5; 2 from 7; two times 5; 2 in 10 2 and 6; 2 froom 8; two tilnes 6; 2 in 12'2 iled 7; 2 fromn 9; two ti mes 7; 2 in 14 2 Itnd 8; 2 from 10; two timnes 8; 2 in 16 2 aind 9; 2 firomn 1; two times 9; 2 in 18.2 and 10; 2 frot 12; two tiines 10; 2 in 20 Sucar sT;oN,.-The object in placing these tables here is tv Ihill the plpil in changiug his mind from one subject to another r.u each line he begins with adulition, passes to subtraction, then multiplication, anid then to division. The difficulty of doing 4 accratel/, and, rapiidly. will iilustrate fully the value of the -ild must d be' r ix i'i. lined that can go through *,les without a mistake. INTELLECTUAL ARITHMETIC. [SEC. IV, 3 and 1; 3 from 4; 3 times 1; 3 in 3 3 and 2; 3 from 5; 3 times 2; 3 in 6 3 and 3; 3 from 6; 3 times 3; 3 in 9 3 and 4; 3 from 7; 3 times 4; 3 in 12 3 and 5; 3 fLom 8; 3 times 5; 3 in 15 3 and 6; 3 firom 9; 3-times 6; 3 in 18 3 and 7; 3 from 10; 3 times ~7; 3 in 21 3 and 8; 3 from 11; 3 times 8; 3 in 24 3 and 9; 3 fi om 12; 3 times 9; 3 in 27 3 and 10; 3 firom 13; 3 times 10; 3 in 30 4 and 1; 4 from 5; 4 times 1; 4 in 4 4 and 2; 4 fron 6; 4 times 2; 4 t~a 8 4 and 3; 4 fromI 7; 7 4 times 3; 4 in- 12 4 and 4; 4 from 8; 4 times 4; 4 in 16 4 and 5; 4 fiom 9; 4 ti mes 5; 4 iln 20 4 and 6; 4 fiom 10; 4 times 6; 4 in 24 4 and 7; 4 from.11; 4 times 7; 4 in 28 4 and 8; 4 from 1.2; 4 times 8; 4in 32 4 and 9; 4 fiom 13; 4 times 9; 4 in 36 4 and 10; 4 from 14; 4 times 10; 4 in 40 5 and 1; 5 fiom 6; 5 times 1; 5 in 5 5 and 2; 5 firom 7; 5 times 2; 5 in 10 5- and 3; 5 fiom 8; 5 times 3; 5 in 15 5 and 4; 5 firom 9; 5 times 4; 5 in 20 5 and 5; 5 firom 10; 5 times 5; 5 in:25 5 and 6; 5 friom 11; 5 times 6; 5 in 30 5 and 7; 5 firom -12; 5 times 7; 5 in 35 5 and 8; 5 from 13; 5 tinles 8; 5 in. 40 5 and 9; 5 firom 14; 5 times 9; 5 in 4'" 5 and 10; 5 from 15; 5 timnes 10; 5 in 5.! B and 1; 6 fonom 7; 6 times; in 6 and 2; 6firom 8; 6 tires 2 6 in:'6' and 3; 6 from 9; 6 times o3; 6,in.l &i: 6 and 4; 6 fiom 10, 6 t:mles 6 and 5; 6 firom 3.; 6 timtes LBS. IV.] INTELLECTUAL ARITI-IMEi'f' 59 6 and 6; 6 from 12; 6 tilnes 6 6 in 36 6 and 7; 6 from 13; 6 tilnes 7; 6 in 42 6 and 8; 6 from 14; 6 times 8; 6 in 48 6 and 9; 6 from 15; 6 times 9; 6 in 54 6 and 10; 6 fromn 15; 6 times 10; 6 in 60.7 and 1I 7 from 8; 7 times 1; 7 in 7 7 and 2; 7 fi om 9; 7 times 2; 7 in 14 7 and 3; 7 from 10; 7 times 3; 7 in 21 7 and 4; 7 from 11; 7 times 4: 7 in 28 7 and 5; 7 from 12; 7 times 5;7 in 35 7 and 6; 7 from 13; 7 times 6;7 in 42 7- and 7; 7 firornm 14; 7 tirnes 7; 7 in 49 7 and 8; 7 fromn 15'; 7 times 8; 7 in 56 7 and 9; 7 from 1.6; 7 times 9; 7 in 63 7 and 10; 7 from 17; 7 tim es 10; 7 in 70 8 and 1; 88 timnes 1;8in 8 8 and 2; 8 from 10; 8 times 2; 8 in 1 6 8 and 3; 8 fi om 11; 8 times 3; 8 in 24 8 and 4; 8 from 152; 8 timnes 4; 8 in 32 8 and 5; 8 from 13 3; times' 5; 8 in 40 8 and 6; 8 fiomn -4; 8 times 6; 8 in 48 8 and 7; 8 fiomn 15; - tirnes 7 8 in 56 8 and 8; 8 from 16; 8 times 8; 8 in 64 8 and 9; 8 fror l1; 8 times 9; 8 in 72 8 and 10, 8 fro'l 18 8 tim-es 0; 8 in 80 9 and 1; 9 froi 10; 9 tinmes 1 9 iln 9 9 and 2; 9 fi'om 11;j 9 times 2; 9 9 and 3; 9 firom. 12; 9 inies 3;. ~9 and 4;9 Iom-fi 13; 9 times 4 ~ t 9:arnd 5; 9 fi in 14 9 ti-mes 5; 9 and 6; fitorl 15; 9 times 6 9 9 aid'7 7; f9 iron 1t; 9 times 7; 9; 9 frotn: 1; 9 tiles 9 flc ~ 60 INTELLWECTUAL ARITHMETIC. [SEC. lV. 10 and 1; 10 from 11; 10 times 1; 10 in 10 10 and 2; 10 from 12; 10 tlmes 2; 10 in 20 10 and 3; 10 from 13; 10 times 3; 10 in 30 10 and 4; 10 fro)- 14; 10 times 4; 10 in 40 10 and 5; 10 fi'dhit15; 10 times 5; 10 ill 50 10 anld 6; 10 firom 10; 10 times 6; 10 in 60 10 and 7; 10 fiom 17; 10 tirnes 7; 10 in 70 10 and 8; 10 from 18; 10 times 8; 10 in 80 10 and 9; 10 fiom 19; 10 times 9; 10 in 90 10 ald 10; 10 fiom, 20; 10 times 10; 10 in 100 11 and I; ll from 12; 11 times 1; 11 in 11 11 and 2; 11 froom 13; 11 times 2; 11 in 22 11 atind 3; 11 friom 14; 11 times 3; 11 in 33 11 anid 4; 1I firom 15; 11 ti mes 4; 11 in 44 I1 mid 5; 11 fiom 16; 11 timlsi 5; 11 in 55 11 alid 6; 11 fron 17; 11 tim-es 6; 11 in 66 ll and 7; 11 from 18; 11 tilnes 7 11 in 77 ll anid 8; 11 fiom 19; 11 times 8; lI in 88 11 and 9;: 11 firom 20; 11 times 9 11 in 99 11 an1 d 10; 11 friom 21; 11 tim-)es 10; 11 it ilO 11 and II; 11 firom 22; 11 times 11; I 1 in 121 11 atnd 12; 11 from 23; 11 times 12; 11 in 1032 12 dand 1; 12 fiom 13; 12 tiiiies 1; 12 in 12 12 anI d 2; 12 from 14; 12 titles 2 12 in 24 12 anid.' 3; 12 fiom 15 12 timnes 3; 12 in 36 12 and 4; 12 fiolml 16; 12 tixies 4; 12 in 48 12 aiid 5; 12 from 17; 12 timnes 5; 12 in 60 6; 12 froni 18; 12 tilnes 6; 12 inl 72 7; 12 frnom 19; 12 tilnes 7; 12 in 84 8; 12 fr'om 20; 12 times 8; 1') in 96 9; 12 friolm 21; 12 tilmes 9; 12 Iu i)08 10; 12 fioiio 22; 12 til-es 10; 12 ai 12( 11 12 frioml 23; 12 timnes 11; I1, in S131 1'2. tilmes 12 ~o a ~oIt INTELLECTUAL ARItHMETIC,.] SECTION FIF TH. LESSON I. UNITED STATES MONEY. 10 mills malke.,. cent. ct. 10 cents..... c dille.. d. 10 dimes... 1 dollar.. 10 dollars.... 1 eagle. E. 1. How many minlls are there in 2 cents? In 8 cents? In half a cent? In five cents? 2. I-ow many cents aie there in 10 mills? In 15 mills? In 65 mills? 3. How many cents are there in 5 dimes? IIn 6 Nw mnany equal parts is the unit dividecl? Which figuire is the numerator?:Which the cdlenomin?tor? 9. In the fi:1u tiolln 1-, il)to how many equal parts is the uinit divifled?'Which is the numerat{or? NWhich the denolni -1iltor? 10. hLeadl the follonwincg fraections: — tlhree eighths. eiht thirteent. 3 three sevenths. 1 fi:-tv-one. sixteenths. 5 five nilths. 9s3 two and 3 eighths. 6 six twelfths. 59 five and 9 sevexnths. 7-5 sevCn fifi;eenlths. (O(-i six land 5 twelftrhs.' 11. In the fiaction, into how many equal plarts is tile unit divided? How mIlany of these parts are taken? 12. In the firaction 3, into how niany equal parts is the unit divided? 1hIow nianny part.s are taken?; ScoGEsrT'oss..When the numerator is less than the leno mllunat'or, the fraction is called a proper fraction. When the numerator is greater than the denominator, the fraction is called an improper fraction. A whole number united with a fraction, is called a vmixed tnumber. 84 INTELLECTUAL ARITHMETIC. I,. A. V~13. In the fraction 5, into how many equal parts is the unit divided? H-low many parts are takenl 14. In the fraction g, into how many equal parts is the unit divided? How many parts are taken? 15. In the fraction 6-, into how many equal parts is the unit divided? How many parts are taken? 16. In the fiaction 7-, into how many equal parts is the unit divided How many parts are take n? 17. In the fiaction R,-, into how many equal parts is the unit divided? How many parts are taken? 18. In the fiaction 2, into how many equal parts is the unit divided? How mlany parts are taklien 19. What figures express one third of one? 20. What figures express five sevenths? 21. What figures express nine twelfths? 22. What figures express twelve qeventeenths? 23. What figures expyess tell thirty-sevenths? 24. What figures express fifteen sevenths? 25. What figures express fourteen twenty-ninths'? LESSON X. Fractional U~wnits. 1. What is the unit of a fraction? It is the whole thing which is divided into equal parts. 2. What is each equal part called? A fractionzal unit. 3. In the fraction - of a pound, what is the unit of the fraction? 1One poLund. What is the, firactional unit? One-eighth of a poulnd. How many fi'actional units are taken? Five. 4. In the fraction A, what is the unit of the fraction? LES. X.] INTELLECTUAL ARITHMETIC. 5 The abstract or simnple unit, one. WVhat is the fractional unit? One fourth. How many fractional units are taken? Three. 5. What is the fractional unit in three fourths? IHow many fractional units in three fourths? 6. What is the unit of the firaction -7 of a dollar What is the fractional unit How many fractional units are taken? 7. What is the unit of the fraction ~? What is the fractional unit? How many are taken? 8. How many firacti)lnal units in i~ What is the unit of the fraction? What the firactional unit? 9. Jamnes and John have each an apple of the saseu size. James cuts his into two equal parits, and gives away one half. John cuts his into three equal parts and gives away two of the pieces. They seek to find what part of a whole apple they have left.* * SUGGcSTION.-This example suggests all the principles cmployed in fiactions. They may be thus stated: 1st. That something regaclded as a -whole, called unity, is the primary base of every friaction: and 2d. That one of the equal parts of unity, called the fractional unit, is the second base of any fractional number. From the nature of Division and Multiplication, we see: 1st. That the firactional unit is as many tines less than uuity as there are units in the denominator: 2d. That the numerator shows how many fractional units ul e taken. 3d. If the numerator be multiplied by any number, the nlml ber of fractional units will be increased as many times as there are Ilnits in the multiplier. 4th. If the numerator be divided by any number, the numlber uf fractional units will be diminished as many thies as the-re are unllits i the divisor. S 86 INTELLECTUAL ARITHMETIC. LSEC. VI. Now, says James, if I cut my half into 3 equal palts, each part will be three times less than before; that is, each part will be one sixth of the entire apple, but I shalli have three times as many parts, co that I shtll still have half my apple. Now, says John, if I cut my third into 2 equal arts, each part will be two times less than befoie; hat is, it will be one-sixth of the entire apple; but I shall have twice as many parts, so that I still have one third of the apple. Now, says John to James, if you have 3 sixths and I have 2 sixths, together we must have 5 sixths of a whole apple. 10. 1t the fiactions I and 1, what is the fractional unit of the first. Wh at of the second? I-Hoo can you reduce them to the same fractional tmlit? By multiplying the numlerator and denominator of the first by 3, and of the second by 2. What is the fractional unit after the multiplieation? ht.ows many friactional iunits arle taken in each fi'action? Ho)w many in both? 11. I-Iow can you reduce firactions to the same fractional unit? By multiplying the numerator and denominator of 6th. If the denominator be multiplied by any number, the friactional unit will be diminished as many times as there are units in the multiplier. 6th. If the denominator be divided by any number, the value of tho fi'actionall unit will be increased as many times as there ui'e units in the divisor. By combining 3 and 5, we see: 7thl. That if the numerator and denominator be multiplied by tile san.me number, the value of the fraction will not be changed. And by combining 4 and 6, -we see: 8th. That if the numerator and denominator be both divided Jby the same number, the value of the firaction will not be changed. LES. X.1 INTELLECTIUAL ARITHMETIC. 8~ eaach fri-ction by such a number as shall make the dei,n nilinltols the same in all. 12. Reduce - anld I, to the same fractional unit?' If we multiply the num erator and denominator of the first firaction by 4, we have -4A; and if we multiply thesecond by 3. we have, in which the fra.. tional unit is rlJ in both. 13. Reduce I and' to the same fitactional unit. 4 5 After reduction, what is the fractional unit, and Iow many fr'actional units in both 14. Reduce - and I to the same fractional unit. After reduction. what is the fractional unit, and how many fri'ational u1nits in both? 15. lRed-uce 1, an3 d 3, to the sam e fractional unit. 2-, 4 After reduction, how many fractional units in each? 16. iRe duce 3 and 5 to the same fractional unit. After reduction, what is the fractional unit? How many in each? 17. Ruclduce 5. r, and 2, to the same fiactional unit. 19. Reduce 4, and 2 to the sane frctional unit. 21. Reduee 3 ad, to the same fractional unit. 219. Reduce 4, 3 and 8 to the same fiactional unit. 20. Reduce anmd 4 to the same fractional unit. 231. Reduce s anld, to the sa me fiactional unit. 22. Reduce 4a and t o the same fraeti,,nal unit. 2' 5, 6 24. In?L5 how iany units 1? 25. In 67 how many units 1? 26. In 495 how many units 1? ~27. In 3-6 how nilany units 1? 28. In 4 7 how rY many units 1? 29. In 24 how nianyv units 1? SG:GEsTIoNs. —There are but three operations which chang tIe units of numbels: They are, 1st. To change integral to fiactiolal units: 2d. To chlange fiactiounal to integral units: and 3d. To change firom one firactional unit to another. The filst two were fully explained in the first seven lessotn of this section: and the third is treated of in this. ;~F INTELLECTUAL ARITHMETIC. [SEC. VL 30. In 227 how many units 12 31. In 39 how many units 1 32. l]i 54 how many units 1? 33.. In 7g how many units 1? 34. How many units I in 3Z 2 35. IHow many units 1 in 49: 36. I-low manv nnits 1 in 6_-3 37. H-ow many units 1 in 47? 38. How many fourths in 1 half? 39. tIow many fourths in 2 halves? 40. How many fourths in 6 halves? 41. H-low many thirds in 12 sixths? 42. HEow in any sixths in I third? 43. How many sixths in 4 thirds? 44. How many sixths in I half? 45. How many sixths in 2 halves? 46. How many eighths in 1 fourth? 47. How many eighths in 1 half? 48. How many eighths in 2 fourths? 49. HIow many eighths in 3 fourths? 50. How manv ninths in 1 third? 51. How many ninths in 2 thirds? 52. 1How nany ninths in 6 thirds? 53. How many tenths in 1 fifth? 54. How many tenths in. 1 half? 55. How many, twelfths in 1 third? 56. How many twelfths in 1 half? 57. How many twelfths in I quarter? 58. I-low many sixteenths in 1 eighth? 59. How many sixteenths in l quarter? (-'0. low many eighteenths in I ninth? 61. How many eighteenths in 1 half? 62. How many eighteenths in 2 thirds? 63. flow many twentieths in 3 halves? 64. I-How many twentieths in 3 quclarters? 65. HTow many twenti-ths in 5 f)Lurths? 66. How many twentieths in 3 fifths? LtS. XI.] INTELLECTUAL ARITHMETIC. 89 67. How many halves in T? In %6_. 68. IHow many halves in 24 2 In - T 69. How many thirds in 2 4 In11 _ 2 70. ITrow many thirds in 8? In i-? 71. H-ow many thirds in 9?In 42 7. 2 Iow many fouirths in 26 In 6 7o. How many fourths in 3-? In 1-6 5. f~low~ many fourths in II In -L 74. I-ow many fourths in 72 In 1 77. H-low mralny fifths ij 5 Z 320 78. Ilow many sixths in 6? In 4 A 79. f-low many sixths in 16? In 4I? 80. How many sevenths in 2? 2o In 54 2 81. How many seveixths in 6 2? In A6 LESSON XI. Valule of nTenbers. 79. hatow is the base if a nunber The0. How primary base of evely number is the UNiT ONE. 2. Whot is an integen, or whole numbe? It is a number which contains the unit one an exact number of times. Thus, three, four, five, &c., are inlteges, or Vahole numbers. 3. What is the fctional number? It is a number which explesses one or more of tir equal palts of unity. T'Phus, 2, 43, S,9 &c., are fractions. 4Th. hat expresses the value of a number either integral or frawhctionale number It is he number of tiwhes which it contains the unit one. 5. numbow any times does, three, four, fie contain one? 3. WTow many times is five greater than oie? I S acnEsTIoN.Dide the numerhltorh and denomilator thy iuch a number as will give the requied fractional unit? 8* 90 INTEILLECTUAL ARITHI-IIETIC. [SEC. VI. 7. [How many times is six gleater than one? Why? 8. Flow mn:ay times is eight greater than one. Why-1? 9. [low many times does one half contain the unik 011.t) 2 On)e-half times. 10. Ilw many times is one half less than ore? Two times. 11. Wlhat expresses the value of a fraction? The nulmber of timles which a firaction coitains 1. 12.'What is thle value of one half? Why? 13.'What is the value of two halves? Why? 14. \hat is the value of three halves? VWhy 1 15. What is the value (f four halves? Why'? 16. What is tile value of one third? Why? 17. What is the value of three thirds? Why I 18. W hat is the value of six thirclds? Why. 19. IH-ow do you find the value of an improper firaction?' BV dividing the numeitttor by the denominator. 20. If ther e is a lielniainder, w hat do you do with it? Write the denomina tor under it, and annex the fracticln to the integrl number. 21. \Vhatt is the value of the flaction seven thirds.'2,. What is the value of nine fourths? Of seventytwo twelfths 2 Of one hundred eighths? Of ninetyfonu' nlinths?2 23. What is the value of seventy-five twelfths! Of sixty-seven eighths? Of eighty-nine eleveuths? Of otie hundred antd twenty tNwelfths? A* ANAL SS. —Since the numerator shows how many firactional uni'ts are taken, and the denominator how many fiactional units nlake 1, it follows, that the numelrator divided by tlle denoulinfator will shlow how many units 1, there are in the firaction. Impriless the pupil, constantly, that every number, whllether integral o1 fiaetionDal, must be compared witfh the ullit one. Also, that the valme of any numiber is expressed by the numbet of times which it contains the unit one. LS5. XI.] INTELLECTUAL ARITHMETIC. p 24. When is a fraction said to be in its ]e;est terms? A fi'action is said to be in its lowest terms when there is no number except 1 that will divide both the numerator and denominator w\ithout a'remainder. 25. How may a fraction be reduced to its lowest terllms? By dividing both the numerator and denominator by the samt-e number.-* 26. Is the fraction 3 in its lowest term:s? Why? 27. What fiaction expresses the lowest termlns of 4 2 28. What fiaction expresses the lowest terms of 8? 29. What fraction expresses the lowest terms ~,f 6? 30. What fraction expresses the lowest terlms of -4r t T 2 30, What are the lowest terms of the frictioun 6? 32. What are the lowest terms of the fractiouJ 7? 33. Whhat are the lowest terms of the friactionl 1s 35. What are the lowest terms of the fraction 8-x' 36. What are the lowest termls of the fraction 1 37. Reduce the fraction -_ to) its lowest teruin. 38. What fraction will express the lowest termso,f 24 8 39. What fraction will express the lowest termns 40. What fraction will express the lowest terms of 6? SucGEs'rooN.