MULTIPLICATION OF FRACTIONS.

173. Multiplication of Fractions is the process of finding a product when one or both factors are fractions. 174. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 3d. A fraction by a fraction.

CASE I.

175. To multiply a fraction by an integer.

MENTAL EXERCISES.

1. How many are 5 times?

SOLUTION.-5 times are 8, which reduced to its lowest terms equals 3. Therefore, etc.

SOLUTION 2d.-5 times equal, or; if 5 times equal }, 5 times A equals 2 times or; hence 5 times equals 3.

10. Since 5 times equals, how may this result be obtained by omitting the analysis?

11. How then may a fraction be multiplied by omitting the analysis?

WRITTEN EXERCISES.

1. Multiply by 4

SOLUTION.-Multiplying the numerator (Prin. 1),

we have 4 times equals 1; or, dividing the denominator (Prin. 4), we have 4 times, or 34.

OPERATION.

Hx4=11 11x4=4

Rule. To multiply a fraction by an integer, multiply

the numerator, or divide the denominator.

10. In multiplying 12 by, by what do we divide 12, and by what do we multiply the quotient?

11. How then may we multiply a whole number by a fraction?

WRITTEN EXERCISES.

1. Multiply 7 by .

SOLUTION.-7 multiplied by one equals 7, hence 7 OPERATION. multiplied by equals of 7, which is t, and 7 multiplied by equals 4 times, which equals 28. Therefore, etc.

2. Multiply 28 by 73.

SOLUTION.-Multiplying 28 by 3 and dividing by 5, we have 28x equals 163; and then multiplying 28 by 7, we have 196; adding 196 to 16, we have 212§. Therefore, etc.

OPERATION.

28

78

5)84

164

196

212 Ans.

NOTE.-This method of multiplying by a mixed number is more convenient than the one usually presented.

Rule.-Multiply the integer by the numerator of the mul tiplier, and divide the product by the denominator; or divide first and then multiply.

CASE III.

177. To multiply a fraction by a fraction.

MENTAL EXERCISES.

1. What is the product of by ?

SOLUTION.- multiplied by one equals, hence & multiplied by equale of, which is, and multiplied by equals 3 times, which is H Therefore, etc.

10. In multiplying

9. Of 5 by 4}?

by, what do we multiply together for the

numerator of the product and what for the denominator? 11. How then do we multiply a fraction by a fraction?

WRITTEN EXERCISES.

1. What is the product of by ?

SOLUTION.- multiplied by one equals, hence multiplied by equals of, which is, and multiplied by equals 7 times, which is. Therefore, etc.

OPERATION.

{x1=4

SOLUTION 2D.-times equals of 5 times 7; 5 times equals (Prin. 1); and of equals (Prin. 3).

Rule.-Multiply the numerators together and the denomi nators together, cancelling common factors.

NOTE.-Practically this case is the same as finding a fractional part of a fraction, but theoretically the two cases are entirely distinct.

63

Ans.

8. 4 × 8 × H? Ans. II. Ans. . 9. 18×88×??

Ans. 10. 12×8?

24.

Ans. 1.

11. 177 × 163?

Ans. 1. 12. 27×21?

Ans. 7. 13. 39 × 331.

14. Multiply% of 18 by 4 of 14.

Ans. 1.

Ans. 104.

Ans. 290.

Ans. 595 †

Ans. 1320.

Ans..

Ans. 1.

Ans. 2210.

MENTAL EXERCISES.

1. How much will 4 yards of muslin cost, at 6 cents a yard? 2. If you give to 8 boys each 24 apples, how many apples will it take?

3. How many dollars does that man give away who gives to each of 10 beggars of a dollar?

4. How much will 15 inkstands cost at the rate of 2 inkstands for of a dollar?

5. What will 12 Letter-writers cost at the rate of 3 for of a dol Ir?

6. If a man can dig a rod of ditch in 7 of a day, how many rods can he dig in 4 days?

7. Multiplying by † is the same as taking what part of a number? 8. Show that a fractional part of an integer equals the product of the integer by the fraction.

9. What will be the cost of 2 yards of cloth at $31 a yard ?

10. What will 7 bushels of shelled corn cost at 25 cents a bushel? 11. A lady bought 4 dozen spoons at $6 a dozen; what did they cost her?

Ans. $3328.

Ans. $1912.

Ans. $43.78.

Ans. $56.

4. 387 sheep, if 1 sheep cost 8 dollars ?
5. 180 tons of coal, at $10.62 per ton?
6. 7294 lb. of sugar, at 6 cts. a pound?
7. 87 barrels of flour, at $63 a barrel?
8. 6 yards of cloth, at $5‡ a yard ?
9. 34 bushels of apples, at $ a bushel?
10. 24 handkerchiefs, at 87 cents apiece?
11. 6 tons of hay, at $67 a ton ?
12. 8 cords of wood, at $44 a cord?
13. 15 barrels of vinegar, at $4‡ a barrel ?
14. 16 quarts of beans, at $0.18% a quart?
15. What is the value of

16. What is the value of

Ans. $33.

Ans. $21.
Ans. $21.

Ans. $46}}.

Ans. $38.
Ans. $71.
Ans. $3.09.

Ans..
Ans. 1.

17. Value of H×?×??

1089/1024

Ans. TIT.

NOTE.-In examples like 15, 16, 17, and 18, resolve the numerators and denominators into their factors and then cancel.

18. Value of ? × j? J × 7 ?? × 8ff × 1998 ?
19. Value of (+18) × &+(1+8)×3?
20. Value of (+1) × ($ +7) × 7 × } } } ?
21. Value of 7×4&+(4}−33)—(5§+6μ3)?

Ans. H.

Ans. 3.

Ans. 3.

Ans. 23488.

22. Find the value of 732-(33}+371) diminished by (61+6,7)×4918. Ans 201 23. Mr. Fish rented a house at $429 a month, taking a lease for 5 years; but disposed of the lease at the end of 31 years; how much rent did he pay? Ans. $1667.

24. A bill of books at retail amounts to $376, but I get a reduction of for wholesale and for cash; what was the exact amount of the bill? Ans. $235.

25. Mr. Alden bought 524 bushels of wheat at 624 cts. per bushel, and sold † at 691 cts., and the remainder at 713 cts. per bushel; what did he clear by the transaction?

DIVISION OF FRACTIONS.

Ans. $4.431.

178. Division of Fractions is the process of dividing when one or both of the terms are fractional.

179. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 3d. A fraction by a fraction.

10. Since÷4=1, how may we divide a fraction by a number that will divide the numerator?

11. Since 4-14, how may we divide a fraction by a number that will not divide the numerator?