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15. What is the difference between 91 and 23.
91 = 912
23 = 21/2
SOLUTION. - We first reduce the fractional parts, and, to a common denominator, 12. Since we cannot take from, we add 1 = to, which makes 19, and 11⁄2 from 11⁄2 leaves. We now add 1 to the 2 in the subtrahend (50). Three from 9 leaves 6. thus obtain 6171⁄2, the difference required.
subtract mixed numbers, reduce the frac
tional parts to a common denominator, and then subtract the fractional and integral parts separately. Or,
Reduce the mixed numbers to improper fractions, and subtract the less from the greater by the usual method.
22. Find the difference between 49 and 754.
23. Find the difference between 2275 and 1963. 24. From a cask of oil containing 31 gallons, 178 gallons were drawn. How many gallons remained? Ans. 137. 25. A farmer, having 450 acres of land, sold 3043 How many acres had he left? Ans. 14518. 26. If flour is bought for $61 per barrel, and sold for $73, what will be the gain per barrel?
27. From the sum of and 31, take the difference between 4 and 51. Ans. 32.
28. What number added to 25 will make 74 ? Ans. 435. 29. What fraction added to 11⁄2 will make 18? Ans. go
135. To multiply a fraction by an integer.
1. If 1 yard of cloth costs of a dollar, how much will 5 yards cost?
2. If 1 gallon of molasses costs of a dollar, how much will 5 gallons cost?
2 × 5= 4 = 13, Ans.
SOLUTION. Since 5, the multiplier, is a factor of 20, the denominator of the multiplicand, we perform the mul tiplication by dividing the denominator, 20, by the multiplier, 5, and we have, equal to 12 dollars. A fraction is multiplied by dividing its denominator (124).
RULE. To multiply a fraction multiply its numerator, or divide its denominator.
Always divide the denominator when it is exactly divisible by the multiplier.
13. If a man earns $8% a week, how many dollars will he earn in 12 weeks?
What will 9 yards of silk cost at of a dollar per yard?
What will 27 bushels of barley cost at of a dollar per bushel? Ans. $235.
136. To multiply an integer by a fraction.
1. At $75 an acre, how much will of an acre of land
Multiplying by a fraction consists in multiplying by the numerator and dividing by the denominator of the multiplier.
7. Multiply 42 by 93.
9. Multiply 156 by 7.
19. At $8 a bushel, what will of a bushel of clover seed cost?
11. If a man travels 36 miles a day, how many miles will he travel in 10 days? Ans. 384 miles. 12. If a village lot is worth 450 dollars, what is of it worth? Ans. 262 dollars. 13. At 16 dollars a ton, what is the cost of 23 tons of hay?
137. To multiply a fraction by a fraction.
1. At of a dollar per bushel, how much will & of a bushel of corn cost?
SOLUTION.-Dividing of a dollar by 4, we have, the cost of of a bushel. A fraction is divided by multiplying its denominator (124). Multiplying the cost of of a bushel by 3, we have
of a dollar, the cost of of a bushel. It will readily be seen that we have found the product of the two numerators, 2 and 3, for a new numerator, and of the two denominators, 3 and 4, for a new denominator.
RULE. I. Reduce all integers and mixed numbers to improper fractions.
II. Find the product of the numerators for a new numerator, and of the denominators for a new denominator.
Cancel all factors common to numerators and denominators.
Ans. 18 Ans. 11. of of },
6. What is the product of, 4, 5, and 1?
by 4 of 1g?
-Canceling like factors in the numerators and denominators, we have left in the numerator 7, and in the denominator 5 × 6 = 30, making for the product.
Fractions with the word of between them are sometimes called compound fractions. The word of is simply an equivalent for the sign of multiplication, x.
of 21 by 1 of 71.
of 16 by 7 of 263.
11. What is the sum of the products of 3, of 4, and
of 8? times
12. What is the value of 24 times of of 14? 13. What is the value of 7 of 1 of 12 times 14. What is the value of of of of 3 of 9 ? 15. What is the product of 12 multiplied by 5 times
16. At of a dollar per yard, what will of cloth cost?
of a yard
Ans. of a dollar.
17. If a man owns of a vessel, and sells
share, what part of the whole vessel will he sell? 18. When oats are worth of a dollar per bushel what is of a bushel worth?
19. What will 73 pounds of tea cost, at of a dollar per pound? Ans. $418.
20. What will 5 yards of cloth cost at of a dollar a yard? Ans. $3.
21. If a man owns of a plot of ground and sells of his share, what part of the plot does he sell?