135. Therefore: When there are mixed numbers, change both to improper fractions, and subtract as in Art. 134: Or, subtract the integral and fractional numbers separately.

11. From 847 take 16. | 13. From 7 take 4

12. From 2463 take 1643.

=

and 71 from, we

NOTE.-Since we cannot take borrow 1, or 21, from the minuend, which, added to; then, from 27, leaves 2. We must now carry 1 to the next figure of the subtrahend, and proceed as in subtraction of simple numbers.

14. From 16 take 58. 16. From 363 take 27.

15. From 26 take 197.

17. From 400 take 3275.

18. From the fraction, take the fraction

OPERATION.

==

NOTE. When the numerators are 1, the difference of the two fractions is equal to the difference of the denominators divided by their II 11X8 product.

8

88

11-8

3

88°

19. What is the difference between and ? Between and? and? and? and?

and ?

MULTIPLICATION.

136. MULTIPLICATION OF FRACTIONS is the operation of taking one number as many times as there are units in another, when one or both of the numbers are fractional.

CASE I.

137. To multiply a fraction by a whole number.

1. If one yard of cloth cost of a dollar, what will 4 yards cost?

ANALYSIS. Four yards will cost 4 times as much as 1 yard. Since 1 yard costs 5 eighths of a dollar, 4 yards will cost 4 times 5 eighths of

OPERATION.

=

8

5X45X4 = 20 = 21.

a dollar, which is 20 eighths: therefore, if 1 yard cost of a dollar, 4 yards will cost 202 dollars.

2d. If we divide the denominator by 4, the fraction will be multiplied by 4 (Prop. II.): performing the operation, we obtain, which = 21: Hence,

OPERATION.
5

× 4 =

=

To multiply a fraction by a whole number,

Multiply the numerator, or divide the denominator.

135. When there are mixed numbers, how do you subtract? Explain the case when the fractional part of the subtrahend is the greater.

136. What is Multiplication of Fractions?

137. What is Case I.? What is the rule?

7. If 1 dollar will buy § of a cord of wood, how much will 15 dollars buy?

8. At cost?

of a dollar a pound, what will 12 pounds of tea

9. If a horse eats of a bushel of oats in a day, how much will 18 horses eat?

10. What will 64 pounds of cheese cost, at of a dollar a pound?

11. At 2 cents a pound, what will 8 pounds of chalk cost?

NOTE. When the multiplicand is a mixed number, multiply the fraction and integer separately, and add the results; or, reduce the mixed number to an improper fraction, and multiply.

12. If a man receives 31 dollars a day, how much will he receive in 15 days?

13. If a family consumes 54 barrels of flour in 1 year, how much would they consume in 9 years?

CASE II.

138. To multiply a whole number by a fraction.

1. At 15 dollars a ton, what will

of a ton of hay cost?

ANALYSIS.-1st. Four-fifths of a ton will cost 4 times as much as 1 fifth of a ton; if 1 ton cost 15 dollars, 1 fifth will cost of 15 dollars, or 3 dollars, and will cost 4 times 3 dollars, which are 12 dollars.

Or, 2d. 4 fifths of a ton will cost 1 fifth of 4 times the cost of 1 ton; 4 times 15 is 60, and 1 fifth of 60 is 12: Hence,

Rule.

Divide the whole number by the denominator of the fraction, and multiply the quotient by the numerator :

Or, Multiply the whole number by the numerator of the fraction, and divide the product by the denominator. NOTE.-Cancel, when possible.

6. If an acre of land is valued at 75 dollars, what is of it worth?

9

7. If a house is worth 320 dollars, what is 1% of it worth?

8. If a man travels 46 miles in a day, how far does he travel in of a day?

9. At 18 dollars a ton, what is the cost of of a ton of hay?

10. If a man earn 480 dollars in a year, how much does he earn in of a year?

CASE III.

139. To multiply one fraction by another.

1. If a bushel of corn costs of a dollar, what will of a bushel cost?

Multiply the numerators together for a new numerator, and the denominators together. for a new denominator.

NOTES.-1. When the multiplier is less than 1, we do not take the whole of the multiplicand, but only such a part of it as the multiplier is of 1.

2. When the multiplier is a proper fraction, multiplication does

138. What is Case II.? 139. What is Case III.?

What is the rule?
What is the rule?

not imply increase, as in the multiplication of whole numbers. The product is the same part of the multiplicand which the multiplier is of 1.

3. If the multiplicand or multiplier, or both, be whole or mixed, the whole number may be reduced to a fractional form, and the mixed numbers reduced to improper fractions; and then the last rule will apply to all examples.

Examples.

1. Multiply by .

2. Multiply by 15.

3. Find the product of,, and 7.

4. Find the product of,, and

5. If silk is worth of a dollar a yard, what is of a yard worth?

6. If I own of a farm, and sell of my share, what part of the whole farm do I sell?

7. At of a dollar a pound, what will of a pound of tea cost?

8. If a knife costs of a dollar, and a slate as much, what does the slate cost?

General Examples.

of 3, by 14. | 4. Mult. 5 of 3 of 3, by 41. 5. Mult. 14 of 5 of 9, by 63.

1. Mult.of

2. Mult.

by of 11.

3. Mult. of 3, by of 15. 6. Mult. of 6 of, by of 4.

139. How do you multiply one fraction by another? When the multiplier is less than 1, what part of the multiplicand is taken? If the fraction is proper, does multiplication imply increase? What part is the product of the multiplicand?