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SUBTRACTION.

MENTAL EXERCISES.

$ L. 1. A boy had of a quart of chesnuts, and gave away . How many eighths had he left ?

2. A man bought so of a barrel of flour, and gave i to some labourers. How much had he left?

3. Two boys together gathered 7 of a peck of strawberries, and one of them took g of a peck, as his share. How many had the other ?

4. from leaves how many ninths ? 5. from leaves how much? : from ? * from 11? from 17 ? from 2 ? from 87%? from 57 ? 1. from 63? from 47?

But we shall often be required to make subtractions between Fractions whose denominators are different. In this case we may reduce them to a common denominator, and then we can subtract as above.

6. Take from 1. is, and is
7. Take ţ from from . 1 from . } from 16.
8. Take 11 from 24. 24 from 57.. 3; from 51.
Let the following be written.
9. From i take.
10. From take. A. H.
11. From take 37. A.
12. From 26 } take 113.

A. 1,92

. NOTE. When mixed numbers occur, perform the subtraction of the whole, and fractional parts separately, if it can be done.

13. From 2013 take 6. A, 1415. 14. From 3711 take 204. A. 17. 15. From 309 take 15]. A. 157. 16. From 21 take 6. A, 14. NOTE. When the Fraction in the subtrahend is greater than that in the minuend, it will be necessary to borrow a unit. Sometimes, likewise, there will be no Fraction in the minuend. In this case, it will also be necessary to borrow.

17. From 22571 take 711), A. 153; 14. 18. From 621 take 497. A. 1231.. 19. From 5251 take 326,11. A. 1981 19. 20. From 2,983 11 take 1,843348. A. 1,13942333. 21. From 214489 take 13317964. A. 893S Hence, to perform Subtraction of Fractions,

A. 13

139

REDUCE THE FRACTIONS TO A COMMON DENOMINATOR, FIND THE DIF. FERENCE OF THE NUMERATORS, AND WRITE IT OVER TÜE COMMON DE.

777 T767

NOMINATOR.

22. A man bought 27; yds. of cloth, and had 19. yds, of it made into clothes. How much was left?

A. 84 yds. 23. A merchant had 56,877 gals. of brandy in a cask, and 7 gals. leaked out. How much had he left?

A. 4947. gals. 24. A man had a lot containing 97} acres of ground, from which he fenced off 147 acres. How many acres were left in the lot ?

A, 8131 Acres.

DIVISION.

MENTAL EXERCISES.

In ?

LI. 1. At } of a dollar a yard, how many yards of calico can be bought with of a dollar? Hom many with ? With A ? With s? With '; ?

With ? ? 2. At of a dollar a bushel, how many bushels of oats can be bought for 4 of a dollar ? How many for ? For $? For ?

3. 1 is contained in i how many times ?
4. are in how many times ? In ?
5. } are in how many times ? In ?
6. & are in how many times? In ?
7. i are in i how many times? In li ? În iu

The pupil will see, that he has been finding how often one Fraetion is contained in another; that is, he has been dividing a Fraction by a Fraction. In the above examples, the denominators are the same. If they were not the same, they might be made so by 9 XLIV. Hence, to divide a Fraction by a Fraction,

I. REDUCE THE FRACTIONS TO A COMMON DENOMINATOR, AND DIVIDE THE NUMERATOR OF THE DIVIDEND BY THAT OF THE DIVISOR.

8. Divide i by š. The least com. den. ($xlv.) is 8. The Fractions, then, are j and. Then, 6:3=2 Ans.

9. Divide by 1. Frac. reduced fi and S. 49- 49

49. Ans.

10. Divide by bys by . by 4.

It will be observed that no use, whatever, is made of the Common Denominator, and that this denominator, is, in fact, lost, in the an. swer. It is very manifest, then, that, it is of no use to find this dee Dominator itself, if we only find the numerators, AS THOUGH we were reducing the fractions to a common denominator. For these numer. ators are the only numbers concerned in the process, Thus,

11. Divide 17 by z's. Find the numerators as though for a common denominator; which is done ($ xliv.) by multiplying 35 into 2=70, and 27 into 1=27. Then 702-27=2=214 Ans.

The answer obtained, is evidently the same we should have found, if we had turned over, or inverted the Divisor, and then, multiplied the two upper numbers together for a numerator, and the two lower for a denominator. 3=11=214 as before. But this process after inverting the Divisor, is exactly like the rule for Multiplication of Fractions. (Š xlix.) Hence, to divide a Fraction by a Fraction,

II. INVERT THE DIVISOR AND PROCEED AS IN MULTIPLICATION.
12. Divide 1: by 18. Ans. Til=1.
13. Divide by 21

Ans. 14. Divide 1873 byg. Ans. on=2007 15. Divide 920's by : . 16. Divide 2 by j. Ans »=6. NOTE. When a mixed number occurs, it is to be reduced to an improper Fraction. (S XXVII.)

17. Divide 53 by 84. Ans. iii.
18. How often is 2 to contained in 31.

Ans. i=i=1,%. 19. How often is 6 contained in 131. 6 may be made an improper Fraction by placing 1 under it, thus f.