-Reducce every fraction to its lowest ternms bCfore pecrforming any other operation. Reduce also every aneowLer t its lowest terms. 92 IDNTELLECTUAL ARITHvIETIC. LSEC. V LESSON XII. Adding Fractional Units. James and John have each an apple of the same size. James cuts his into 4 equal parts and gives away 2 parts. John cuts his into 5 equal parts and gives away 3 parts. They seek to find what part of an apple each has left, and what part both together have? Now, says James, if I cut each of my fourths into 5 equal parts, I shall have 10 parts; that is, 10 twentieths of an apple: and says John, if I cut each of my fifths into 4 equal parts, I shall have 8 parts; that is, 8 twentieths of an apple; hence, James has 10 twentieths, and John 8 twentieths, and together they have 18 twentieths. 2. What is the unit of the fraction l W't What is the fractional unit in 18 twentieths of an apple? How many fractional units are taken? If the numerator and denominator be each divided by 2, what. does the fraction become? What, then, is each fractional unit? IIow many are taken? 3. What is necessary in order that two or more fi'actions may be added together. That they have the same integral unit, and the same firactional unit. 4. When no integral unit is named, what unit is understood? The abstract unit one. 5. tHave the fiactions 1 and - the same integral uiiit? What is it? Have they the same fractional unit'? What is it? d. I-low many fractional units in the first? Itow manly in the second? How many in both? 7. What is the sum of the fiactional units in 1 snd 2? 8. What is the sum of 3 and 2? LES. XI.]j INTELLECTUAL ARITHMETIC. 9. What is the sum of -, 2 and 32 10. What is the sum of -1, 4 and a,? 11. What is the sum of 7, 5, and 6? 12. What is the sum of 5, A, and 2-? 13. What is the sum of 3,a 93nd? 14. What is the sum of 3, 5, and 7 15. What is the sumr of,6, 6, and 7 1 16. What is the sum of, 3 and 5 17. What is the sum of 3, A, -, and ~?' 18. What is the sum of 6 7 8,and 9 19. What is the sum of 3 4 6 and 7? 20. What is the sum of 4, 3, a, kd L 21. What is the sum of 62 -, and 7? 22. What is the sum of -, and 1 2 i, 7 7, and z 7 7 23. What is the sum of' 8, and? 24. What is the sum of I -a and 1,2 9 9' 9 9 252. What is the sum of I, 3 5, and 3? 2' 2' I 2'I 2 26). What is the sum of 5 6 - 27. What is the sum of 8 3 4 and I1 28. What is the sum of 3,, 7 and 9 29. What is the sulm of 5 6 9 anld 1- 2 30. Add together 2 and 3.. First, I is equal to 3-, and ~ is equal to 2; ther, plus C- are equal to 631. W hat is the sum of I and 4 32. What is the sum of 5 and 4? (Reduce to thiitieths.) 33. What is the sum of 2 and 3? 34. What is the sum of 3 and' A. 8 2 4 35. What is the sum of 4 and -3? 36. What is the sumn of 2 and 5,? 37. What is the sum of 5- and 3-'. 38. What is the sum of 4 and 3? 39. lW hat is the sum of s and A 2 90. WhSatt is the slum of g- and 4? 41. What is the sum of -- and 3. 94 INTELLECTUAL ARITHMETIC. [SEC. VI. 42. What is the sum of 2- aild 34 and 5 12 13. What. is the sumn of 23 and 44 and 66? 44. What is the sum of I 2 and 64 and 7260? 45. What is the su1m of 4 2 and 5 and 2.7 40i. What is the sum of 94 and 5 and 747. What, is the sum of 5s and 4 - and 5 48. What is the sum of 94 a-nd 75 and. 7? 49. What is the sum of 5-8 and 41-9- ald 89? 50. What is the sum of 64 and 8 - anld 1 0 ~ 51. What is the sum of 5a- and 8 8 atnd 2- 9 2 52. What is the sum of 5 and 2~ and 3? QUESTIONS. I John buys a top for one sixth of a shilling, a stick of candy for one twelfth of a shilling, and a piece of india-rubber for one third of a shilling: what does the whole cost him?t 2. James pays 3 of a dollar for a pair of gloves, and 3 of a dollar for a handkerchief: how much do they cost hin? 3. Nancy buys a work-box for 7 of a dollar, a pair of gloves forl3 of a dollar, and a comb for -;- of a dollar: how much do they all cost? 4. Jane buys a yard of ribbon for 2 of a dollar, a gold pin for - of a dollar, and an inkstand for 3 of a dollar: how much did she pay in all 5. Williaim buy-s a kite for R of a dollar, and a string for -4 of a dollar: how much did he pay? 6. Three ducks cost } of a dollar, two fowls 2 of a dollar, and two geese 1- of a dollar: what is the entire cost? * SBUoEOSTIONs.-Add the integer numbers separately, and unite the sum to the sumi of the fractional parts. { ANALYSIS. — of a shilling is 12r of a shilling. 2 of a shilling is 2 " 3 of a shilling is A2 aence the whole costs is - - - -72 of a shilling. LES. XII.] INTELLECTUAL ARITHTMETIC. 95 7. Three sheep cost 17 of a dollar, a calf of a dollar, and a lamb 5 of a dollar: what is the entire cost? 8. Three yards of cloth cost 4 of a dollar, a handkerchief 7 of a dollar, and a pair of gloves ~ of a dollar: what is the entire cost? 9. A father paid 3 quarters of a dollar for his own breakfast, one third of a dollar for his son's, and a quarter of a dollar for his daughter's: how much did he pay in all? 10. A merchant sold 3! yards of cloth from one piece, 2- yards from another, and 5- yards from another: how much did he sell in all?* 11. If a turkey costs I of a dollar, a goose I of a dollar, and 2 chickens - of a dollar: how much will the whole cost? 12. James spends 6~ cents for candy, 12-1 cents for a top, and 51 cents for a slate: how much does he spend in all? 13. A man travelled X2 miles the first hour, 3-, miles the second, and 41 the third: how far did he travel in the three hours? 14. What amount was paid for 4 weeks board, the board for the first week being 51 dollars, for the second 316 dollars, for the third 4~ dollars, and for the fourth 4'5 dollars? 15. A man paid 2 of a dollar for a breakfast, 16 of adollar for dinner, 2 of a dollar for supper, and 3- of a dollar for lodging: what did he pay for the days' entertainment? 16. Jance paid 32 cents for tape. 6- cents for needles, and 4-1 cents for ribbon: how much did she pay in all? 17. If 5 yards of muslin cost 7 of a dollar, 9 pairs of stockirgs 23 dollars, and 2 pairs of boots 91 dol lars, what will be the whole cost o6 INTELLECTUAL ARITHMETIC. [SEC. VY. 18. A laborer sawed wood for 4 hours, and was to have 36 cents a cord. Thle first hour he saw-ed 3. of a cord, the second 2 of a cord, the third } of a cord, and the 4th hour 13 of a cord: how much ought hie to receive? LESSON X II. Subtractiing Fraelions. 1. What is necessary in order that one fraction muay be subltracted friom another? ThPat both fia'ctilns have the same integral and the same fl-a'etiolal unit. i r. Flom A subr]trlca 27. Silnce the integer unit is the cabstract unit one, and ttle fractional unit in both, is -}, the difference is ftond by sutbtracting 2 sevenths from 4 sevenths, which leaves 2 sevenths. 3. What is tile difference between W6h anid t 5 4. Whait is the difference betwseeu -8T'aid a, -' 5. YWhat is the diffietlce betmee 2e, It' 2 1 6. What is the differience between 1I and 1? 7. -What is the diL-aerence between 2-9 aidl - i 8. What is the diffrence betweetl 36q- asd I7 2 19 VNhfat is the diifference betwel en 42 lAldl i 2 10. What is the -litrenee betxxecn 14 7- 1 12 II. WYhat is thc dcifeleioelnlce betxween 2'9 e ild -]21 12. What is the diffeirence ebitwee 1 6 ai_,', I 13. \W hat is the difmere!lce between'3-;a.i 14. \What is the difdierence betxween 6 amlj i? i 2~ d duce both to the frlactiolnl i uiiit one si'wth On-1e half is equatl ti i4 and cnn thud to -2 heces their difirenl elce is equalr 1 to i 6 15. What is the di fi-tence between 14 7!mn.' 13. XWhat is the ditllerence tbetween -3 atl d- 27 17. What is the difi-erenee betweenA, idc 6 -A LES. XIII.1 INTELLECTUAL ARITHMETIC. 97 18. What is the difference between -_6 and a 19. What is the difference between 5 and S- 2 20. W hat is the difference between 11 alid -9 2 21. WThat is the c ifference hetween - T aid -9 2 22, What is the difibrence between I 1 and 4.4 2 13 52 23. Frorm 2- take I1* 24. What is the difference between 5':Ind 2.d 25. What is the diftilenee betwenll il 3' ~ - 26. Whtat is the difelrence between 19 mind x-.- 27. What is the diflln fence between z 2 ancdi 1- 2 28, What is the difference between 17 Mnd 2' 2 QUESTIONS. 1. If you give I of an orange to one boy and 4 to mlother, how much more do you give to one that: to the other? 2. If I have B of a dollar and give ~ dollar for a knife, how much would I have left? 3. WY illiamn had - of a dollar and gave 4 of a dollar to a beggar, hw l muchLi had he left 2 * ANAL-YSIS. First: 2l are equal to 1, equal to; an s1- are equal to equal; hence, the differeuce is.1 It is genetraaly best to subtriaet the inteigral and fi ctionai l uum erI' Sel)'ae'rtel.y' thus, in examiple-25, 3-1, less 2 we may say 3 less 2 equals I; - less - equals 2, equals, hence, the tru differ ence is 1 —. If llhe firactional part of the subtrahend is of greater value ~thlan the fractionail pI>'t of the minuend, take one of the integltl' units of the mlinuend and add it to tho fi actional part, aud thel. subtratet. Thus, in example 24, - is greater thlan -: hence, we tare I unit froln 5 whichlh aded to cake; d tos le akes - and then adding I to the next figure in the subtratl end, wlhich is the same as taking 1 froin the minuendi, we have 3 fn'om 5 leaves 2; therefore, the ditifelence between 5} and 23 is 2-'-. 98 INTELLECTUAL ARITHMETIC. [SEC. VI. 4. B travels 45 of a mile in the same time that C tr~avels -: which travels the farthest and how much? 5. A merchant sells -- of a barrel of sugar from a barrel 7- full: what part was there left? 6. A tailor cut -} of a yard of cloth from a piece containing 1 — yards: how much was there left? 7. John pays 6 of a shilling for a knife, and ~ of a shilling for a top: for which does he pay the most? Ho' miuch? 8. Four pounds of tea cost L5 dollars, and twenty pounds of sugar {: which costs the most How in u ch. 9. A farmer buys a calf, for which he pays X dollars, and a lamb, for which he pays 5 dollars: for which does he pay the most? How Imuch? 10. Janmes' shoes cost 15 dollars, and his vest 7. d.ollars: what is the difference of their cost? 11. A man earned in 4 dclays of a week, 537 dollars, and paid 13 dollars for his board, the three other days: how much should he receive? 12. From a piece of cloth which was 12-~ yards long, 3:- yards is cut: how many yards sre ]eft? 13. If from a box of sugar containing 188 pounds, 6- pounds are taken, how much will be left? 14. A grocer bought 1 6 bushels of beans, and after selling 532 bushels, how many has he left? 15. Jane is 156 years old, and Nancy is 9- years old: how many years is Jane older than Nancy? 16. A draper cuts 56 yards of cloth fiom a piece 2l11 yards long: how much is left? 17. A grocer purchases a box of eggs, for which he paid 37 dollars, and sold them for 5A- dollars: how mnuchi did he make? 18. A grocer bought a pair of chickens for 8 of a dollar, and sold them for ~ dollars: did he make or lose, and how much? LES. XIV.] INTELLECTUAL ARITHMETIC. 99 LESSON XIV. J.Multiplication of Fractions. 1. How many are 3 times 2? Here the fractional unit is o)ne third, and there are two fractional units in the expression, which being taken 3 times, gives six thirds, or 2 for the prolduct. 2. How many are 5 times 6-2-?* 8 times 4? 3. How many are 2 times 12 6 timhes 2 4. How many are 8 times -a 7 titles - 2-A 5. How many are 4 times -? 3 times }' 6. How many are 7 times -42 6 times 2-O? 7. fIow many are 3 times 2- 5 times 3'? 8. How many are 8 times 97? 4 tirnes 6- 2 9. How many are 4 times 6 -? 3 times 9~ 2 10. How many are 10 times 5'.2 9 timnes 64 L 11. How many are 7 times 63 9 times 9 12. How many are 5 times 34. 8 tines 6`' 13. HIow many are 7 times 4.? 6 times 3 -? 14. If the denominator of a fiaetion be divided by a whole number, how will the fraction be affected? It will be increased as many times as there are units in the divisor. (See Anialysis, page 86.) How then may a fraction be multiplied by a \N hole number. Either by multiplyi)ng the numerator, or dividing the denominator by the multiplier. 15. Wh'at is the product of 2 by 2? both ways. 16. What is the product of 3 by 4? both ways. 17. What is the product of a by 7? both ways, 18. What is the product of 8 by 4? both ways. 19. What is the product of f by 9? both ways. SUGGESTION.-Let the fraetional and integral units be multiplied separately: thus, 5 times 3 sixths are 15 sixths, equal to 2 and a half; and 5 times 6 are 30, to which add 2-, giving 322 for the product. 100 INTELLECTUAL AR.ITHMETIC. [SEC. VL 20. W7T-hat is the produact of by 8? both ways. 21. What is the product of 6 by 3? both ways. 22. What is the product of 8- by 8? both b ays. 23. Wthat is the product of { by 4; both ways. 24. What is the product of 4- by 5? both ways. 25. If 3 h-e multiplied by 4, what is the product 2(6. W hat is the cost of -3 of a yard of cloth at dollars a yard? Whhat is 4 of 6? What part is it of 1'?t W27. xht is the cost of ~ boxes of raisius, at ddl.Hais a box? 28.o What par$t of 1 is ~ of2? What is ofI 29. What is the value of -2 of -? W hat par of I? 30. What is the value of 2,} of 1.? 31e. vWhat is the value of. of 3 of ~ q o2. What is the value of o ef 2 o3. How maIny timles 1 is "of 2 34. How many timnes 1 is 2s-; of 1I-? 35. hTi ow many times I is 3~ of 2 36(. IHtw wmnany times 1 is 52 of 6? 37. I-ow many times I is -1 of ~ of - of 38. How many times I is I of 3 of 2 39. h-Tow manly times 1 is - of 2~ of 3? 40. How many times 1 is A of I of 7. 41. It-ow rmany times 1 is 3 of 1 2? * The world or signifies multiplication. 3 ANALYSIS.-If the price per yard, - of a dollar, be multipi.4i by -A, the umnber of yards, the product will be the cost. Now, if 1 yard of cloth cost -. of a dollar, 4- of a yarld lwi cost orne fourlth as mnuch, that is, 5of, a dollar, and 41 will cost $ tiines as Imuch as 4, that is, 24, or. of a dollar. Hence: The product of two fractions is found by multiplyinf ths'v-umeraeors and denominators togethler. LES. XIV.] INTELLECTUAL ARITHMETIC. 101 QUESTIONS. 1. What will 5 yards of cloth cost, at $1-l a yard'! 2. Whart will 4 pounds of tea cost, at 5,4 shilli[igs pounld'! 3. WLhat will 7 dozen of apples cost, at 11 cents a dozen? 4. W\hat will 9 dozen of oranges cost, at 37 Shi inItlg s a dozenu5. What xxill 5 pairs of shoes cost, at 2- a pa 6, Whlat.t will 10 hats cost, at 3 7 dollars a liece. 7. WMhat will 4-. yards of cloth cost., at $7 a yartd? S. Whlat will be the cost of 94 yards, at i a yard? 9. What will 12 pounds of coffee cost, at 11~ ce2nts a pond? - 10. What will 9 sheep cost, at $25 a head' 11. What will 8 calves cost, at $-1 a hea d?i 12. What will 11 quills cost, at 1-i cents apiece? 13. What will 10 yards of carpet cost, at 4 yard 14. What will 9 pairs of gloves cost, at $3- a pair 9 15. WVWhat will 6 pairs of fowls cost, at - of a cldolla pair? 16. What will 9 pairs of boots cost, at 5: dollars a, pair 2 17. What will be the cost of 6 yards of cloth, at'4- dollars a yard? 18. \Whitt is the product of 2- of 2 by 19. Boulght 2 of I- yards of cloth at 1- of -- oI 4 dollars a yard: what did it come to? 20. If raisins are 2- dollars a box, what will be the )st of 16 boxes? 21. W'hat will be the cost of 11 hats, if 1 hat cost 8~ dollars? 22. If I pair of shoes cost $25 dollars, what will be the cost of 7 pairs. 102 INTELLECTUAL ARITHMEIC. [SmC. IV. 23. What will 12 penkuives cost, at $1 apiece? 24. What will 8 lb. of tea cost, at $1,-5 a pound? 25. What will 124 lb. of butter cost, at 12 cents a pound? 26. What will 20 bushels of wheat cost, at $8 a bushel? 27. What will 7 chickens cost, at 8 apiece? 28. What will 9 turkeys cost, at $11 apiece? 29. What will ]2 geese cost, at $4 apiece? 30. If 1 man can earn 1~ dollars -a day, how much call 5 men earn? 31. If 1 yard of cloth cost 23 dollars, how much will 24 yards cost? 32. A grocer bought I -of a box of raisins for which he paid 21 dollars a box: what did they cost him? 33. If i of a sheep is worth 3 dollars, what will be the cost of 2 sheep? 34. What will 4 of a yard of cloth cost, at 7 dollars a yard? 35. If it cost 2I dollars to build 1 rod of wall, how much will it cost to build 31 rods? 36. James gave John 1- apples and had 3 times as many left, how many. had he. at firstS? 37. If a yard of calico cost - of a shilling, what will 2~ yards cost? 38. What will 12 yards of cloth cost, at 24 dollars a yard? 3'. What will be the cost of 6 turkeys at I1- dollars apiece? 40. If 2 yards of cloth cost 8- dollars, what will ] 2 yards cost? 41. James gave 84 d-imes for 10 sheets of drawings..paper: lhow- mu-ch was that a sheet' W42. What will be the cost of a bushel of wheat, if S bushels c,,st 123- dollars LES. XV.] INTELLECTUAL ARITHMETIC. 103 LESSON XV..Dividing Fractions. 1. What is the quotient of 6 divided by 2?* 2. What is the quotient of 2 divided by 2? both ways. 3. What is the quotient of 3 divided by. 3. botb ways. 4. What is the quotient of 8 divided by 3? both ways. 5. What is the quotient of -Oo divided by 5? both way s. 6. What is the quotient of 1-4 divided by 7? both ways. 7. What is the quotient of - divided by 4? both ways. 8. What is the quotient of 18 divided by 9? both ways. 9. What is the quotient of 1 2 divided by 6? both way s. 10. What is the quotient of 8 divided by 5? 11. What is the quotient of A divided by 3? 12. What is the quotient of 2-4 divided by 4? 13. What is the cquotient of; divided by 9? 14. What is the quotient of 6 divided by 29 15. What is the quotient of 1 divided by 8. ANALySIS.-Here the fractional unit is 5, and theire are 6 taken. Now, if we divide the 6 fractional units by 2, the qrotient will be 3; that is, 3. Again, if we multiply the denominator by 2, we shall habw a%, in which the fractional unit is -l~, (which is the half of,) and since the number taken is the same, it follows that % is on half of 6-. Hence: A fraction may be divided by a whole number, either bS dividing the nzumerator or mul.tiplifing the denominator by the 4ivisor. 