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Subtract the number denoting the numerator from that denoting the denominator, under the remainder write the denominator, and adding one to the whole number in the subtrahend, subtract the sum from the minuend.

NOTE. When the subtrahend is a mixed number, we may reduce it to an improper fraction, and change the whole number in the minuend to a fraction having the same denominator, and then proceed as in Art. 230.

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Ex. 1. From 8 take 4.

FIRST OPERATION.

=

From 83
Take 44

=

Rem.

15

Ans. 333.

We first reduce the fractional parts to a com815 mon denominator, and obtain as their equivalents and. Now, since we cannot take 3 from 1, we add 1, equal to 3, to the in the minuend, and obtain 5g. From taking, we have left, which we write below the line, and carry 1 to the 4 in the subtrahend, and subtract from the 8 above as in subtraction of simple whole numbers.

SECOND OPERATION.

=

From 83
Take 44

=

59
24

295
35

Rem.

35

= 168
35
127 =

In this operation, we reduce the mixed numbers to improper fractions, and these fractions to a common denominator. We 33 then subtract the less fraction from the greater, and, reducing

the remainder to a mixed number, obtain 33, as before. Hence, in performing like examples, we may

Reduce the fractional parts, if necessary, to a common denominator, and subtract the fractional parts and the whole numbers separately. Increase the fractional part of the minuend, when otherwise it would be less than the subtrahend, before subtracting, by as many parts as it takes to make a unit of the fraction (Art. 208), and carry 1 to the whole number of the subtrahend before subtracting it. Or,

Reduce the mixed numbers to improper fractions, then to a common denominator, and subtract the less fraction from the greater.

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14. From a hogshead of wine there leaked out 79 gallons; what quantity remained?

Ans. 55 gal.

15. A man engaged to labor 30 days, but was absent 511⁄2. days; how many days did he work?

16. From 144 pounds of sugar there were taken at one time 17 pounds, and at another 28 pounds; what quantity remains? Ans. 97 lb.

17. A man sells 9 yards from a piece of cloth containing 34 yards; how many yards remain ? Ans. 24дyd.

18. The distance from Boston to Providence is 40 miles. A, having set out from Boston, has travelled of the distance; and B, having set out at the same time from Providence, has gone of the distance; how far is A from B?

Ans. 28m. 19. From of a square yard take of a yard square.

233. To subtract one fraction from another, when their numerators are alike.

Ex. 1. From take 4.

7 3

=

7 X 3

=

OPERATION.

Ans.

4, difference of the denominators.

21, product of the denominators.

We first find the product of the denominators, which is 21, and then their difference, which is 4, and write the former for the denominator of the required fraction, and the latter for the numerator. By this process the fractions are reduced to a common denominator, and their difference found. Hence, to subtract one fraction from another, whose numerators are a unit, we may

Write the difference of the denominators over their product.

2. Take from 3.

OPERATION.

Ans. T

Difference of the denominators mul

tiplied by one of the numerators, (7 — 3) × 2 Product of the denominators,

3 x 7

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We multiply the difference of the denominators by one of the numerators for a new numerator, and the denominators together for a new denominator, by which process the fractions are reduced to a common denominator, and the difference of their numerators is found. Hence, when the given fractions have their numerators alike and greater than a unit, we may

Write the product of the difference of the given denominators, by one of the numerators, over the product of the denominators.

EXAMPLES.

from 1, 1, 1, 1, 1, 4; 1' from 1's, 16, 1‍5.
from 1, 1, 1, 1; 73 from 1, 3, 1, †, §.

3. Take

4. Take

5. Take

from 1, 1, 1; † from 1, 1, 1, 1.

6. Take

from 16, 1, 3, 4, 1, 8, 4, §, §.

7. Take

from ; from 1⁄2, }; † from §.