Ans =V=2. Note. It will be best usually to reduce the divisor, and, sometimes the divid. end to its lowest terms before dividing:

20. Divide is by 14. 15. Ans. j=1}. 21. Divide iby i=. Ans. =2.

A. 1

6 7 338 861044 30 36 36 4 6 3 2 301

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MENTAL EXERCISES.

À LII. 1. A man bought cotton cloth, to the amount of 5 dollars, at į of a dollar a yard. How many yards did he buy ?

?. A boy bought marbles at į a cent apiece, to the amount of 12 cents. How many marbles did he buy?

3. How many fifths are there in 4? In 5? 4. How many sevenths are there in 3? In 4? 5. A man put 6 barrels of beer into kegs that held 3 of a barrel each. How many kegs did it take?

6. How many times ? in 10 ? in 9 ? in 8? in 4 ? in 6 ?

7. How often is fin 6? in 5? in 3? 4. in 6 ? 8. How often is ; in 11 ? , in 9? 4 in 7? 4 in 4? In the above examples, it is required to find how often a Fraction is contained in a whole number, that is, to divide a whole number by a Praction. The quotient will be seen to be larger than the dividend, and it ought so to be whenever a proper Fraction is the divisor; for (XLI.) the smaller the divisor, the greater the quotient. There are twice as many half-pints in a pail of water, as there are pints. If I divide & whole number by 1, the quotient will be the same whole number. Thus 1 is in 8, 8 times; 1 is in 12, 12 times, &c. If I divide by a number less than 1, then, I ought to obtain a greater quotient. Thus ) is in 8, 16 times ; is in 12, 24 times, &c.; in which cases, I have a quotient, twice as great as my dividend.

9. At of a dollar a piece, how many spools of thread can be bought with 2 dollars?

First find the ninths in 2=49. Then say , are in 18, 9 times. Ans.

Hence, to divide a whole number by a Fraction,

MULTIPLY THE WHOLE NUMBER BY THE DENOMINATOR OF THE FRAC. TION, AND DIVIDE THE PRODUCT BY THE NUMERATOR.

11. Divide 24 by it. It is best to reduce the Fraction to its lowest terms, ti,=4. Ans. =28.

12. Divide 84 by on. Ans. 90%. 13. Divide 22 by 2 =V. Ans. 8.

NOTE.The divisor, if a mixed number, must always be reduced to an improper Fraction.

21. Divide 37 by 43A. 8 P. 53 by 8. A. 64. 127 by 9. 948 by 84 1,847 by 15 23

To this case belong such questions as these; 4 is of what number; 6 is of what number, &c.; that is, all questions where a certain part is given, to find a whole.

14. 125 is of what number? Ans. 225.

Perhaps it may be well for beginners, to accustom themselves to solve these examples by analysis, thus. If 125 is 5 ninths, what is one ninth ? Ans. 25. If 25 is }, what is ? Ans. 225.

15. 178 is of what number? Ans. 623.
16. 256 is of what number? Ans. 352.
17. 1,874 is is of what number? Ans. 14,055.
18. 29,864 is of what number? Ans. 175,451.
19. 425,763,891 is gids of what number?

Ans. 23,444,046,633 20. 7,587,648 is 24 of what number? 21, 1,343,826 is fit of what number?

MENTAL EXERCISES.

LIII. 1. A lady, having of a yard of ribbon, cut it into 5 equal parts. How much was there in each part?

2. Thomas had f of a pint of chesnuts, and distributed them equally among 4 of his companions. How many had each?

3. A man divided / of an acre of ground into 7 equal parts. How much was there in each part ?

4. A boy divided of an orange among 5 of his companions. How much did he give each?

5. A grocer put of a hogshead of brandy into 6 casks, putting an equal quantity into each. How much did he put in each?

6. Divider by 6. by 3. by 5. 4 by 4. 4 by 2. by 8. by 4. by 2. i by 11. by 2. by 4. } by 3. by 4. i? by 6. 13 by 4.'13 by 3. 1 by 2.

In the above examples, a whole number is the divisor, and a fraction, the dividend, that is, it is required to divide a fraction by a whole number. Let the following be written.

7. Divide you by 3. Dividing the numerator divides the value. ($ xli.) Ans

.. 8. Divide by 27. A. IT: 145 by 9. A.

9. Divide 3 by 2. 2 will not divide the numerator, but ($ xli.) multiplying the denominator divides the value. A. 2017

10. Divide 314 by 4. A. 08. by 5. A. 75. Hence the rules,

I. DIVIDE THE NUMERATOR OF THE FRACTION BY THE WHOLE NUM. BER.

II. MULTIPLY THE DENOMINATOR OF THE FRACTION BY THE WHOLE NUMBER.

11. Divide 789-64-37, by 7,896,437.- A. 77675777 12. Divide by 8.

8=4X2. We may divide by these factors successively. ($ xxix.) 1 ::-2=25. is to 4=1Ans. This method is often convenient.

13. Divide 373 by 9. A. Tito 1114 by 32. A. Tito. 344 by 81. A. zł

14. Divide 14 ; by 7. Divide the whole number and Fraction separately. Ans. 2.

15. Divide 575 by 25. Ans. 23 zit. 16. Divide 65 } by 7. If you divide the whole number by 7, you obtain

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OR,

16

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