104 INTELLECTUAL ARITHMETIC. [SEC. VI 16. What is the quotient of - divided by 4? 17. What is the quotient of,-32 divided by 3? 18. WThat is the qulotient of 19 divided by 3? 19. Wfhat is the quotient of 24 divided by 12? 20. What, is the quotient of'3- divided hy 6? 21. What is the quotient o f 5-3 divided by 7? 22. What is the quotielnt of 94 divided by 5? 23. What is the quotient of 124 divided( by 8 24. What is the quotient of 95 div ided by 9? 25. What is the quotient of 5 6 dividled i- 7?2 26. What is the quotient of 8-3 dividedl by 43? 27. What is the quotient of 94 divxied bv 7? 28. What is the quotient of 63 di x idd by 9 2 29. What is the qllotient of 1 ] div-ided by 23 2 30. What is the quotient of 2 - divided cty 8' 31. What is the quotietnt of c- dividcid by 2? 32. W That is the q(litient of 3 divicdd by x3 33. What is the quotient of 251 divided by 1 1? 34. What is the qllotient t of' () 40 divided by 3,2 2 35. What is the quotient of 83 divided by,2 36. What is the quoLtient of" 5 divided by 13 2. 37. What is the qtlotient of 64 divided by 2 7 38. What is the quotient o(f 5-4 divided by 7 39. What is the quotieiit of, divided by 21? 40. WVhat is the quotient of 74 divided by 2-6 2 41. What is the quotient (of 83- dix ided by (i6 I 42. What is the quotient of 5 l divided by 51 43. What is the quotient of 8-~ divided by 21 44. What is the quotieint (f 34 divided by 6 12 2 *SUGGESTION.'-Reduce both to sixths:. is equal to 6 and 2- ae equal to f-. Now, since the firactional unit is the samne in both, the qiuotieu will be found by dividincg the numlerators: Hence, to dlividle nue firaction by another, Reduce them to the saome fractiontal unit, and then divide the awmerator of the dividend by the numerator of the divisor. LES. XV.| INTELLECTUAL ARITHMETIC. l0oI 45. What is the quotient of 6- divided by 3? 46. What is the quotient of 8- divided by 2. 47. WVhat is the quotient of 44- divided by 6 2 48. What is the quotient of 24 divided by 3? 49. What is the quotient of d divided by 2a? 50. What is the quotient of 84 divided by 6 —' 51. What is the quotient of 34 divided by 2T O 52. What is the quotient of 63 divided by 22? 53. What is the quotient of 34 divided by 84' 54. What is the quotient of 94 divided by 34 55. How many times is 2 cOntailled inl 82 56. hIow many times is ~ contained in 2? 4 is contained in 1 how many times? In 2 how many timres? How many times are 9 contained? 57. How many times is 4 contained in 8? 58. How many times is 24 contained in 3? 59. How many times is 4 contained in 5 2 60. H1ow many times is - contained in 2-. 61. How many times is 3- contained in 7? 62. How many times is 24 contained in? 63. How many times is 2- contained in -? QUESTIONS. 1. If 3 yards of cloth cost $10i, how much does it cost a yard? 2. If six pounds of tea cost $48-, what does it cost a pound? 3. If John gives 94 cents for 7 tops, how much do thiley cost him apiece? 4. If' 7 pounds of sugar cost 9 of a dollar, how much is it a pound? * ANALYSIs. —One is contained in -, 4 tiIne. But - is con4 tiamed in 4, 5 times as many times as 1; that is,?5 times. But 2 fifths is contained half as many times as; that is, times: Iencee, to find the quotient of one firaction divided by another, 1ivert the terms of tle divisor and n multiply. 4 106 INTELLECTUAL ARITHMETIC. [SEC. VI. 5. If 4 pounds of coffee cost $17-, how much does it cost a pound? 6. If 7 oranges cost 93 cents, how much do they cost apiece? 7. If three and three fourths yards of cloth cost $114~, how much does it cost a yard' 8. If James can walk 14 miles in 7 of a days how far can he walk ill one day? 9, If John can buy 9 lemons for 10} cents, how much do they cost him apiece? 10. If 9 eggs cost 10; cents, how much do they cost apiece? 11. If 71 bunches of grapes are worth 224 shillings, how much are they a bunch' 12. If 54 bushels of potatoes cost $2,3,, how much do they cost a bushel? 13. If nine baskets of peaches cost $124, how much are they a basket? 14. If 8 lambs cost $124, how much do they cost apiece? 15. If 93 pounds of cheese cost $3, how much does it cost a pound? 16. If 4 barrels of flour cost 24 dollars, what will 1P cost? 17. James has 3 oranges and 5 playmates: he wishes to give 3 of an orange to each, how must he divide the oranges, and how many oranges will he have left? 18. If one man consumes 11 pounds of meat in a day, how many men would 8a pounds supply? 19. A man distributed 173 pounds of bread among 8 persons, how much does he give to each? 20. If 12 horses consume 284 tons of hay in a winter, how much is consumed by each horse? 21. If 3 of a barrel of flour will last a family 30 days, how long will 2 barrels last themn LES. XV.] INTELLECTUAL ARITHMETIC. 10O 22. A farmer has a field containing 8a acres; if a man can mow 1I acres in a day, how many men will it take to mow the field in 1 day? 23. If 9 bushels of apples cost 41 dollars, how much will 17 bushels cost? 24. If 7 pounds of butter cost 10 shillings, how much is that a pound? 25. If 3 pounds of butter cost 71 shillings, how much will 12 pounds cost? 26. If 5 yards of cloth cost 11i dollars, what will 8 yards cost? 27. If 8 yards of cloth cost 42} dollars, how much will 4 yards cost? 28. If 1- dollars will buy 2 yards of cloth, how many yards will 6 dollars buy? 29. How many times is 22 contained in 1.2 - 30. How many pounds of tea can be purchased for 63 dollars, if it cost A dollars a pound? 31. If a turkey costs 15 dollars, how many can be bought for 124 dollars? 32. If calico is worth J of a dollar a yard, and 71uslin 7 of a dollar a yard, how much calico must be given for 9 yards of muslin? What is the cost of 9 yards of muslin? 33. A tailor bought a piece of cloth containing 122 yards, for which he paid 38 dollars: what did it cost him a yard? 34. If you give 152 dollars for a cow, and sell her for 2 more than she cost, how much more do you receive for her than you gave? 35. If 6 men can do a piece of work in 153 days, Lh:w long will it take 1 man to do it? 2 men? 3 men? 36. A man divided 25 ofa dollar among his children, giving _o of a dollar to each: how many children were there. 108 INTELLECTUAL ARITHMETIC1 [SEC. VIL SECTION SEVENTH. LESSON I. Comparison of Numbers. 1. What part of 2 is I? How many times does 1 ontain 2?* 2. What part of 3 is 2? What is the ratio of to 2 3. What part of 6 is 3? How many times does 8 contain 6? 4. What part of 12 is 4? What is the ratio of 12 to 4? 5. What part of 4 is 1? What is the ratio? 6. What part of 3 is 2? What is the ratio 2 7. What part of 8 is 5? What is the ratio? S. What part of 20 is 10? What is the ratio? 9. What part of 30 is 20? What is to ratio? 10. OWhat part of 50 is 10 2 What is the ratio? 11. What part of i is 2? What is the ratio? 12. What part of 4 is 2? What is the ratio? 13. What part of A is 3. What is the ratio'? S SuGGEESTloNs.-To find what part any number is of a nulri ber less than itself, we divide the less by the greater, and the quotient shows the part. This quotient is called the ratio. Thus, what part of 2 is 1; it is 1 divided by 2, which is i. If we wish to find how many times one number is greater than another, we divide the greater by the less, and the quotient shows the number of times. This quotient is also called the ratio of the two numbers. Thus, how many times is 12 greater than 2: it is 12 divided by 2 times greater, which is 6 times. But the general question, " How many times," has no ref'eratnce to the relative value of the numbers. Thus: what part of 2 is 1 Or, how many times is 2 contained in 1 i One half or one half times? IHow many times is 2 contained in 41'avise, or 2 times. MtL. ] INTS LECTUL ARITRHmTETO. I 14. What part of 3 is. \VWhat i the ratio t 15o What part of s- is -? {- What is the ratio 16. What part of is What is the ratio 17. What part of 31 is 39. What is the ratio 18. What part of 4- is 31- W W Ehat is the ratio 19. What part of 3~ is 14. t What is the radtio? 20o. What pat of 6{- is 5f. What is the ratio 1. What part of 8 is of 2,. What is ti. ratio? 22. Whet part of0 is a of4- Wht is the 23. What part of 15 is 4'4. What part of 45 is 12s 25, Whal part of o is I 204. What is the ratio of t2 t 7T 7. What is the ratio or.t _ to t 28 What is thei ratio of 3 to Itl. $ 29. What is the ratio of 9 to - of,T 10. What part of 7i- is li of 3j- I 1. Vs st parsto 30 dollars is 5 dar 1. 6 dootllarst I0 dollar- s 2 dollars. 2. \ T'rt part of 56 bu sels is $ bushel-s! T busihels.'4 bushels?.28 bushels.T o. Wha, pasrt 25 yards is a of It! yards? 4. A taleor has L2 yards of cloth; -e cputs offt 3 yards for a coat, 1{ yards for fo, pair of pantalo uso I.- of at yard for a vest: - what part of the ctoth did. he use 2 iW/hat part was leAft?' 5. A itarmer lhas 6 bushels of whea-t ii his barn;'e sells 20 busheis to oue inat, 15 to n.o ther, aEnd 1.0 to anmther what part of the wi etat has e I let f 6. Mr. Dix buys. orf ~ of M of a M Jones' filPr hic. cat iNs 100 acres iwhat part of the f'm, doe, buy? s IW many r mrt4 s ari e th ref IC te 10 110 INTELLECTUAL ARITHMETIC. [SEC. VIL. 7. James gave 5 dollars for his hat, and his father gave 25 dollars for a coat: what part of the cost of the coat was the cost of the hat? 8. What part of a barrel of flour is I of ] of it I What part of the whole barrel is the remainder? What will each part cost if flour is 8 dollars a barrel? 9. William having a pine-apple said he would give 2 of - of it to the one that could tell how much that would be: how much did the individual receive? What part of the pine-apple was left? 10. A merchant had 44 barrels of flour and sold. of it: how much had he left 2 11. A lady paid A of 53- dollars for a pencil and 15 dollars for a ring: how many times as much did she pay for the pencil as for the ring? 12. One man can build 5-5 rods of wall in one day, and another can build 31 rods: what part is the second of the first 2 13. Albert is 93 years of age, John is 5 of Albert's age: how old is John 2 14. James bought a kite for 161 cents, which was 21 times as much as he paid for his top: how much did he pay for his top 2 15. A farmer bought a calf for 4z dollars, and a pig for 14 of that sum: what did he pay for the pig? 16. A box of raisins cost 3} dollars, at tne rate of i of a dollar a pound: how many pounds were there in the box? 17. If 14 oranges are worth as much as 34 pine. apples, how many oranges is one pine-apple worth 2 18. Mary gave j of 3- dollars for a bonnet, wvhich was I of what Jane paid for hers: how much did Jane pay 2 19. If a father can do 1* times more work in a day than his son, how many days' work of the father will be equal to 18 of the son's? LES. II.] INTELLECTUAL ARITIIMETIC. 1 l 20. A gentleman gave 200 dollars to his three sons; to the first he gave - of it, to the second < of 2 times as much, and the rest to the, third: how much had each? 21. If a boy can run 80 rods in eight minutes, what part of a mile can he run in 16 minutes? 22. James bought -1 pound of candy, and gave 4 of it to Mary: what pa,t of a pound did he give her X 23. A vessel sails 150 miles one day, and is retarded - of 4 as much on the second: what part of 150 miles did she sail the second day? LESSON II. Comparison of Numbers. 1. Two is one half of what number?' 2. Four is one third of what number? 3, Five is one fifth of what number? 4. Two and one half is one-fourth of what number? 5. One and one half is one sixth of what number? 6. One and 2 thirds is one-third of what number?t 7. Three and one third is one third of what number 2 8. Ten and five sixth is one sixth of what numher? 9. Twelve is-one fifth of what number a 10. Eight and three fourths is one fourth of what rtimber? 11. Nine and one sixth is one sixth of what num. ber? 12. Five and one tenth is one tenth of what numiher? * ANALYSIS.-Two is one half of 2 times 2, which are 4 therefore, 2 is one half of 4. f ANrALSIs.-One and 2 thirds is one third of 3 times one and 2 thirds, which are 5: therefore, one and two thirds is one third o f. 112 ITTLLECOTUAL ARITIMET1C. [Se-,. yvu. 13, Five and is one fifth of what number? 14. ASix and three fourths is one eighth of what number? 15. Twelve and one sixth is one sixth of hat numoer? 16. Two and three fourths is one twelfth of what number? 17. Three and six ninths is one ninth of what number? 18. Six and Z is one twelfth of what number? 19. Eight is 2 thirds of what number?2 20. 12 is 3 fourths of what number? 21. 16 is 4 fifts1 of what number? 22. 30 is 6 sevenths of what number? 23. 32 is 8 ninths of what number? 24. 15 is 5 ninths of what number? 25. 12 is 4 sevenths of what number? 26. 22 is 11 twelfths of what number 27. 27 is 9 tenths of what number? 28. 21 is 7 ninths of what number? 29. 15 is 3 sevenths of what number? 30. 24 is 8 elevenths of what number? 31. 16 is 5 times what number?t 82. 48 is 6 times what number? 33. 39 is 3 times what number? 34. 56 is 9 times what number I 35. 21 is 5 times what number? 36. 75 is 8 times what number? 37. 54 is 8 times what number? 38. 95 is 9 times what number? A'NALYsIS. —Sinee 8 is 2 thirds of some number, one half o! 8, which is 4, is one third of the same number; but 4 is one third of 3 times 4, which are 12; therefore, 8 is 2 thirds of 12., t AxALvsis.-16 is 5 tines ~- of 16, w'eh is 83 therefore,'.I is 5 times Si. LES. II.] INTELLECTUAL ARITHMETIC. 113 QUESTIONS. 1. Two thirds of nine is one half of what number? 2. Two sevenths of fourteen is one third of what number? 3. Three twelfths of thirty-six is one eighth of what number? 4. James gave nine cents for a slate, which was three fourths of his money: how much had he 5. A man bought a cow, for which he paid $30, which was three fifths of his money: how much had he 8. 6. A lady was married at twenty years of age, which was the half of eight fifths of the age of her husband: how old was the husband? 7. In a pasture are 45 sheep, which is three fourth the numbelr of cows. in the same pasture: how many cows are there? 8. John gave 36 cents for a knife, which was six sevenths of what he gave for a sled: how much did he give for his sled? 9. If a manl can make six and two ninths rods of fence in one day, how much can he make in 12 days? 10. Two men agreed to build a wall; one man built 16 rods, which was four fifths of what the other built: how much did the second build, and what was the whole length of the wall? 11. A man willed one half of his property to his wife, one third of the remainder to his daughter Mary, and one sixth to his son'James: how much of it was left for his only remaining son John? 12. John gave one and 3 fourth cents for a Ieach which was one third of what he gave for an orange what did the orange cost him? 13. Charles gave ten and five sixths cents for Ilis kite, which was five times what he paid for his top: how much did he pay for his top? How much fir both? 10* 114 INTELLECTUAL ARITHMETIC. LSEC. VII 14. WVilliam gave eight and five ninths cents for a pencil, which was one fourth the cost of his penknife: what did his penknife cost him. 15. A farmer paid four and three fourths dollars for a calf, and one fifth as much for a pig: what did the pig cost him? 16. A farmer bought a calf for three and one sev. enth dollars, which was one seventh of what he paid for a cow: what did the cow cost him? 17. A man bought a pair of boots for six and I dollars, and a coat which cost him three and a hal times as much: what did the coat cost him? 18. If 3 sevenths of a barrel of flour cost 6 dollars, what will 5 sevenths cost? What will the whole barrel cost? 19. A grocer bought 5 boxes of raisins for two and three fifths dollars a box, and a barrel of sugar, which cost one half of two thirds as much as the raisins: how much more did the sugar cost him than the raisins? LESSON III. Comparison of VNumbers. 1. 25 is 5 eighths of how many times 7'1* 2. 63 is 7 ninths of how many times 8? 3. 36 is 4 sevenths of how many times 6? 4. 45 is 5 sixths of how many times 5? 5. 84 is 7 eighths of howl many times 9? 6. 29 is 3 ninths of how many times 10. 7. 42 is 7 thirds of how many times 5? 8. 64 is 8 fifths of how many times 3? 9. 32 is 3 eighths of how many times 11 * ANALYSIS. — If 25 is 5 eighths of some number, 1 eighth is 1 fifth of 25, which is 5. If 6 is 1. eighth, 8 eighths are 8 times 5, which are 40. As many as 7 is contained times in 40:' is eontained in 40, 5 and 5 sevenths times; therefore, 26 is 6 eighths of 5 times 7 and 5 sevenths of 7 LES. III.] INTELLECTUAL ARITHMETIC. 115 10. 75 is 5 eighths of how many tilmes 6 11. 21 is 7 fifths of how many times 4? 12. 40 is 4 sevenths of how many times 3 13. 6 sevenths of 21 is 3 fifths of what number?* 14. 8 ninths of 27 is 4 sevenths of what number? 15. 3 fourths of 40 is 6 tenths of what number? 16. 5 eighths of 64 is 8 ninths of what number? 17. 2 thirds of 42is5twelfthsof what number? 18. 7 tenths of 80 is 8 ninths of what number 2 19. 4 fifths of 60 is 6 sevenths of what number 2 20. 5 sixths of 48 is 7 eighths of what number?2 21. 3 sevenths of 21 is 2 thirds of what number? 22. 4 ninths of 81 is 5 eighths of what number 2'23. 7 fifths of 45 is 9 sevenths of what number? 24. 5 ninths of 36 is 4 fifths of how many times 5 2 25. 4 sevenths of 56 is 8 ninths of how many times 7 2 26. 3 fourths of 24 is 6 sevenths of how many times 4 2 27. 5 eighths of 48 is 5 ninths of how many times 9? 28. 4 thirds of 30 is 8 elevenths of how many times 8? 29. 5 tenths of 72 is 4 ninths of how many times 6? 30. 11 twelfths of 84 is 7 ninths of how many times 10? 31. 6 fifths of 50 is 4 thirds of how many times 12 l *ANALYSIS.-6 sevenths of 21 is 6 times 1 seventh of 21: 1 seventh of 21 is 3, and 6 sevenths is 6 times 3, which are 18I If 18 is 3 fifths of some number, 1 third of 18 which is 6 is I fifth of the same number; 6 is 1 fifth of 5 times 6 which is 30: therefore, 6 sevenths of 21 is 3 fifths of 30. VERIFICATION.-One fifth of 30 is 6, and 3 fifths of 30 are 18 out 6 sevenths of 21 are 18: therefore, 30 is the number soughL NOTRE.