126. REDUCTION OF Whole or MIXED NUMBERS TO IMPROPER FRACTIONS.

ILL. EX. Change 27 to an improper fraction.

RULE. To reduce a mixed number to an improper fraction;— Multiply the whole number by the denominator of the fraction; to that product add the numerator; and write the result over the denominator.

10. Reduce 584, 368, 875, to ninths.

11. Add 784 to 916, and express the answer in sevenths.

12 Reduce 7 X 98 to eighths.

13. Reduce (15—8) × 16 to fifths.

14. Reduce 8692 to a fraction whose denominator is 25. 15. Reduce 367 to an improper fraction.

16. Change 4567§ to ninths, 43862 to elevenths. 17. Change 36864 to an improper fraction.

For Dictation Exercises, see Key.

127. REDUCTION OF IMPROPER FRACTIONS TO WHOLE OR MIXED NUMBERS.

ILL. EX.

OPERATION.

29 = 45, Ans.

times.

Change 29 to a mixed number.

Hence the

There are in 1 unit, in 22 there are as many units as & is contained times in 22, which is 48

RULE. To reduce an improper fraction to a mixed number; Divide the numerator by the denominator.

328

For Dictation Exercises, see Key.

128. MULTIPLICATION OF FRACTIONS BY WHOLE

NUMBERS.

As multiplying the numerator of a fraction multiplies the number of parts, their size remaining the same, and dividing the denominator multiplies the size of the parts, their number remaining the same (Art. 119), it follows that,

To multiply a fraction by a whole number, we may either multiply the numerator by the whole number, or divide the denominator.

The latter method is preferable when the denominator can be divided without a remainder, as it gives the answer in lower

terms.

We might have cancelled in the first operation, and thus have

obtained the same result as in the second; thus,

7X4

7

$

72

17. If one yard of cloth costs of a dollar, what will 17 yards cost?

18. If a ton of coal costs of an Eagle, how much will 15

tons cost?

19. Required the cost of 28 pounds of candles, at of a dollar a pound.

20. Multiply 2567 by 18.

21. Multiply 37633 by 21.

For Dictation Exercises, see Key.

Ans. 46233.

129. MULTIPLICATION OF WHOLE NUMBERS BY FRACTIONS.

ILL. EX. Multiply 8 by §.

X 5

OPERATION.
4 X 5

3

3

8 multiplied by 5 is 5X 8; if it is =63, Ans. multiplied by §, a number one sixth as large as 5, the product must be one sixth as large as if 5 had been the mul

tiplier, or of 5 × 8. The expression then becomes 8 times 5, divided by 6; after cancelling, 4 X 5 20

3

3

= =63, Ans. Hence the RULE. To multiply a whole number by a fraction; — Multiply the whole number by the numerator of the fraction, and divide that product by the denominator.

3. 385 by 25.

8

6. 87036 by 27.

7. What cost of 1 ton of hay, at $12 a ton?

OPERATION.

8. What cost 9. What cost 10. What cost 11. What cost 5 12. What cost 38

If 1 ton of hay costs $12, of a ton will cost of $12, and § of a ton will cost 5 times of $12. Cancelling, we have $1.50 X 5: = $7.50, Ans.

of an acre of land, at $100 an acre?
of an acre of land, at $150 an acre?
of a gross of pens, at $.96 a gross?
cords of wood, at $7.56 a cord?
hogsheads of molasses, at $18.80 a hogshead?

13. What cost 25 firkins of butter, at $12.60 a firkin? 14. What cost 63 yards of flannel, at $.54 a yard?

For Dictation Exercises, see Key.

130. MULTIPLICATION OF FRACTIONS BY FRACTIONS. ILL. Ex. Multiply by .

OPERATION. 4

5X8 20

6X9 27

3

Ans.

multiplied by 8, is 8 times; if it be multiplied by, a number one ninth as large as 8, the product must be one ninth as large as if 8 had been the multiplier, or one ninth of 8 times. 5 X 8 after cancelling, 3x9 6X9;

The expression then becomes
Hence the

5 X 4 20
27'

Ans.

RULE. To multiply a fraction by a fraction;- Multiply the numerators together for a new numerator, and the denominators

for a new denominator.

NOTE.-Reduce mixed numbers to improper fractions before multiply.

ing by fractions.

boxes of raisins, at 1

dollars a box?

of a dollar a pound? at 2% of a dollar a

15. What cost 103 tons of coal, at $71⁄2 a ton? 16. What cost 5 pounds of coffee, at 17. What cost 17 pounds of pork, pound?

For Dictation Exercises, see Key.

131. REDUCTION OF COMPOUND FRACTIONS TO SIMPLE

FRACTIONS.

ILL. Ex., I. If 1 dollar buys of a yard of cloth, how much will of a dollar buy?

of a dollar will buy of of a yard.

of

of a yard is

of a yard,

there

which may be shown by dividing of a yard into 4 equal parts. The whole yard can thus be divided into 4 X 5, or 20 equal parts; fore, 1 part will be of the whole yard. If of =26, of & muse be, and of must be 3 times, or 2.

Ans.

4 X 5 20

Ans.

of is a compound fraction (Art. 116), and is reduced to a simple fraction by multiplication. Let the explanation be similar to the second explanation of the illustrative example.

5. of 2 of of 31

7 8

6. 11 of 3 of 113 of 60 — what?

For Dictation Exercises, see Key.

132. DIVISION OF FRACTIONS BY WHOLE NUMBERS.

As dividing the numerator of a fraction diminishes the number of parts, their size remaining the same, and multiplying the denominator diminishes the size of the parts, their number remaining the same (Art. 119), it follows, that

To divide a fraction by a whole number, we may either divide the numerator by the whole number, or multiply the denominator.