8. Take

9. Take

10. Take

11. From

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take ; take 20; 11⁄2 take 1; 1' take

12. Take from ; & from ; & from ; & from 3.

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MISCELLANEOUS EXAMPLES IN ADDITION AND SUBTRACTION OF FRACTIONS.

of a

1. A benevolent man has given to one poor family cord of wood, to another of a cord, and to a third of a cord;

how much has he given to them all?

Ans. 2

cords.

2. I have paid for a knife $ §, for a Common School Arithmetic $, for a slate $, and for stationery $ §; what did I pay for the whole ?

3. R. Howland travelled one day 20 miles, another day 19 miles, and a third day 22 distance travelled?

miles; what was the whole.

Ans. 62 miles.

4. I have bought 6 tons of anthracite coal, 19 tons of Cumberland coal, and 33 tons of cannel coal;

quantity purchased?

5. There is a pole standing the remainder above the water; water?

what is the whole Ans. 30 tons.

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6. F. Adams, having a lot of sheep, sold at one time of them, and at another time of the remainder; what portion of the original number had he then left? Ans..

7. From a piece of calico containing 31 yards there have been sold 11 yards, 9 yards, and 3 yards; how much remains?

8. From a cask of molasses containing 843 gallons, there were drawn at one time 4 gallons, at another time 11 gallons; at a third time 26 gallons were drawn, and 1⁄2 of 71⁄2 gallons returned to the cask; and at a fourth time 13 gallons were drawn, and 3 gallons of it returned to the cask. How much then remained in the cask? Ans. 35gal.

9. A merchant had 3 pieces of cloth, containing, respectively, 192 yards, 36 yards, and 33ğ yards. After selling several yards from each piece, he found he had left in the aggregate 713 yards. How many yards had he sold? Ans. 18.

MULTIPLICATION OF COMMON FRACTIONS.

234. MULTIPLICATION of Fractions is the process of multiplying when the multiplier, or multiplicand, or both, are fractional numbers.

NOTE. If the multiplier is less than 1, only such a part of the multiplicand is taken as the multiplier is of 1. Therefore, the product resulting from multiplying a number by a proper fraction is not larger, but less, than the multiplicand.

235. To multiply when one or both of the factors are fractions,

Ex. 1. Multiply by 9.

FIRST OPERATION.

18 X 9 93 = = 34 Ans.

18

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It is evident that the fraction

is multiplied by 9 by multiplying its numerator by 9, since many as before, while the parts

the parts taken, 63, are 9 times as
into which the unit of the fraction is divided remain the same.

SECOND OPERATION.

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It is evident, also, that the fraction

3 Ans.is multiplied by 9 by dividing its denominator by 9, since the parts into

which the unit of the fraction is divided are only as many, and consequently 9 times as large, as before, while the parts taken remain the same. Therefore,

Multiplying the numerator or dividing the denominator of a fraction by any number multiplies the fraction by that number (Art. 217).

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SECOND OPERATION.

14. 3

42÷7

= 6 Ans.

of 14

6, as before.

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By multiplying the whole number, 14, by 3, the numerator of the fraction, we obtain 42, a product 7 times as large as it should be, as the multiplier was not 3, a whole number, but, or 37; hence, we divide the 42 by 7; and thus obtain Therefore,

Multiplying by a fraction is taking the part of the multiplicand denoted by the multiplier.

3. Multiply by %.

OPERATION.

× Ans.

=

Ans. .

To multiply by is to take of the multiplicand, 7. Now, to obtain of 7, we multiply the numerators together for a new numerator, and the denominators together for a new denominator (Art. 226). Therefore,

Multiplying one fraction by another is the same as reducing compound fractions to simple ones.

When either of the factors is not a fraction, as in examples first and second, it may be reduced to a fractional form, and then the operation may be like that in the last example. Hence the general

RULE. Reduce whole or mixed numbers, if any, to improper fractions. Multiply the numerators together for a new numerator, and the denominators together for a new denominator.

NOTE. When there are common factors in the numerators and denominators, the operation may be shortened by cancelling those factors.

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