-With the correct answer all the questions may be reversed and similarly analyzed: thus, 30 is 6 thirds of 6 sevenths of what number? Again, 30 is 5 thirds of how many sevenths of 21 110 INTELLECTUAL ARITHMETIC. [SEC. VIL 32. 9 elevenths of 88 is 8 fifths of how many ti'mes 6? 33. 2 thirds of 75 is 5 sixths of how many times 91 34. 5 halves of 24 is 3 fourths of how many times 5 35. 5 ninths of 72 is 4 thirds of what number? 36. 6 sevenths of 56 is 5 ninths of what number! 37. 28 is I fifth of how many times 9? 38. 42 is 7 twelfths of' how many times 12? 39. 5 ninths of 108 is 10 elevenths of how many times 10? 40. 6 fifths of 35 is 7 twelfths of how many times 5? 41. 81 is 9 fourths of how many times 8 2 42. 7 thirds of 36 is 9 tenths of what number? 43. 7 twelfths of 96 is 8 ninths of how many times 5? QUESTIONS. 1. A boy gave 5 apples to one of his companions, which was one third of all he had: how many had he 2:2. A man bought a watch for 24 dollars, and sold it for four fifths of what it cost him: what did he receive for it, and how much did he lose by the bargain? 3. Charles gave away 12 apples to 3 of his cornplanions, which was 3 of ~- of all he had: how many hald ne, and how many did he give eachl 4. If 3 fourths of a hundred weight of sugar cost 12 dollars, what will a hundred weight cost? How nany barrels of cider at 2 dollars a barrel will pay fur it? 5. A man sold a horse for 60 dollars, which was 5 sevenths of what he cost-him: how much did he cost him, and how much did he lose? When he bought him he paid in cloth at 6 dollars a yard: how many yards of cloth did he give? 6. A pole is - in the water, 1 in the mud, and 14 feet out of the water: how long is the pole 2 LES. III.] INTELLECTUAL ARITHMETICG 117 7. John s age is 23 William's, and the sum of their ages diminished by 5 is equal to 70: what is the age of each? 8. Mr. Wilson gave 200 dollars to his neice, which was I of;- of his property, and the remainder equally tc his 4 sons: how much did each receive? 9. How many yards of cloth, at 4 dollars a yard, must be given for a hogshead of sugar, if four sev. enths of it cost 48 dollars 2 10. There is a pole standing - in the water, i of the remainder in the mud, and 6 feet above the water: how long is the pole? 11. A staff 5 feet long casts a shadow of 3 feet: what is the length of a pole that casts a shadow of 24 feet the same time of day'?* 12. A man can build 56 rods-of wall in a certain time, another man can build 8 rods while the first builds 7: how much would the second build in the same time? 13. Two boys counting their marbles, one said he had sixteen. The other said, 3 eighths of yours is exactly 2 sevenths of mine: now if you will tell me how many I have, I will give you the difference between yours and mine: how many had he? 14. A man being asked how many sheep he had, said he had them in three pastures: iii the first he sad 42, which was 7 eighths of what he had in the second; and that 5 thirds of what he had in the second was just 4 times what he had in the third: how many had he in each field 2 15. A gambler lost 3 fourths of his money in play; he then won 30 dollars, which was 5 sixths of what he lost: how much money had he when he began to play? * ANALYsIs.-Since the shadow of the staff is 3 fifths the length of the staff, the shadow of the pole must be 3 fifths the length of the pole; then if 24 is 3 fifths of some number &c. 118 INTELLECTUAL ARITHMETIC. [Se. Vn. 16. A tailor cut off 3 fifths of a piece of cloth, he then cut off 4 yards, which was one third of the remainder: how many yards were there in the piece? 17. A gentleman left to his eldest son 300 dollars, which was 3 fourths of what the second son had, and twice the second son's share was just four times what thle third son received: how much was the fathe worth? 18. James being asked how many credit marks he had, said: if 1 third of the number be taken firoml 1 half of the number, the remainder would be 2- times 4: how many credits had he? 19. Three fourths of 40 is 5 sevenths as many dollars as Mr. C. paid for his horse: what was the cost of the horse? 20. A person being asked his age said, that 3 fourths of 80 was 6 sevenths of 5 times his age: what was his age? 21. Bought 45 yards of cloth and sold 4 ninths of it for 20 dollars, which was 5 sixths of what the whole cost: what would be the gain on the whole, at the same rate? 22. A merchant bought 6 barrels of flour at 54 dollars,, which was 9 eighths of what it cost him: what did it cost him a barrel? LESSON IV. 1. 3 sevenths of 56 is 8 ninths of 3 times what number.* * ANA.LYSIS.-Three sevenths of 56 is 3 times 1 seventh of 56. One seventh of 56 is 8, and 3 sevenths of 56 is 3 times 8 which are 24. Since 24 is 8 ninths of some number, 1 eighth of 24 which is 3 is 1 ninth of the same number: 3 is 1 ninth of 9 times 3 which is 2'. Now, 27 is 3 times 1 third of 27 which is 9; therefore, 3 sevenths of 56 is 8 ninths of 3 times 9, or 27. LES. IV.] INTELLECTUAL ARITHMETIC. 119 2. 5 sixths of 54 is 5 eighths of nine times what numberl 3. 4 fifths of 30 is 3 fourths of 8 times what number? 4. 2 ninths of 81 is 3 elevenths of 4 times what number? 5. 5 eighths of 56 is 7 ninths of 6 times what number? 6. 4 thirds of 36 is 6 fifths of 10 times what number? 7. 9 tenths of 90 is 6 fourths of 8 times what number? 8. 6 halves of 30 is 9 tenths of 20 times what number? 9. 7 ninths of 108 is 7 twelfths of 8 times what number? 10. 5 eighths of 32 is 4 fifths of how many sixths of 18*P 11. 6 sevenths of 56 is 8 ninths of how many fourths of 24? 12. 8 fifths of 40 is 4 thirds of how many sixths of 36? 13. 4 thirds of 36 is 8 twelfths of how many fifths of 45? 14. 2 fifths of 75 is 5 sevenths of how many ninths of 54? 15. 3 halves of 40 is 6 twelfths of how many tenths of 80? 16. 7 ninths of 72 is 8 fifths of how many thirds of 24? 17. 6 elevenths of 44 is 3 tenths of how mary fourths of 32? * ANALYsIs. —The same as in the preceding examples until you obtain the second number, which in this example is 25.'1en, 25 is how many sixths of 18? 1 sixth of 18 is 3, and 3 is contained in 25 8 and 1 third times; therefore, 5 eighths of 82 is 4fifths of 8 and 1 third times 1 sixth of 18. 120 INTELLECTUAL ARITHMETIC. [SEC. VII 18. 5 sevenths of 77 is 11 twelfths of how many eighths of 56? 19. 10 thirds of 24 is 8 halves of how many twelfths of 36? 20. 12 fifths of 45 is 9 tenths of how many eighths of 64? 21. 5 sixths of 54 is 3 fourths of how many sevenths of 42? 22. 4 fifths of 30 is 4 sevenths of how many times 2 thirds of 21?* 23. 6 thirds of 27 is 6 ninths of how many times 1 tenth of 90' 24. 8 ninths of 63 is 7 twelfths of how many times 4 eighths of 24? 25. 7 sixths of 54 is 9 tenths of how many times 2 ninths of 45? 26. 5 halves of 24 is 5 eighths of how many times 4 sevenths of 28? 27. 9 tenths of 70 is 7 sixths of how many times 3 ninths of 27? 28. 4 thirds of 36 is 4 ninths of how many times 2 sevenths of 42? 29. 5 ninths of 72 is 4 fifths of how many times 5 twelfths of 60? 30. 7 eighths of 64 is 8 sevenths of how many tines 3 eighths of 32? 31. 10 fourths of 36 is 9 thirds of how many sevenths of 63? * NOTE-In connection with the answer, a reversed stateilient of the examples in this lesson may be made, giving two other propositions, to be solved by a similar analysis. Thus, in Example 22, 3 being the answer we have, First. 8 times 2 thirds of 21 is 7 fourths of 4 fifths of what number I which will give 80.,SecoTnd. 3 times 2 thirds of 21 is 7 fourths of how many fifths of 80? which will give 4 fifths of 30. This will give the pupil the benefit of three examples in one LES. III.] INTELLECTUAL ARITHMETIC. 121 32 6 sevenths of 77 is 11 twelfths of how many fifths of 50? 33. 8 ninths of 81 is 6 fifths of 4 times what number? 34. 9 tenths of 100 is 5 halves of 8 times what inumber? 35. 3 sevenths of 84 is 4 ninths of how many times 4 ninths of 45. QUESTIONS. 1. Two boys comparing their ages, one said he was fifteen years old; the other said, 4 fifths of your age is just 3 halves of my age: what was his age? 2. A farmer had a certain number of sheep which he put in two fields; in one field he had 28, and 6 sevenths of them was 4 ninths of 2 times what he had in the second: how many were there in the second field? 3. A man pays 300 dollars a year for benevolent objects: - of this sum is equal to - of 2 times the amount of his personal expenses: what are his personal expenses? 4. A farmer sold a number of cows and had 12 left, which was. of the number sold; if the number sold be divided by 3- of 91, the quotient will be 5- the nuns ber of dollars he received per head: how much did he receive apiece for his cows? 5. The insurance on a house is 600 dollars, and 4 of that is * of 4 times the value of the furniture: what is the furniture worth? 6. A man bought a horse for 100 dollars, A of what the horse cost was 3 of what he paid for a carriage how much did the carriage cost him? 7. 4 of A's age is A of B's, and 3 tinmes B's is k (of C's: how old are A and E,if C is 24 years old? 8 Fort Plain is 56 miics fi'om Aibany, 5 of this distance is 4 times * the distance from Albanv to Rochester: what is the distance to Rochester'l 11 1 S2 INTELLECTUAL ARITHMETIC. [SEC. VII. 9. The contents of a certain store cost 1,000 dollars, and -2 the entire cost is 4 of 3 times what the cloths cost: what was the cost of the cloths? 10. James has a certain number of marbles; John has 6 as many less 3, and William has 4 as mnally as ollhn less 7; William has 5 marbles: how many have ohn and James? 11. A house is worth 600 dollars, and z- of its value is I of 2~ times the value of the farm on which it stands: what is the value of the farm? 12. Buffalo is 325 miles from Albany, and - of this distance is 3a times 2 the distance from Rochester to Buffalo: what is the distance? 13. A boy being asked his age said, that 9 years was 3 years more than J- times 47 of his age: what was his age? 14. A man had 5- of his money stolen from him; the thief was caught, but not until he had spent I of it, the remainder, ($50), was given back: how much nmoney had he at first? 15. A and B engaged in play with equal sums of money, B gained 40 dollars, which was 2 of 3 times what he commenced with: how much had each when they began to play? How much had A left? 15. A farmer sold a horse for 96 dollars, which was A times 5 what he paid for him: how much did he pay for him 2 LESSON V. Comparison of the units of Denominate Numnbers. 1. Four mills are what part of a cent? 2. Five cents are what part of a dollar? 3. Three dimes are what part of a dollar? 4. Thirty-six cents are what part of a dollar? LlS. V.] INTELLECTUAL ARITHMETIC. 123 5. Three dollars is what part of an eagle? 6. How many cents in I of a dollar? 7. HIow many dimes inll of a dollar? 8. Itow many mills in -30 of a dollarS 9. How many dollars in -4 of an eagle! 10. What part of a pound is 1 shilling? 11. What part of a pound is 6 pence? 12. What part of a shilling is 5 pence? 13. What part of a shilling is 3 farthings? 14. What is the value, in pence, of - of a shilling? Of8 of a shilling? Of i of a pound? 15. What is the value, in shillings and pence, of % of of a pound? Of of ofof a pound? 16. Seven pence is what part of a pound? 17. Eleven pence is what part of a shilling? Of fi 18. What'part of a pound is 6 shillings? 7 shil rings? 12 shillings? 13 shillings? 14 shillings? 15 shillings? 19. What part of a pound is 3s. 8d.? 4.. 6d.' 2s. 7d.? 5s. 9d.? s. Sd.? 4s. 9d,? 20. What part of a shilling is 8-d.? 63d.? 9~d.? 21. What part of a pound is 20z. 4oz.? 6oz.. 9oz.? 120z.? 22. What part of a quarter is 81b.? What part of a quarter is 121b. 141b. 1 81b.. 20lb.? 23. What part of' czt. is 3 quarters? 2 quarters I quarter? 24. What part of 1cwt. is 5l6b.? 271b.? 951b.? 751b.? 681b.? 25. What part of a ton is Scwt.? 12cot.? 14cwt.? 16cwt.? 19cwt.? 26. What part of a ton is 651b.? What part of a ton is 951b.? 35016b.? 10001b.? 27. What part of a foot is 5 inches? 7 inches? 9 inches? 4 inches? 10 inches? 11 inches? 28. W'hat part of a yard is 1 foot? What part is I foot of 2 yards? What part of 5 yards is 1 foot? 124 INTELLECTUAL ARITHMETIC. [SEC. VIL. 29. What part of a yard is 1 foot? What part of a yard is 2 feet 2 30. What part of a furlong is 1 rod? 3 rods. 5 rods? 8 rods? 10 rods? 12 rods? 15 rods? 31. What part of a mile is 7 furlongs? 5 fur,'ongs? 4 furlongs? 3 furlongs? 2 furlongs? 6 furlongs? 32. What part of a mile is 20 rods? 30 rods? 40 rods? 50 rods? 160 rods? 33. What part of a square foot is 12 square inches? 48 square inches? 100 square inches? 34. What part of' a square yard is 2 square feet? 7 square feet? 8 square feet? 5 square feet? 35. What part of a square yard is 3 square feet? 6 square feet? 1 square foot? 4 square feet? 36. What part of a rood is 4 square rods? 8 square rods? 9 square rods? 7 square rods?2 37. What part of an acre is 4 square rods.? 10 square rods? 40 square rods? 100 square rods? What part of an acre is 3 roods? 38. What part of a quart is 1 pint? 2 pints is what part of 8 quarts? 3 pints is what part of 5 quarts? 39. In wine measure, what part of a quart is I pint'? What part of a quart is 2 gills? 3 gills? 5 gills? 40. What part of a gallon is 3 quarts? 3 pints? 3 gills. 2 quarts? 5 pints? 5 gills? 41. What part of a hogshead is I gallon'? 2 galIons.? 8 gallons? 9 gallons? 42. One pint in dry measure is what part of a quart? What part of a peck? What part of a bushel? 43. Three pecks is what part of a bushel? What part of a chaldron? 44. Five minutes is what part of an hour I LES. V.] INTELLECTUAL ARITHMETIC. 125 8 seconds is what part of a minute? What part of an hour 2 45. Three hours is what part of a day? What part of a week'? 46. Four days is what part of a week? What part of a month? What part of a year? 47. Five years is what part of a century Whal part of a century is 30 years? 40 years? 6 years? 48. What part of 3 days is 5 hours? 6 hours.' 7 hours?. 9 hours 10 hours? 49. What part of 7 months is 9 weeks? What part is 8 weeks. 6 weeks? 50. Four minutes is what part of a day? Of an hour? Of a week? Of a month? QUESTIONS. 1. What will 3V yards of cloth cost at 4 dollars a yard? 2. If 4 bushels of wheat be divided equally among 5 men, how much will each receive? 3. What will 5- bushels of wheat cost at 21 dimes a peck? 4. What will 2t yards of cloth cost at 21 dimes a nail? 5. At 2- dimes a yard, what will be the cost of 13- yards of muslin? 6. What will 122 barrels of wine cost at 7 dollars a barrel? 7. A piece of cloth containing 16- yards is equally divided between 3 persons: how much has each ol l 8. If the twelve months were of equal length, how many days would each contain? 9. If a man travel 21 miles in - of an hour, how far will he travel in 5 hours 10. If wheat is one dollar a bushel, how much will I quart cost? 11* 126 INTELLECTUAL ARITHMETIC. [SEC. VII. 11. If cloth is 8 dollars a yard, what will 1 nail cost? 12. If wine is 4 dollars a gallon, what will 3 pints cost? 1 quart? 2 gills? 13. What will 9 yards of cloth cost at three cents a nail? At 5 cents? At 7 cents? 14. If 1 quarter of a yard of cloth costs 3- dollars, what xvill 4 yards cost? 15. If a man spends - of a dollar in 3- of a day, how much will he spend in 2 weeks? 16. If -3 of a hogshead of wine cost 54 dollars, what does the wine cost a gallon? 17. If a man earns 13 dollars a week, how much &oes he earn in each of the 6 working clays? 18. If a man earns 13 dollars a day, how much does he earn in a month of 26 working days? 19. If hay is 15 dollars a ton, how much is that per Icwt.? For 1 quarter? 20. If 10 pounds of hay cost 51- mills, how much will a ton cost? LESSON VI. Per Cent and Per Centage. 1. What is 1 per cent of 1 dollar? What is 2 per cent of 1 dollar? 3 per cent? 4 per cent? 5 per cent?* 2. What is 4 per cent of 50? 3. What is 6 per cent of 200 dollars? 4. What is 4 per cent of 150 dollars? Of 200 dollars? * SUGGESTION.-Per cent means by the hundred. Thus, 1 par eent, 2 per cent, 3 per cent, 4 per cent, &c., of any number or LEtS. vi.] INTELLECTUAL ARITHMETIC. 127 5. What is 9 per cent of 300 dollars? Of 400 dollars? 6. What part of 1 is 3 per cent of 30? Of 4 per cent of 25? 7. What is 6 per cent of 60 dollars? Of 50 dollars? 8. What is 8 per cent of 70 dollars? Of 90 dollars? 9. What is 4 per cent of 250 dollars? Of 45 dollars? 10. What is 9 per cent of 40? Of 50 Of 60 11. What is 3a per cent of 100 dollars? Of 40 dollars? 12. What is 21 per cent of 200 dollars? Of 60 dollars? 13. What is 4 per cent of 600 dollars? Of 30 dollars? 14. What is 3 per cent of 25 dollars Of 36 dollars? 15. A person has 250 dollars, and takes out 2- per cent: how much will he have left? 16. What is 8 per cent of 15 dollars? Of 25 Of 30? Of 45? thing, means that the number or thing is divided into 100 equal parts, and that 1, 2, 3, 4, &c., of these parts are taken. The number of parts taken, determines the rate per cent. Thus, the rates above, are 1, 2, 3, 4, &c., per cent. The part of the number taken, is called the per centage. Thus, when the thing is 1 dollar, and the rate 1 per cent, 1 cent is the per centage; if the rate is 2 per cent, the per centage is 2 cents, &c. Observe, If any number be divided by 100, the quotient uill be 1 per owt of that number. Hence, to find the per centage of any number, for any rate per cent..Multiply the given number by the rate per cent and cut oq two figures fromt the right hand of the product, which is equivalezid to dividing by 100, 128 INTELLECTUAL ARITHMETIC. LSEC. VM 17. Whut is 10 per cent of 20 dollars2? Of 25? Of 35? Of 40? 13. What is the per centage of 75 dollars, at the rate of 6 per cent? 19. What is the per centage of 80 dollars, at the rate of 5 per cent? 20. What is the 8 per cent of a piece of cloth, measuring 50 yards 2 21. What is 3- per cent of a piece of muslin, measuring 75 yards? 22. What is 20 per centage of a box of shoes, coon taining 250 pairs? 23. What part of a number is 5 per cent of that number'124. Forty per cent is what part of any number? 25. What is 25 per cent of 60 2 Of 50.? Of 40? * ANALYSIS-Five per cent of any number, is 5 hundredths of that number: hence 5 per cent of a number being 51 of that number - - 2 of that number. The following table shown tile per centage in terms of the number: 5 per cent equals - = -7I of the number 6 per cent equals 6= of the number 10 per cent equals -Ao - ~-, of the number: 12- per cent equals 25 - - of the number. 15 per cent equals 95r = 3 of the number. 20 per cent equals 20 _= of the number. 28 per cent equals 2-o = - of the number. 830 per cent equals 1O ~ -- - of the number. 337 per cent equals 100 = - of the number. 40 per cent equals f1s-~ = 5 of the number. Y-v -- of the number. 40 per cent equals 4O = ~of the number'75 per cent equals 15 _ = 1of the number. 100 per cent equals I o 1 the number. LES. VI.] INTELLECTUAL ARITHMETIC. 129 26. Whatis 10 per cent of 60 Of 40? Of 15? 27. What is 5 per cent of 40 Of' 80? Of 100? 28. What is 15 per cent of 40 dollars? Of 80 dollars 2 29. What per cent of any number is ~ of it? what per cent is I of it?* 30. Five is what per cent of 20?t 31. Six is what per cent of 18? Of 24? Of 30, 32. Ten is what per cent of 50? Of 302 Of 40? Of 60? 33. Three is what per cent of 12? Of 15? Of 24 X Of 36 2 34. Forty is what per cent of 80? Of 20? Of 10? 35. Fifty is what per cent of 200? Of 60? Oi 100? 36. Seven is what per cent of 49? Of 21? Of 56 37. Eight is what per cent of 562? Of 64? Of 84?' * ANALYSIS.-Since 1 of a number equals 2o, it follows that - of a number is equal to 20 per cent; and we may find the 20 per cent by adding two O's to 1, and then dividing by 5. Hence, we see, that having written the per centage in the forln of a fraction, if we add two cyphers to the numerator and then divide, the quotient will express the rate per cent. Therefore, the rate per cent, when the per centage is l, is 100 divided by 6, which gives 16-i per cent. f ANALYSIs.-Five is what part of 20? (see lesson VI, page 89.) 5 is of 2 of 20; of 20; but ~ is 25 per cent: hence, 5 is 25 per cent of 20. Hence, to find the rate per cent when the per centage and number are known: Divide the per centage by the number. 130 INTELLECTUAL ARITHMETICO. SEC. V#* QUESTIONS. 1. A grocer purchased a bag of coffee at ten cents a pound: at what price must he sell it a pound, in order to make 10 per cent? What must he sell it Fir, to make 25 per cent? 50 per cent? 2. If a piece of broadcloth, containing 30 yaids; cost 5 dollars a yard, what must it be sold for to gain 20 per cent? What will be the profit? 3. A grocer bought 10 barrels of flour, at 8 dollars a barrel: what must they be sold for, to gain 25 per cent? 4. If sugar is bought at 6 cents a pound, what per cent will be gained if it be sold at 7? 5. If a barrel of flour cost 8 dollars, what must it be sold for, to gain 5 per cent? 6. A merchant finds that a lot of goods, which cost 60 dollars, is damaged, and he sells them at a loss of 15 per cent: what does he get for them? 7. The price of a book is 80 cents; but being sold to a friend, a discount is made of 20 per cent: what is paid for it? 8. A piece of cloth, which cost $45, is somewhat damaged, and is sold at a discount of 33~ per cent: what is paid for it? 9. A merchant buys a chest of tea, for which he pays 85 dollars; but finds it injured, and sells it at 20 per cent loss: how much does he get for it? 10. If a grocery merchant buys sugar at 6 cents a pound, and sells for 8, what per cent does he make? 11. A merchant buys a barrel of sugar for $60 and sells it for $80: what was the rate per cent and what the percentage? 12. A grocer buys sugar at 5 cents a pound: what must he sell it for to make 60 per cent? 13. A grocer buys sugar at 8 cents a pound: what must he sell it for to make 25 per cent 2 LESr -II.] INTELLECTUAL ARITHMETIC. 131 14. A grocer buys sugar at 6 cents a pound and sells it at 9: how much does he make per cent? 15. A grocer buys a bag of coffee at 12 cents a pound: what must he sell it for a pound in order to net 162 per cent? 16. A bag of coffee is bought at 10 cents a pound, nd being injured, is sold at 8 cents a pound: what was the loss per cent? 17. If flour cost 9 dollars a barrel, what must it be sold for to give 10 per cent. profit? 121 per cent profit? 18 per cent? 18. If molasses costs 30 cents a gallon, what must.t be sold for to yield a profit of 20 per cent 2 33j per cent? 19. What is 25 per cent of 6? Of 9? Of 10? 20. Nine is what per cent of 36? Of 54? LESSON VII. Of Interest. 1. Interest is an allowance made for -the use of money, and is generally reckoned at so much per cent for each year on the sum loaned, which sum is called the principal. The allowance, or per centage, is called the interest, and the principal and interest, together, are called the amount. 2. What is the interest of $100 for 1 year, at 1 per cent? At 2 per cent? At 3 p&a cent? At 4 per cent? 3. What is the interest of $150 for 1 year, at 1 per sent 2 At 2 per cent? At 3 per cent? 4. What is the interest of $200 for 1 year at 2 per cent? At 5 per cent? 5. What is the. interest of $160 for 1 year, at 5 per eent? At8 per cent?. 132 INTELLECTUAL ARITHMETIC. [SEC. VIL 6. What is the interest of $200 for 2 years at 5 per cent.* 7. What is the interest of $200 for 1 year at 6 per cent? 8. What is the interest of $150 for 1 year at 8 per cent? 9. What is the interest of $300 for 2 years at 4 per cent? 10. What is the interest of $500 for 2 years at 6 per cent? 11. What is the interest of $90 for 3 years at 2 per cent? 12. What will be the amount if $120 be put at in. terest for 2 years at 6 per cent? 13. What will be the amount if $60 be put at iln. terest for 3 years at 3 per cent 2 14. If $80 be put at interest for 2 years at 4 per cent, what will be the amount? 15. What will be the amount of $70 for 2 years at 5 per cent? 16. What is the interest of $320 for 4 years at 3 per cent? 17. What is the interest of $260 for 3 years at 6 per cent? 18. What is the interest of 125 dollars for' years at 7 per cent 2 19. What will be the amount, if $300 be put at interest for 2 years, at the rate of 4 per cent? * ANALYSIS.-The interest of 1 dollar for 1 year at 1 per cent is 1 cent; and for any number of dollars, as many cents as there are dollars in the principal. Hence: Find the interest of the principal for 1 year, at one per cen& antd thetn multipwlv by the time, in years, and by the rate per centz the product will be the interest. lahs, the interest of $200 for 1 year, at 1 per cent, is $2: len, $ 2 X 5 =$20, the interest for 2 years, at 5 per cent. W The eanths of a dollar may be read dimes, and the htuidrledths, cetjts. L>. VII,] INTELLECTUAL ARITHMETIC. 132 20. What will be the amount, if 8250 be put at interest at 6 per cent for 3 years? 21. What will be the amount, if $500 be put at interest for 2 years at 4 per cent? 22. What will be the amount, if $55 be put at interest for 5 years at 7 per cent? 23. What will be the interest of $85 for 2 years at 7 per cent? 24. What will be the interest of $75 at 6 per cent for 5 years 2 25. What will be the interest of $275 at 6 per cent for 4 years? 26. What will be the amount of $175, after drawing interest for 3 years at 5 per cent 2 27. What will be the, interest of $160 for 2 years at 8 per cent? 28. What will be the interest of $375 for 2 years at the rate of 5 per cent? 29. What will be the amount of $350, drawing interest for 2 years at the rate of 4 per cent 2 30. What will be the amount of $95 for 2 years at the rate of 5 per cent? 31. What will be the amount of $86 for 3 years at the rate of 3 per cent? 32. What is the interest of $150 for 3 years at the rate of 6 per cent? 33. What is the amount of $360 for 5 years at 2 per cent'? 34. What is the amount of $240 for 3 years at 3 per cent 2 35. What will be the amount of $120 for 4 years at the rate of 5 per cent? 36. What will be the amount of $240 for 3 years t 5 per cent? 12 134 INTELLECTUAL ARITHMETIC. [SEC. vU LESSON VI. Interest for parts of a Year. 1. What is the interest of 200 dollars for 3 months, at 5 per cent?* For 5 months? 2. What is the interest of 60 dollars for 4 months, at 5 per cent? 3. What is the interest of $40 for 6 months at 0 per cent? 4. What is the interest of $20 for 9 months, at 3 per cent? 5. What is the interest of $15 for 10 months, at 6 per cent? 6. What is the interest of $6 for 8 months, at 7 per cent?' 7. What is the interest of $12 for 11 months, at 8 per cent? 8. What is the interest of $25 for 10 months, at 9 per cent? 9. What is the interest of $60 for 5 months, at 6 per cent? 10. What is the interest of $84 for 11 months at 10 per cent? 11. What is the interest of $96 for 7 months, at 9 per cent? 12. What is the interest of $72 for 5 months, at 7 per cent? 13. What is the interest of $144 for 11 months, at 9 per cent? " ANALYSIS.-Find the interest for 1 year, at 1 per cent, which is $2; then, as 3 months being 4 of a year, the interest for three months will be J of 2, or i a dollar; then multiply by 5, the rate per cent, and we obtain 21 dollars, or 2 dollars and fifty cents, for the interest. For 5 months, we have f152 of 2 dollars, which is 7- of one dollar, which being multiplied by 5, gives 2 X or 16 dollars. LES. VII.] INTELLECTUAL ARITHMETiC. 135 14. What is the interest of $60 for 6 days,* at 9 per cent? 15. What is the interest of $84 for 9 days, at 7 per cent? 16. What is the interest of $24 for 16 days, at 8 per cent?t 17. What is the interest of $48 for 17 days, at 6 per cent? 18. What is the interest of $50 for 6 days, at 8 per cent? 19. What is the interest of $96 for- 11 days, at 7 per cent? 20. What is the interest of $40 for 15 days, at 9 per cent? 21. What is the interest of $144 for 8 days, at 5 per cent? 22. What is the interest of $60 for 18 days, at 10 per cent? 23. What is the interest of $72 for 10 days, at 9 per cent? 24. What is the interest of $132 for 5 days, at 6 per cent? 25. What is the interest of $42 for 20 days, at 9 per cent? 26. What is the interest of $12 for 19 days, at 10 per cent? 27. What is the interest of $36 for 21 days, at 10 per cent? 28. What is the interest of $15 for 25 days, at 8 per cent? * ANALYSIS.-The interest of $60 for 1 yeat, at 1 per eent is 60 cents; and for 1 month, is 5 cents, and for 1 day is - cr oi f 1 cent; and at 8 per cent., it is * X 8 = 1 -i ]3 cents t OnsEavATION-Observe that 16 days is'1 of a month, and 1 lay over; then as the interest at 1 per cent is 2.cents a month, t6 days gives 1 cent, and-L of 2 cents, or I-j- of a cent: hence T1- of a cent multiplied by 8, gives 8T9 of a cent. (36 INTELLECTUAL ARITHMETIC. [SEC. VnI 29. What is the interest of $20 for 2 years a months and 6 days, at 7 per cent? 30. What is the interest of $30 for 3 years 9 months and 10 days, at 6 per cent? 31. What is the interest of $24 for 4 years 0 months and 20 days, at 8 per cent 32. What is the interest of $36 for 1 year and 5 days, at 8 per cent? 33. What is the interest of 60 dollars for 2 years t months and 25 days, at 6 per cent? 34. What is the interest of $200 for 2 years lO months and 10 days, at 5 per cent? 35. What is the interest of 60 dollars for 2 years 2 months and 3 days, at 8 per cent? 36. What is the interest of 48 dollars for 3 years 8 months and 10 days, at 2 per cent? 37. What is the interest-of $72 for 3 years 0 mnonths and 5 days, at 4 per cent? 38. What is the interest of $84 for 2 years 4 months and 3 days, at 6 per cent? LESSON VIII. Analysis of Questions. 1. If 1 yard of cloth cost $2, how much will 4 yards cost at the same rate? 2. If 5 yards of cloth cost $10, what will 8 yards cost at the same rate? 3. If 4 yards of cloth cost $9, what will 16 yards cost at the same rate?* ANALYSI.-Thil, and similar examples, may be done by tim analysis on page 63; or they may be done thus: The ratio of 4 yards of cloth to 16 yards is 4; that is, 10 yards of cloth is 4 times as much as 4 yards, and therefore, will aost 4 times as much? If 4 yards of cloth cost $9, 16 yards will cost 4 times 9 dol iaes, which are 86 dollars. Bi. VIII.] INTELLECTUAL ARITHMETIC. 137 4. If 6 men consume a barrel of flour in 2 months how much will they consume in a year 2 5. If a man travels 60 miles in 5 days, how far will he travel in 30 days? 6. If 4 men consume I barrel of flour in 20 days, how much would 32 men consume in the same tire? 7. If 3 barrels of flour cost $14, how much will 9 barrels cost? 8. If 41b. of sugar cost 64 cents, what will 131b. cost? 9. If A of a piece of cloth costs $84, what will - pieces cost? 10. If 4 of a barrel of cider cost lo of a dollar, what will 2 of a barrel cost? 11. If 9 bushels of oats will feed 4 horses 5 days, how long will 36 bushels feed them? 12. if 3 paces of the common step be equal to 2 yards, to how many yards will 18 paces be equal? 13. If 5 yards of cotton cloth are equal in value to 2 yards of linen, how many yards of linen will 20 yards of cotton buy? 14. If 81b. of coffee is of the same value as 3t1. of tea, how many pounds of tea will 241b. of coffee buy? 15. If 8 oranges are worth 24 cents, how much are 2 oranges worth? 16. If 4 apples are worth I1 oranges, and one orange is worth two lemons, how many lemons will 12 apples buyv? 17. If 5 baskets of peaches are worth $83, how much will 8 baskets be worth. 18. If a man travel 7 miles in two hours, how fai will he travel in 14 hours? 19. Two men start from the same point and travel in opposite directions; one at the rate of 38 miles an hour, and the other at the rate of 41 miles an 12* 138 INTELLECTUAL ARITHMETIC. [SEC. YI hour; how far apart will they be at the end of 4 hours? 20. If, in the last question, the men were to travel in the same direction, how far apart would they be at the expiration of 4 hours? 21. If 61b. of butter cost 14 shillings, how much will 151b. cost? 22. If 4 turkeys cost $24, what will 16 turkeys cost? 23. If 9 yards of broadcloth cost $271, how much will 27 yards cost? 24. If 51b. of loaf sugar cost $3-, how much will twenty-five pounds cost 2 25. If 2 tons of hay cost 191 dollars, what will 8 tons cost? 26. If 3 pairs of boots cost $214 dollars, how much will 18 pairs cost? 27. If 3 horses eat 6-1 bushels of oats in 2 days, how much will 12 horses eat in the same time? 28. If a family consume 143 barrels of flour in 2 months, how much will they consume in 4 months? 29. If 7 bushels of wheat cost $64, how much will 14 bushels cost? 30. If a family consume 2:q bushels of grain in 34 weeks, how much will they consume in 61 weeks? 31. If 34 yards of cloth cost $62, what will 14 yards cost? 32. If 3 pairs of shoes cost $54, what will 9 pairs cost? 33. If a man travels 9 miles in 24 hours, how far will he travel in 74- hours? 34. If 9~ pounds of tea cost 12g dollars, how much will 28 pounds cost? 35. If 8 yards of broadcloth cost 17 dollars, what will 16 yards cost? L,.S. Ix.] INTELLECTUAL ARITIMETIC. 139 LESSON IX Analysis of Questions Continueda 1. If a man can build a wall in 1 day, how long will it take. two men to build it? 2. If 2 men can build a wall in 4 days, how long will it take 4 men to build it? 3. If 4 horses, in two days, eat 5 bushels of oats, how much will 6 horses eat in 4 days?* 4. If a barrel of flour last 15 men 20 days, how long will it last 25 men If 5. If 6 men consume 241b. of beef in 5 days, how much will 9 men consume in ten days? 6. If 6 horses eat 21 tons of hay in 2 weeks, how much will 16 horses eat in 1 weeks? 7. If 8 men can build a wall 6 days, in how many days can 12 men build it? 8. If a certain amount of provisions will last 2 families of 5 persons each 3 weeks, how long will the same provisions last 5 families of 6 persons each? 9. If 2 men, in 5 days, can Muild 160 feet of wall, how long will it take 4 men to build 192 feet of wall? 10. If 5 men can do a certain work in 6 days, how long will it take 3 men to do 5 times that work? * ANALYSIS —Four horses will eat as much in 2 days, as 8 hoises in 1 day; and 6 horses will eat as much in 4 days as 24 horrses in 1 day. Now, 24 horses is 3 times 8 horses: if 8 horses eat 5 bushels of oats in 1 day, 24 horses will eat 3 times as much, which are 15 bushels: therefore, 6 horses in 4 daya (equivalent to 24 horses for 1 day) will eat 15 bushels. T Fifteen men will eat as much in 1 day as 1 inan will eat in 16 (days and 15 men will eat as much in 20 days as 1 man in 300 days. Now, the same provisions will last 25 men only one-twenty-fifth as long as they will last 1 man; that is, so many days as 25 is contained times in 300, which are 12. 140 INTELLECTUAL ARITHMETIO. [SEC. VrL ] 1. If a man travels 48 miles in 2 days, travelling 6 hours a day, how far will he travel in 3 days, travelling at the same rate, 5 hours a day 2 12. If 12 dollars' worth of provisions will supply 9 men 4 days, how much will it cost to supply 21 men for 5 days 2 13. If 2 men consume 21b. 4oz. of flour in 1 day, kow much will 8 men consume in 4 days? 14. I-ow many sheep, at 3 dollars a head, must be given for 5 cows at $18 apiece? 15. A man, failing in trade, pays his creditors 3 shillings on the dollar, while another pays an equal sum by paying two shillings on the dollar: what is the ratio of their debts? 16. If ~ of a bushel of oats will feed 2 horses half a day, how many will be required to feed 4 horses 4-& days? 17. If a barrel of flour will serve a farnily of 6 per. sons 31 weeks, how much will serve a family of 9 persons 43 weeks? 18. If 5 men can cut 30 cords of wood in 3 days, how much will 4 men cut in 8 days 2 19. If 3 men can mow 15 acres in 2+ days, how long will it take 11 men to mow 44 acles? 20. If a farmer can plough 9 acres in 4 days, with one team, how much can he plough with two teams in 8 days? 21. If a pasture of 8 acres will feed 3 horses for 2 months, how many acres will feed 4 horses 5 months? 22. If the wages of 3 men for 7 days are 21 dollars, what will be the wages of 9 men for 11 days? 23. If a man travels 121 miles in 5 hours, how far will he travel in 4 hours, at the same rate 2 24. A man loses, at play, - of his money, after which he gives away i of the remaindelr, and finds that he has 8 dollars left; how much had he at firsts LES. X. I NTELLECTUAL ARITHMETIC. 141 25. How many yards of cloth at 44 dollars a yardL must be given fur 9 yards at 3 dollars a yard? 26. If 4 tailors can make 8 pair of pantaloons in 2 days, how many will 3 tailors make in 5 days? 27. If 5 horses consume 2414 tons of hay in a winter, how much will 10 horses consume? 28. James being asked how many marbles he had, replied, that if 3 of the number be divided by 2, and the quotient subtracted from one half the number, the remainder would be 12: how many had he? 29. If 9 men can do a piece of work in 4a days, how many men should be employed to do the same work in 7 days? 30. If 3 horses eat 34 tons of hay in 2 months, how much will supply 5 horses for 4 months? LESSON X. 1'b find the parts, knowing the whole and the proportion of the parts. 1. James bought an orange and a melon, for which he paid 8 cents. He paid three times as Inuch for the melon as for the orange: what did he pay for each?* 2. Charles bought a whistle and a top, for which he paid 12 cents. He paid five times as much for the whistle as for the top: what did he pay for each? 3. What number added to itself will give a sum equal to 20? 4. What number added to twice itself will give a number equal to 15? * ANALYSIS.-Jamies paid one equal part of the wehole sum fo, the orange, and 3 equal parlts for the melon: hence, he paid 4 equal parts for both. Then, the whole sum which he paid, (8 cents), divided by the number of equal parts, (4 equal parts), will give 1 part. which is 2 cents, what he paid for the orange; and 2 cents multiplied by 3, will give 6 cents, what he paid for thse melon. 5 142 INTELLECTUAL ARITHMETIC. [SEC. VII. 5. What number added to five times itself will give a number equal to 30? 6. John bought an apple, a peach, and an orange, or which he paid 6 cents. He paid twice as much for the peach as for the apple, and as much for the orange as for the apple and peach together: what did e pay for each? 7. A man bought a horse, saddle, and bridle, for which he paid 90 dollars. He paid twice as much for the saddle as for the bridle, and four times as much for the horse as for the saddle and bridle together: what did he pay for each? How many parts are there? 8. The sum of the ages of James, Charles, and John, is 44 years. James' age is one half Charles and one third John's: what is the age of each? 9. A farmer has in his garden apple trees, pears, and peaches; in all 72. He has twice as many pear as apple trees, and three times as many peaches as pear trees: how many has he of each? 10. A person distributed 36 cents amongst three beggars, a father, mother, and son. He gave the mother twice as much as the boy, and the father twice as much as the mother and boy together: how much did he give to each? 11. A farmer has 72 sheep in four lots. In the second he has twice as many as in the first; in the third as many as in the second; and in the fourth twice as many as in the third: how many has he in each? 12. Mary, Jane, and Nancy, gather 144 apples from the orchard. Jane is to have twice as many as Mary, and Nancy is to have three times as many as Mary and Jane together: how many will each have 1 13. Divide 12 into two such parts that the second shall be double the first. Into how many equal parts is 12 to be divided? LEd. X.1 INTELLECTUAL ARITHMETIC. 143 14. Divide 21 into three such parts that the second shall be double the first, and the third double the second. 15. James asked John how many miarbles he had? John replied, if you will give me twice as many as i now have, and William will give me 5 times as many as I should then have, I would have, in all, 36: how many had he?. 16. Mr. Parsons bought 4 pounds of coffee, a pound of tea, and a yard of cloth, and paid in all $16. He paid twice as much for the tea as for the coffee. and 5 times as much for the cloth as for the coffee: what did he pay for each? 17. Divide 36 into four such parts that the second shall be 3 times the first, the third 5 times the first, and the fourth 9 times the first. 18. In a pasture there are seven times as many sheep as cows, and twice as many lambs as cows; in all, 40: how many of each sort? 19. A father, mother, and son, receive 108 cents for a day's work. The mother receives twice as much as the son, and the father as much as the mother and son together: what does each receive? 20. In a school of three departments, there are 150 pupils. In the first department there are one third as many as in the second, and in the second one half as many as in the third: how many are there in each? How many equal parts of the whole school in each department? 21. Divide 82 into four such parts that the second shall be 4 times the first, the third 3 times the second, and the fourth 2 times the third. 22. A sloop employed in carrying bricks to New York, carries 40 thousand at a load. She is loaded twice and urloaded once, the first week-the second week, she is unloaded twice and loaded once; and so 144 INTELLECTUAL ARITHMETIC. L[SEC. VIT. on for the season: how much does she average a week? 23. James being asked what he had been about during the day, replied, that he had been ciphering 4 hours and done 82 sums. That in the second hour lie did 4 times as many as in the first; in the third hour, three times as many as in the second; and in the fourth, 2 times as many as in the third: how muany did he do in each hour? LESSON XI. To divide NVunbers into proportional parts. 1. Divide the number 18 into two parts, such that the ratio of the parts shall be the same as 2 to 4.* 2. Divide the number 28 into two parts, such that their ratio shall be the-same as 5 to 9. 3. Divide the number 34 into two parts, such that the first shall be eight ninths of the second. 4. Divide 34 into two such parts that the first shall be one and one eighth times the second. 5. Two men bought a piece of muslin containing 30 yards; one paid $2 and the other 83: how many yards belonged to each? 6. Two men hired a pasture for $24. One pastured 5 horses and the other 3: how much should each pay? 7. Two men hired a pasture for $72. One pastured 3 horses for 5 weeks, and the other 7 horses for 3 weeks: what proportion should each pay? 8. A father divides 84 cents between John and Charles, giving 5 cents to John and 7 to Charles each ANALYsIS.-Thela are 6 units in the sum of 2 and 4. If 18 itc divided into 6 equal parts, each part will be 3. Two of theme parts must form the first number, and 4 of them tlh secamd. Hence, the numbers are 6 and 12. LES. XL I INTELLECTUAL akRf RMET'C. 145 tine, till the whole was distributed: how muach did tae give to each? 9. Three persons buy a piece of cloth contaiining 48 yards. The first puts in 5 dollars, the second 9, and the third 10: what was each one's share? 10. James has 72 marbles; he gives 3 to Wilillham and 5 to John, each time, until none are left: how nmany does he give to each. 11. William has 9 cents and John 7, and they buy 06 apples: how many apples should each havet 12. AMr. Wilson fails in business and pays - of hh; debts: how much will Mr. Squires receive to whom. (Le owes $108? 13. A grocer weighs out 24 pounds of -sugar to 2 customers, giving.2 pounds to one as often as he gave: of a pound to the other: how much did he give to each? 14. A draper divides a piece of cloth coistaining 36 yards, betwee n 2 persons, giving 2- yards to the one every time that he gave 3~ yards to the othero how much did each receive? 15. A man distributed 78 cents amnong 8 beggars, g of whorm were men and It were women. lie gave twice as much to each womran as to each man: htow much did he give to each? 16. James and John start from the same place, travel the samna way, and take steps of equal length James steps 4 times while John steps but 3: how far will they be apart when the distances travelled by both is 14 miles? 17. James and John start friom the same place, travel the same way, and take steps of equal length; James steps 4 times while John steps but 3: how fatr will each have travelled when they are 3 mntiites apart? T 146 INTELLECTUAL ARITHMETIC. [SEC. VII. 18. Two men agree to do a piece of work for which they are to receive $88; the first sends 4 hands for 3 days, and the second 5 hands for 2 days: how much should each receive? 19. A person met three beggars, a boy, a mother and father, and distributed 84 cents among them. For every 5 cents he gave the boy he gave the mother 7 and the father 9: how much did he give to each? 20. Three persons hire a pasture, for which they pay $56. The first puts in 2 horses for 3 weeks, the second 5 horses for 2 weeks, and the third 9 horses for 1- weeks: how much ought each to pay? 21. Charles has 5 marbles and John' 9, and they agree to share their winnings or losses in the same proportion. After several games they find that they have won 42: how are they to be divided? 22. A and B enter into partnership: A puts in 7 dollars, and B 11: they make 9 dollars by the operation: how should it be divided? 23. A and B enter into partnership: A puts in 6 dollars for 2 months, and B, 5 dollars for 3 months: they gain 81 dollars: what is the share of each? 24. Two barrels of flour, costing 12 dollars, are consumed by three persons; the first ate fiom theml 2 months, the second 3 months, and the third 5 ntonths: how much should be paid by each? 25. Three persons hire a pasture for sheep, for which they pay 12 dollars. The second puts in twice as many sheep at the first, and the third three times as many as the first; but the sheep belonging to the first man are in twice as long as those belonging to the second, and three times as long as those belonging to the third: how much should each pay? LES. XII.] INTELLECTUAL ARITHME:TrIC. 147 LESSON XII. Analysis of Questionzs. 1. The suim of two numbers is 10 and their difference 4: what are the numnbers?* 2. The sum of two numbers is 16 and their dif. ference 8: what are the numbers? 3. James and John together have 24 marbles, and the difference between thciir marbles is one fourth of the sum: how many had each? 4. A tailor in measuring two pieces of cloth found their difference to be 6 yards, and also that this dif ference was an eighth part of the cloth in both pieces: how much was there in each piece? 5. James and John have 16 marbles, and James has 4 more than John: how many has each? 6. Nancy has 6 more pins in hei cushion than Jane, and together they have 30: how many has each? 7. John, in a week recited 10 lessons more than Charles, and together they recited 24: how mlany did each recite? 8. A farmer bought an equal quantity of sugar and coffee, and then gave a cheese for 20 pounds of coffee, when it appeared that he had in all 50 pounds of sugar and coffee: how much had he of each? 9. A ralan being asked how much money he had, said, that he had only dollars and dimes, and that he had 72 pieces in all: that the number of dollars less the number of dimes, was one-twelfth the sum of the pieces: how much money had he? * ANALYSIS.-If the two numbers were equal and their sum 10, each number would be 5. Now, if you take 1 from one of the 5's you have the number 4, and if you add it to the other you get the number 6; if you do the same for the two last nullLers, you get the nunmbers 3 and 7, whose difference is 4: that is, the greater of two numbers is equal to half their sum plus'alf their diference, and the less is equal to half their sum minus half their diference. 148 INTELLECTUAL ARITHMETIC. [SEC. VIL 10. A farmer has 10 more sheep than he has cows, he loses 3 cows and 6 sheep, when he finds that he has 17 of both kinds remaining: how many had he at first of each kind? 11. A father being asked his age, replied, that 7 years ago his age was double that of his son's, and now that the sum of their ages was 89: what was the age of each? 12. A man and son engage to work 20 days, the son to receive 3 dimes a day less than the father: at the end of the time they receive 42 dollars: how much of this sum did each earn, and what did each receive per day 2 13. John being asked how many marbles he had, replied, that 19 was 3 more than 4 of the number: how nmany had he? 14. Lucy being asked her age, said, that her sister Jane was 6 years old when she was born, and that now the sum of their ages was 20: what was the age of each? 15. A farmer had as many sheep as hogs, and after losing 12 of his hogs, his sheep and hogs amounted to 04: how many had he at first of each kind? 16. A man bought a vest, for which he paid 21 dollars less than he paid for his coat, and for the two together he paid 33 dollars: what did he pay for fach? 17. Mr. Wilson sold his cow for 30 dollars, which was jo of what she cost himn: what did he give for her? 18. A gentleman bought a coat and hat, for which lie paid 27 dollars, and the cost of the hat was ono eighth of the cost of the coat: what was the cost of each? 19. If- 5 of a number is 3 less than what is the iumber? LES. XIII.] INTELLECTUAL ARIThMETIC. 149 20. Two men have 60 dollars between themn; if the,.ne having the largest sum gives the other 5 dollars, their money will be equal: how much had each? 21. A man bought a cow and a calf, for which he paid 36 dollars, paying 5 times as much for the cow as for the calf: what did he pay for each? 22. A man bought a pairi' of boots for 6- dollars, which was 3 of what he paid for his coat: what did his coat cost? 23. A man paid 2- dollars more for his pantaloons thllan for his vest, and for both -he paid 14} dollars: what did he pay for each? 24. A market woman bought 36 eggs; for I: of them she paid 2 cents for three eggs, and the remain der she bought at the rate of 4 cents for 3 eggs: for what must she sell them that she may make 6 cents? LESSON XIII. Separate and Concurring Causes. 1. If Charles can do a piece of work in 2 days. what part of it can he do in I day? If you denote the work by 1, what will denote the part which he can do in 1 day? 2. If James can do a piece of work in 3 days, what part of it can he do in I day? What part in 2 days? 3. If a family consume 12 pounds of sugar in a week, how much will they consume in 1 day? IIo,w nmuch ir. 3 days? 4 days? 4. If Jamnes can do one fifth of a piece of work in 1 day, how long will it take him to do the entire work'l If he can do I in a day, how long will it take him to do the work? How long if he can do 4I If he can do i, how long? 5. John can do a piece of work in 2 days, and Charles can do the same work in 3 days: 13* 150 INTELLECTUAL ARITHMETIC. [SE(C VIL. WThat part of the work can each do in a single day? What part can both do in I day; and itr what timne can the work be done by both of them working together? * 6. A can do a certain piece of work in 3 days, B can do it in 5 days: What part of it can each do in 1 day? What part can both do in I day, and in what time can the work be done by both working together? 7. A cistern is to be filled by two pipes. One can fill it in 2 hours and the other ill 5: What part of the cistern can each fill in 1 hour? What part of it can they both fill in 1 hour? In what time can they fill the cistern running together i S. A cistern is to be filled by three pipes. The first can fill it in 2 hours, the second in 3, and the third in 4: What part of it will each fill in 1 hour? What part of it will they all fill in 1 hour? In what time will the three fill it, running together? 9. John can do a piece of work in 2 days, and John and James together, can do it in 15- days: ill what time can James do it alone?ft * ANALYsIs.-Denote the work to be done by 1. Then -T will will represent the part which John can do in 1 day; and * will Represent the part which Charles can do in 1 day; and the sum of 1 and ~ will represent what both can do in 1 day, viz.: 5- of the entire work. Then, the number of times which 1 (the work to be done) contains 6, viz.: 1- times, shows the numnber of days in which they can do the work together. Many similar examples may be done in the same manner. f ANA rSIS.-If John can do the work in 2 days, he can do 1 of it in 1 day; and if James and John together, can do the work in 1 - days, that is 6 days, they can do 1 divided by f, that, is ~F of it in 1 day: then, James can do 5 of the work less } of the work, that is, 3 of the work, in 1 day; therefore, he canl do the whole work in three days. I.Es. XIII.] INTELLECTUAL ARITHMETIC. 151 10. A can do a piece of work in 3 days, A and B together, can do it in 1l days: in what time can B do it working alone? 11. A cistern can be filled by one pipe in 2 hours, and by 2 pipes in 1- hours: how long will the second pipe require to fill it if running alone? 12. A cistern can be filled by 3 pipes in 1 I hours, one of the pipes can fill it in 2 hours, and another in 3: how long will it require the third to fill it, if running alone? 13. A man and his wife usually drank a gallon of beer in 12 days; but when the man was fiom'home it lasted his wife 30 days: how many days would the man require to drink it? 14. A quantity of flour will last one family six weeks, and the same flour will last another family 3 weeks. It is found that one-third of the flour is spoiled: how long will the remainder' last both families? 15. If two families consume a quantity of provisions in 13 weeks, and one family alone would consume the same provision in 4 weeks: how long would it last the other? 16. A can mow a field in 1 day, B in 2 days, and C in 3 days: in what time can they all mow it, working together? 17. A can mow a field in 2 days, B can mow it in 3 days, but by the aid of C, they can mow it in 6 of 1 day: how long will it take C to mow it alone? In what.time can A and B mow it? In what time can A and C mow it? 18. Three carpenters can finish a house in 2 months; two of them can do it in 2~ months: how long will it take the third to do it alone? 19. A, with the assistance of B, can build a wall 2 tfeet wide 3 feet higTh and 30 feet long in 4 days, but with the assistance of C they can do it in 21- days: 152 INTELLECTUAL ARITHMETIC. [SEC. VI[ What part of it can A and B build in 1 day? What part of it can they all build in 1 day? What part of it can C alone, build in 1 day? In how many days would C build it alone LESSON XIV. Analysis of Questions. 1. A laborer engaged to work for 16 days on these conditions: For every day he labored, he was to receive 4 shillings, and fobr each day that he was idle he was to pay 2 shillings for his board; at the end of the time he received 52 shillings: how many days did he work, and how many days was he idle?* 2. A carpenter took an apprentice on these terms: he paid his father, at the end of each month of 26 working days, 3 shillings a day, for every day the boy worked —charged him 20 shillings for clothes, also, I shilling for board, for every idle day; at the end of the time there was 30 shillings due him: how many days was he idle? 3. A merchant bought 50 yards of calico, some of which was damaged, on these terms: He was to pay 3 dimes a yard for all that was perfect, and 1 dime a yard for all that was injured; at the settlement he paid 12 dollars: how many yards were injured? 4. A grocer purchased 30 fowls, a part turkeys and a part of them chickens; for the turkeys he was tc pay 11 dimes a piece, and for the chickens 4; he paid in all, 24 dollars and 60 cents: how many were there of each kind? * ANALY-Ss.-Had he labored the 16 days. he would have re ceivecl the 64 shillings. But he received only 52 shillings: hence, lie lost 12 shilliuns by idleness. But as he paid 2 shillings a day for his board, and lost 4 shillings a day in wages, he lost, in all, 6 shillings a day: therefore, the esmaber of days he was idle will be expressed by 12 divided by 6, giving 2 idle days: therefore, he worked 14 days. LES. XIV.] INTELLECTUAL ARITHMETIC. 153 5. A farmer hired a father and son to do 20 days work between them: the father was to have a dollar for every day he worked, and the son 75 cents; at the end of the time, the amout paid was 17 dollars: how many days did each work? 6. A yard stick is broken into 2 parts, the shorter of which is A the length of the longer: what is the length of each piece?* 7. A piece of cloth of 40 yards in length is cut into 2 pieces, such, that the smaller piece is 7 of the larger: What is the length of each piece? 8. What number is that to which if 3 of itself be added, the surm will be 32? 9. A coat and vest cost 24 dollars, and the vest cost 7- as much as the coat: what was the cost of each? 10. A cow and calf are worth 56 dollars, and the calf is worth r- of the cow: what is the valueof each? 11. A pole 16 feet long stands in the mud, water, and air. The part in the mud is I of the part in the water, and the part in the air is equal to the other two: what is the length of each part? 12. There is a fish weighing 72 pounds. His head weighs twice as much as his tail, and his body weighs as much as his head and tail together: what is the weight of each part? 13. Divide the number 52 into two such parts, that.{ of the larger part shall be equal to the less part. * ANALYSS.-The fractional unit, in this question, is one fifth the length of the longer piece. There are 5 of these units in the longer piece and 4 in the shorter: hence, there are 9 in the whole stick, which is 36 inches long. The value of the fraetional unit, is, therefore, 36 divided by 9, or 4 inches. Hence, one of the pieces is 20 inches, and the other 16. All simiat questioms are solved byfindinfg thefractional unit. 154 INTELLECTUAL ARITHMETIC. LSEC. VIL 14. A tailor has 48 yards of cloth in 2 pieces; i of the longer piece is equal to 2 of the shorter: how iany yards in each piece?* 15. A fiarmer bought pigs and sheep, in all 33: 2 of the pigs was equal to - of the sheep: how many were there of each kind? 16. Three fourths of a son's age is equal to on fourth the age of the faither, and the sum of their ages is 80 years: what is the age of each? 17. There are 125 sheep in two fields, 4 of the number in one field being equal 1* times the number in the other: how many in each field? 18. A man after counting his gains at play, found that he had increased his money by 2} of -, and that he then had 42 dollars: how much had he at first? 19. The difference of two numbers is 6, and the less number is I of the greater: what are the numbers?t 20. A father's age is such, that ~ of it is equal to 1- the age of his son, and the difference of their ages is 36 years: what is the age of each? 21. What number is that which being diminished by the difference between 4- and 3 of itself leaves a remainder equal to 34? 22. A flag staff 52 feet long, is so broken by the wind, that ~ of the top piece is equal to - of the piece left standing: how long are the pieces? e SUGGESTION.-If - of the shorter piece equals 2 of the longer, then of the shoter equals of thhe longer, and the whole of the shorter piece equals -6 of the longer. Always find the unit of the smaller in terms of the larger qmvlber. t ANALYSIr.-Suppose the greater number be divided into 8 equal parts; two such parts, are found in the lesser number: hence, the difference between the numbers is equal to 1 of the greater, which is 6: therefore, the greater number is 18 and the )less 12. LES. Ir.] INTELLECTUAL ARITHMETIC. 155 23. Jamnes was asked how many marbles he had, and replied, I have 55, black and red, and. of the black make just as many as ~ of the red. Pray how many have I of each sort? 24. A laborer engaged to work 20 days, and was to receive 9 shillings for every day he worked, and pay 3 shillings a day for his board, every day that he was idle; ill settling, he received 84 shillings: how nany days did he work? 25. A church, including the steeple, is 188 feet high. If the height of the steeple is equal to I the height of the body of the building, Nwhat is the height uf each? LESSON XV. Analysis of Questions. 1. William chases Henry, who is 42 feet in advance, around a circular walk of 100 feet. Their steps are each 3 feet, but William takes 6 steps while Henry takes but 5: how many steps must William make to overtake Henrry? 2. A hare is 25 of his own leaps before a greyhound, which is pursuing him. The greyhound makes'2 leaps while the hare makles 5; but 1 leap of the greyhound is equal to 3 of the hare's: how many leaps will the greyhound make before he overtakes the hare 2* 3. James is in pursuit of John. and 12 of John's steps behind him. James steps 3 times while John * ANALYSIS.-Since the greyhound makes 2 leaps while the hare makes 5, in the time that the greyhound makes 1 leap the hare will make 2A leaps. But 1 leap of the greyhound is equal to 3 leaps of the hare; hence, every time the greyhound jumps he will gain on the hare ~ of a hare's leap: therefore, he must make 50 leaps to overtake him. 156 INTELLECTUAL ARITHMETIC. [SEC. VI1 steps 4 times; but Jamnes' steps are twice as long as John's: how many steps must James make to overtake himl? 4. A-Mary is 24 years old and Jane is 4: how rr any years must elapse before Mary's age will be iust double Jane's.* 5. A father is 60 years old and his son is 35: how long since the age of the father was double that of the son? What was then the age of each? 6. A amother is 36 years old and her daughter 12: how long before the age of the nlother will be double that of the daughter? What will then be the age of each? 7. A mother is 48 years old and her daughter 30: how long since the age of the daughter was half that of the another? What was then the age of' each? 8. A mother is 48 years old and her daughter 10: how long before the age of the daughter will be onethird that of the mother?2t What will then be the age of each-? 9. A mother is 54 years old.and her daughter 20: how long since the age of the daughter was one-third that of the mother? What was then the age of each? ANALYSIs.-At Jane's birth, 4 years ago, Marly was 20 years old, atnd Jane's agre was 0. Twenty years from that date, Mary will be twice as old as Jane: Hence, filfJary's age be dirninished by Jane's, anid the rernainder be rlltiplied by 2, the product will denote Afary's age whent it is clouble Jane's. 1 ANALYis.-At the birth of the daughter, the mother was 88 years old. What number added to 38 years will give a sumn equill to 3 times the number added? If 3 times the number added is equal to 38 plus the number, then, twice the number added will be equal to 38' and the number mIust be one half of 38, or 19 years. Hence, i' the daEughtr's age be subtraccted froi the mother's, acnd the renainbder divided by 2, the q7lolient vwill be the daughter's age, whten it is one-third the iother's. If the difference be divided by 3, the quotient will be the daughter's age when it is one-fouth the mother's; and so on. LES. XVI.] I TELLECTUAL ARITHMETIC. 157 10. A fathei's age is 45 years, and his son 9: how long before the age of the son will be one fourth that of his father i What will then be the age of each? 11. A father is 54 years old, and his son 30: how long since the age of the father was 4 times that of di( son? What was then the age of each 3 LESSON XVI. Analysis of Questions by means of Unity. WE have said, that any number, regarded as a whole, may be called UNITY. What may any number be called, when it is re. garded as a whole? 1. What number added to twice itself will give a sum equal to l12.* 2. What number is that which added to thiree times itself will give a sum equal to 24? 3. What number is that which added to half itself will give 6'3 4. What number added to half itself will give 9? 5. What number added to one-fourth of itself will give 20 6. What number is tbfat which added to twice itself, and the sum to 3 times itself will give 30? 7. What number added to half of itself, and to one fourth of itself will give 28 3 8. James being asked his age, said, I am half the age of my father, and the sum of our ages is 60 years: What is the age of each? * ANALYSIS.-Call the number sought, unity. Then by the conditions of the question, unity plus twice unity is equal to 12. But muity plus twice unity is equal to 3 times unity, which is equal to 12. Then, if 8 times unity is equal to 12, once unity is equal to 12 divided by 3; which is 4. SuGcES'noN.-Let the pupil see if 4 wvill fulfil the conditiols: 4 + 4 X2 = 4 + 8 = 12. Let every question of the lesson b!e analyzed and verified in a similar way 14 158 INTELLECTUAL ARITHMETIC. [SEC. VII. 9. What number is that to which if its * be added, the sum xwill be 35? 10. A man being asked his age,, replied, if to my age you add one-third of it and then' of it, the sum will be 57: How old was he 2 11. John being asked how many marbles he hlad, aid, that one half of what he had, increased by l, and diminished by ~, was equal to 14. 12. Divide the number 12 in two such parts, that one shall be three times the other? 13. Divide 24 into 3 such parts, that the second shlall be 3 times the first, and the third 4 tinmes the first: What are the numbers? 14. Divide 32 into 3 such parts, that the second shall be 4 times the first, and' the third 3 times the first: What are the numbers? 15. Divide 27 into two such parts, that the second shall be 5 of the first 16. Divide 20 into three such parts, that the second shall be one half of the first, and the third, one third of the second. 17. James asked Robert how many marbles he had. Robert answered, "If you will give me half as many as I now have, and afterwards give me I of what I shall then have, my number will be 54." How many had he? 18. A young chap asked an old gentleman his age, who replied, "When you was born, I was 6 of my present age: one third of your age plus one fourth of it, is equal to 7 years. Can you now tell how old I amll 2. 19. A tailor buys a piece of cloth for 6 dollars a yard, and a piece of equal length for $2 a yard, and sells the whole at $4 a yard: does he make or lose? 20. John and James together have 45 marbles: if John's is equal to 2 of James, how many has each ] IEs. XvII.] INTELLECTUAL ARITHMETIC. 159 21. A market woman bought geese and turkeys; 36 in all; one seventh of one sort was equal to one half of the other: how many of each kind? 22. A pole 72 feet long has one half as much in the mud as in the water, and twice as much in the air as in the mud and water together: how many feet in each? 23. What number added to ~, to I, and to i of itself will give a sum equal to 50 2 24. Mr. Wilson bought a hat, a coat, and a cloak; he paid for the coat I1 as much as for the hat, and for the cloak,three times as much as for the coat; and for all he paid 42 dollars; how much did he pay for each? LESSON XVII. Analysis of Questions by means of Unity. 1. What number is that to which if 10 be added, the sum will be 16.* 2. James being asked how many marbles he has, replied, if to ~ of what I have you add 8, the sum wil be equal to 18: how many had he? 3. What number is that whose half exceeds it. third by 4? 4. What number is that, to which if I of itself and 4 be added, the sum will be equal to 22? 5. What number is that, which being added to on, third of itself, and to 3 times itself, will give a suir equal to 26? 6. What number is that, which being diminished by its half and its third, the remainder will be equa to 42 * ANALYSIS.-Denote the required number by unity. Then by the conditions of the question, unity plus 10 equals 16. But if' unity plus 10 equals 16, unity must be equal to 16 minus 10, or 6. For, if the same number be subtractedfrom two equal c;umere, lthe remainders will be equal. 160 INTELLECTUAL ARITHMETJC. [SEC. VII, 7. What number is that, to the one fourth of which if 10 be added, the sum will be equal to 20? 8. What number is that to which if its one fourth be added, and the sum diminished by 7 will leave 13 for a remainder? 9. A. tailor cuts a piece of cloth 21 yards long into 3 pieces; the second contains 1 of the first minus 5 yards, and the third is equal to 1 of the first plus 5 yards: how many yards are there in each piece.* 10. One fourth of William's age is equal to one half of John's, and -the sum of their ages is 24: what is the age of each? 11. If one half of Charles' age equals one sixth of John's, and the sum of their ages is 16, what is the age of each 3 12. Divide 15 into two such parts, that one eleventh of the first shall be equal to 1 of the second. 13. A watch and seal are together worth 64 dollars, and the watch is worth 7 times as much as the seal: what is the value of each? 14. A mother divided 56 pins between Jane and Nancy, so that one fifth of Jane's was equal to one half of Nancy's: how many had each? * ANYsis.-Denote the first part by unity. Then unity = the first part, of unity - 5 = the second part, and v of unity + 5 = the third part; and since the sum of the parts is equal to 21 yards, unity 3 unity- 5 +- of unity +5 =21; that is, I of unity = 21; and X of unity = 3, or unity 12. Hence, the pieces are 12, 1, and 8 yards. LES. XVII.] INTELLECTUAL ARITHMETIC. 161 15. John gave one third of his marbles to William, and then gave half of what he had left and 4 more to Charles, after which he had 8 remaining: how many had he at first?. 16. James and John together, have 50 marbles; 3 times James' number is equal to 7 times John's: how many has each? 17. The sum of two numbers is 16, and the greater is 3 times the smaller: what are the numbers? 18. Two partners in trade have made a profit of 99 dollars, and agree to divide it so that the second shall have 4 dollars every time the other has 5: what was the portion of each 2 19. A school of 88 scholars has three classes; the second contains H1 times as many as the first, and the third twice as many as the second: how many scholars in each class 2 20. If - of John's marbles is equal to l of James', and together they have 56, how many has each? 21. A piece of cloth is divided into 3 parts; one piece is 4 yards long, which is one eighth of the length of the other two, but of these two pieces the longer is 3 times the shorter: what is the length of each piece 2 22. Two persons, A and B, at a tavern, spend 80 cents, of which twice what A spends is equal to 3 times what B spends: how much is spent by each? 23. The sum of the ages of two persons is 56 years, and twice the age of the elder is 6 times the age of the younger: what is the age of each'1 24. A man has 66 fowls, and after selling a-part of them found that what he had left was twice the number sold: how many did he sell, and how many had he left? 25. A man after spending a part of his money at a.tavern, found that what he had left was one fifth of 14* 162 INTELLECTUAL ARITHMETIC, [SEC. VI1 what he had spent, and remembering that he had 84 cents at first, wished to know how much he had left 26. A man sold a horse for 140 dollars, by which he gained R- of what the horse cost him: what did he give for the horse? 27. John anid Charles receives 18 cents for premiiiums, t school. If three times John's money be subtracted from 3 times what both receive, the remainder will be 24: how much does each receive? 28. A fish weighs 64 pounds. The head weighs 3 times as mu-ch as the tail, and the body weighs as much as the head and tail both: what is the weight of each part' 29. If a shadow 15 feet long is cast by a stick 10 feet long, standing vertically, what will be the length of' a stick or pole, like placed, which casts a shadow 24 feet long, at, the same time of day? 30. If a shadow 10 feet long is cast by a stick 5 feet high, what will be the length of a stick, standing in a like position, which casts a shadow 16 feet in length, at the same time of day? 31. A tailor cut 6 coats firom a piece of cloth, after which it, measured 24 yards; he then cut 5 pairs of pantaloons, which took -} as much as the coats, when it was found that one half of the piece was left: how many yairds did the piece contain? 32. A. farmer has two bins for grain, which together, hold 60 bushels; if he diminishes the greater by (. of its capacity and increases the less by the same amnoull; the two will then hold an equal number of bushels: how mrnany bushels does each hold? LES. xv11I.] INTELLECTUAL ARITHMETIC. 163 LESSON XVIII. Promiscuous Questions. I. A man sold a barrel of flour for $4, which was ] of what it cost: how much profit did he make? 2. If - of a piece of broadcloth, containing 18 yards, cost 32 dollars, how much was that a yard? 3. A man sold a cow for 18 dollars, which was 6 of what she was worth: what was her true valLue-? 4. A man spends 5 of his mnonthly income for a hat, and twice as much for a coat, and has $10'left: how much does he receive a month? 5. A man pays I of his daily wages for board, a.d i for his clothes, and at the end of the week has saved $1': what are his wages a day? 6. A pole stands 1 in the air, 1 in the water, and 3 feet in the mud: how long is it? 7. The body of a fish is three times as long as hlis head, and his tail is 2 feet, which is one fourth the length of the head and body: -what is the length of each part, and of the entire fish? 8. A person gave -- of his money to each of 5 persons, and had 4 cents left: how much had he at first? 9. John gave Charles 4 times as manly apples as he gave to William, and to William-1 as miany as he had left, which was ten: how many had he at first? 10. What will be the cost of a bag of coffee, if 2 of it cost $4S? 11. Mary gave i of her money to Jane, and i to Eliza, and had 3 cents remaining: how much had she at first 12. If 3 men can do a piece of work in 8 days, how long will it take 12 men to do the samne worlk? 13. It 5 men can do a piece of work in 71 days, how long will it take nine men to do the same worlk? [low long 12 men? 18 mnen? 36 rle.l? 722 men 1(64 INTELLECTUAL ARITHMETIC. [SEC. VII. 14. A man would give 3, i, and k of the money in his pulrse to the oldest, the second, and youngest son; now, if he has fifty-two dollars left, how much will each receive? 15. James and John, on comparing their marbles, fild that together, they have 112; and that if James gives a of his to John, they will have an equal num. ber: how many has each? 16. James, John, and Charles together, have 120 marbles; if James gives - of his to John, each will ha.,ve 1: as many as Charles: how many marbles has each?2* 17. James, John, and Charles together, have 800 marbles; if James gives 12 more than X of his to John, each will have I as many as Charles: how many has each? 18. In a fruit orchard, one third of all the trees are apples, I are pears, I are cherries, and 26 are plums: how many of each sort, and how many in all? 19. A person being asked the time of day, said, that the time past 12 o'clock x. (that is noon), was { the time past the previous midnight: what was the timle? HIIow many equal parts (fourths), from 12, midnight, to the required time? fEow nmany of these parts between midnight and noon? What is the value of each part? 20. I-ow will you divide half a water-lmelon among 3 boys, so that the second shall have twice as much as the first, and the third twice as much as the second1: whllat part of the whole melo!e will each boy have? NolE.-After James has given one sixth of his marbles to John, each ivill have one fourth of 120, or 30, and Charles will have one half of 120 or 60. Then, what number diminished by $ leaves 30? LES. XIV.] INTELLECTUAL ARITHMETIC. 165 21. If 2 of a barrel of flour will last a family of 5 persons 10 days, how long will a barrel last a family of 25 persons? 22. If 5- baskets of peaches are worth $8[,' how many potatoes at 25 cents a bushel, will one basket buy? 23. If 25 bushels of oats will serve 5 horses for 7 days, how many will serve 7 horses for 9 days? 24. A farmer has his sheep in 3 partures. In the first he has - of his flock, in the second I, in the third i, and 6 over: how many has he in all? 25. A tailor has a piece of cloth, from which he cuts enough for a suit of clothes, and finds that he has 28 yards left, which is just I of 2 of his piece: how many yards in the piece? 26. A man starts on foot from Albany, for New York, and at the end of the first day finds that he has 125 miles yet to travel, which was just 5- of the whole distance: what was, the whole distance? 27. A grocer bought an equal number of lemons and oranges; he paid 9 cents for every 2 oranges, and 7 cents for every 4 lemons: what must he sell them at a piece to make 100 per cent? 28. A grocer bought a certain number of eggs at the rate of 2 for 5 cents, and an equal number at the rate of 3 for 7 cents, and sold them at 3 cents a piece, by which he made 21 cents: how many eggs did he buy? 29. A man and his wife consumed ten pounds of meat in 3 days. The man alone would have consumed it in 5 days: what part of the meat did the -oman consume? 30. If two men can dig 32 bushels of potatoes in I day, working 8 hours a day, how long will it take 3 men, working 9 hours a day, to dig 54 bushels? 166 INTELLECTUAL ARITHMETIC. L[SEC. VII. 31. If 5 men can build a fence 20 rods long in 8 days, homw long will it require 16 men to build it? 32. What number is that fiom which if 2 be subtracted 2 of the remainder will be 4? What number is that a of which is 4? If 2 be added, what number? 33. After paying away 3 and - of my money, I had 10 dollars remaining: how much had I at first? 34. A person after spending - of his money, and then 3 of the remainder, had 8 dollars left: how much had he at first? 35. A man at play lost - of his money, and the next night lost ~ of the remainder, when he found that he had but $12 left: how much had he at firsts 36. John is 20 paces ahead of Charles, but Charles takes 2 steps while John takes 1: how many steps will John make before he is overtaken? How much does Charles gain at each step? 37. If John is 15 steps in advance, and Charles makes 3 steps while John makes 2: how many steps will Charlesmake before he comes up? 38. John being asked how many marbles he had, said, 1 have but four, fbr James first took 4 of all I had, and Charles then took 3 of what was left: how mnany had he before he lost any? 39. A man bought a harness, a carriage, and a pair of horses. The horses cost 70 dollars more than the carriage, and the carriage 50 dollars more than the harness, which cost 40 dollars: what was the whole cost? 40. A tailor cut a piece of cloth into 3 parts: the first part contained 2 yards more than the second; the second contained 4 yards less than the third; and the third was found to contain 12 yards: how many yards in the piece? 41. A tailor cut a piece of cloth into 3 parts: the LES. XIV.1 INTELLECTUAL ARITHMETIC. 167 first part contained -1 of the piece and 4 yards over; the second contained - of the piece wanting 2 yards; and the third contained -A of the piece and 6 yards over: how many yards in the whole piece, and how many in each part? 42. A man paid 3 of his year's bill; after which he paid I of what was left, and yet owed 12 dollars: h}ow much was the bill 2 43. A man spends in a pleasure trip - of his money at a hotel, -I of it in a railroad fare, and 14 dollars in carriage hire and other expenses: how much did he spend in all? 44. Mr. Wilson spends A of his money at a tavern, aind then goes to a grocery and pays 2 of the remainder for coffee and tea; he counted what he had left, and found 2 dollars: how m-uch had he at first 2 45. A young lady had a portionl at marriage. She expended A of it in furniture, gave 6 dollars to each of two sisters, and had 28 dollars left: how much had she at first? 46. James has half as many marbles as Charles tend 10 over; William has half as many as James, and Charles has as many as James and William: how many has each? 47. A tailor wishes to divide a piece of cloth containing 50 yards, into two such parts that one part shall be 51 times the other: how many yards in each piece? 48. A farmer has a field containing 30 acres of land, and wishes to divide it into 3 such fields that she second shall be double the first, and the third equal to the sum of the other two: how much in each of the new fields? 49. A father buys 70 marbles for John and Henry; and wishes that John should have 2-~ times as many as Henry: how must he divide them? ~G ( INTELLECTUAL ARITHMETIC. [sEO. VIl. 50. A tailor's bill amounts to 96 dollars; 4 articles were charged, vests, pantaloons, and coats; there was 3 times as much charged for pantaloons as for vests, and twice as much for coats as for vests and pantaloons: how much was charged for each? 51. A tailor having a piece of cloth, cut it into two parts, one of which was 2 yards less than one half the piece; he then cut the smaller piece into two equal palrts, when lie found that each part contained 14 yards: how many yards in the piece? 52. A mason. built a wall in 2 days. The first day he built 3- of it and 2 rods over; the second day he built t of the remainder and 3 rods more: what was the length of the wall? 53. A horse is worth 4- times the saddle, and both are worth 110 dollars: what is the value of each? 54. A father divided a farmll of 130 acres between 3 sons; the second was to have 14 times as rmuch as the first, and the third to have 2 as rmluch as the second: what was the share of each? 55. A merchant in settling up his cash accounh found that if he had - and I more, that he would still need 4 dollars to malie $70: how much had he? 56. The difference between 2 a:nd 3 of a number is 8 less than I of the number: what is the number? 57. The difference between -5 and i- of a number is 5 greater than 41 of the number: what is the numnber; 58. A mcan on foot is 35 miles in advance of a man pursuing himn on horseback; the footman travels -1 miles an hour, and the horseman iides 7: how It ng befolre the footman will be overtalen? 59. A dog pursues a fo.x, which is 10 rods in advance; while the fox ruins 3 rods, the dog runs 5: how rluny rods will the dog run before overtaking tli fox. LIS. XVIII.] INTELLECTUAL ARITHMETIC. 169 60. A cistern is filled by a pipe which runs 5 gallons a minute; while the water is discharged by a leak, at the rate of 2 gallons a minute: if the cistern holds 120 gallons, how long will it take to fill it, and how many gallons will have run out? 61. The minute hand of a clock moves 12 times as fast as the hour hand, and moves over one space on the face, in five minutes: how long will it take the minute hand to overtake the hour hand when it is one space behind? 62. A poultry-yard contains geese, ducks, and 15 turkeys; if there were 10 more ducks the number would be equal to that of the geese and turkeys; and the number of the geese is equal to -5 the number of the ducks: how many are there of each sort 2. 63. A father distributed a sum of money between his three sons, thus: to John he gave - of the whole and 9 dollars over; to Reuben he gave 15 dollars; and to William he gave the remainder, which was. of the sum that he gave to his other two sons: how much money did he distribute? 64. A piece of cloth containing 86 yards, is cut into two parts: -2 of the whole piece is equal to I of the smaller: how many yards must be added to the less piece to make the two pieces equal? 65. A person being asked his age, replied: 0 f my age added to 20 equals my age diminished by 5. 66. A man bought a horse and a colt; 2~ times what he paid for the colt equalled 1 I times 50 dollars, which he paid for the horse: what did he pay for the colt, and how much more for the hor'se than the colt' 67. A farmer bought a cow and a sheep, and paid 40 dollars for both. I-e paid for the sheep one half what he paid for the cow, less 8 dollars: what did he pay for each? 15 170 INTELLECTUAL ARITHIMETIC. [SEC. VI1. 68. James has 8 dollars more money than John; 5- times this difference equals 1A- times James': how much has each? 69. Charles after eating -4 of his chestnuts gave iway ~ of what he had left, and then had 16 remainuig: how many had he at first? 70. A and B enter into trade together; A puts ill 2 dollars every time B puts in 5: Ai's money remains in 12 months and B's 8; they make a profit of 128 dollars: how should it be divided between them? 71. James went out hunting, and shot one of every 5 squirrels which he saw; had he seen 10. more, and killed in the same proportion, he would have brought hoime 6: how many squirrels did he see? 72. Two men hired a pasture for $33, and agreed that the pasture of 2 cows should count for 1 horse: one pastured 4 cows and 2 horses for 3 weeks, the other 2 cows and 3 horses for 2 weeks: how much should each pay? 73. If 40 dollars be divided between two persons, so that one shall have 3 dollars every time that the other has 2, how much will each receive? 74. In a school of 44 pupils, there are 13 times as many girls as boys: how many of each? 75. A man bought a horse, saddle, and bridle: he paid 2 times as much for the saddle, as for the bridle, and 11 times as much for the horse as for saddle and bridle both; in all he paid $108: how mluchl did he pay fkor each? 76. A merchant sold 63 yards of cloth at 4 dollars a yaid, and took his pay in equal quantities of rye tland wheat, the former at 50 cents, and the latter at 1 a bushel: how much wheat did he receive? 77. Find the ages of three persons, knowing that the age of the second is equal to twice the age of the first, and the age of the third five times the age of LES. XVIII.] INTELLECTUAL ARITHMETIC. 171 the first and second, and that the sum of their agcs is 90 years. 78. A drover after selling 5 of his flock of sheep, finds that if he had sold 4 less he would have sold just I of his flock: how many had he? 79. A boy being asked his age, replied: if to my age y:ou add 2 of it, 3 of it, and 14 years, you will have a sum equal to 3 times my age. 80. A hare pursued by a hound, runs 13 times as far as the distance between them when the pursuit commenced, and the hound runs 28 rods before overtaking the hare: how far was the hare in advance when the pursuit began 2 81. A firmer buys a pig, for which he pays 3 dollars, and also a sheep and a cow; the cow cost 3 times as much as the pig and sheep, and the sheep cost 5 times as much as the pig: what was the cost of each 3 82. A garrison of 300 men, has provisions for 6 months, at the rate of 16 ounces a day: how much must the allowance be diminished to last 8 months? 83. If a staff 3 feet high, casts a shadow 6 feet in length: how long is a pole which casts a shadow 20 feet long, at the same time of day? 84. If a stick 12 feet long casts a shadow 2 feet long, what is the length of a pole which casts a shadow 9 feet long, at the same time of day? 85. A man's coat cost him 1a- times as much as his pantaloons, his pantaloons cost 6 dollars less than his coat: what was the cost of the pantaloons? What was the cost of the coat 86. James, John, and Charles together have 150 marbles; Charles has X as many as John; if J ohln gives Charles 30 of his, they will all have the dinme ullh, " how many has each? 172 INTELLECTUAL ARITHMETIC. [SEC. VII. it in'2i- hours, and the other in 3U-: in what time will they fill it running together? S8. A person hired a man and two boys. To the nman he gave six shillings a day, to one boy four, and to the other three; at the. end of the time he paid them 104 shillings: how long did they work? 89. A cask of wine leaked out one quarter, after which one third of the remainder was drawn, when the cask was found to contain 30 gallons: how much did the cask hold? 90. A market woman bought a certain number of eggs at 3 for 2 cents, and an equal number at 5 for 4 cents. She paid for both lots 44 cents: how much did her eggs cost her apiece, and how many did she buy? 91. A market woman bought a certain number of eggs at the rate of 4 for 3 cents, and sold them at the rate of 5 for 4 cents, by which she made 4 cents. What did she pay apiece for the eggs? What did she make on each egg sold? How many did she sell to make 4 cents? 92. A market woman bought 36 fowls, of three different sorts, for which she paid 84 shillings. She bought half as many of the first sort as of the second, and three times as many of the third sort as of the first, and paid 1, 2, and 3 shillings apiece for each sort: howii many of each did she buy? 93. If James can weed his father's onions in 9 hours, and John in 12 hours, how long will it take both, working together, to weed them? 94. A lady wishes a dress, and does not know'~ hether to buy silk or muslin. The silk costs 9 shil lings a yard, and the muslin 3. If she purchases the silkxE will cost 72 shillings more than the muslin: l..\....-.... vanrrl Ai;d1 sheAP need? LfS8. XVIII.] INTELLECTUAL ARIT'IMETIC. 173 lhe worked he wvas to receive 5 dimes, tnd for every day he played he was to pay 2 dimes tvr his board. When he came to settle he received 87 dimes: how many days did he work 96. How many small cubes, 1 inch on a side, man be sav ed out of a cubic block, 2 feet on a side, allow. ng ne waste in saw'in.'97. If A caln do Iof a piece of work in 2 days, and 13 can do ~1 of it i-4-: days, how much of it can each do in 1 day, and hWw long will it take both to do it, working together i 98. A man having a goose, pig, and calf, was asked the value of them. He said that the three were worth 30 shillings, that the goose was worth one third as much as the pig, and that the calf was vworth 1-4 times as much as the goose and pig togethier: what vwas the value of each? 99. James is 10 of his own paces behind John, and in pursuit of him. James steps 3 times,,lwhile John steps 4 times; but James' steps are twice as long as John's: how many steps must James make to over take him? 100. Two families bought a barrel of flour tog, ther, for which they paid $85 and agreed that each child should count half as much as a grown person. In olle family, there were 3 grown persons and 3 childreln; atld in the other, 4 growln persons and 10 children; the first family fed from the flour 2 weeks, a1nd the seoond 3: how much ought each to pay? 15*