MULTIPLICATION OF FRACTIONS.

THEOREM.

a

The product of two proper fractions is less than either fraction.

For, if a number is multiplied by one, the product is the same as the number multiplied. If the multiplier is greater than one, the product is greater than the number; and if the multiplier is less than one, the product is less than the number.

In the multiplication of two proper fractions, each factor is less than one; hence the product is less than either fraction.

PROBLEMS.

f

1. Multiply s by 1.

Ans. x1= 2. Multiply by 2.

Ans. f x 2 = $. 3. Multiply by to

Ans. x = t. It is evident that f multiplied by 1 = $, that if x 2 is $, and f x for 1 time f is f; and, as alternating the factors does not change the product, therefore, 1x=$, . 2 xf = 4, and $ x= t.

$x 4. Multiply } by .

Ans. } *=. CoR. 1.-In multiplying by a fraction the numerator is a multiplier and the denominator a divisor.

COR. 2.-In the multiplication of fractions, the product of all the numerators is the numerator of the product; and the product of all the denominators is the denominator of the product.

a

COR. 3.-The product of two improper fractions is greater than either fraction.

EXAMP.LES.

1. Multiply 1, 5, 1, 4, 4, , 5, % 14, 14.

1x xx xx xx xx xx = 1. By analysis, 1 of j = }, $ off = 1, 1 of 4 = }, { of te

: 4 = , of 4 = 4, 4 of 3.= }, $ of f = t; } of % = tos

1 to of H = ti, ti of = . 1

11 2. Multiply }, Hf, and }; thus,

119

= 315

3. Multiply 16 and 1.

X = , product.

I 4. Multiply 353 by 9.

1 x 9 = 2 = 63
35 x 9

321% = Product. Axiom 7.—If any number be both multiplied and divided by the same number, the value of the original number is not changed.

COR. 1.-If the multiplier is greater than the divisor, the product is greater than the number multiplied, but if the multiplier is less than the divisor, then the product is less than the multiplicand.

Cor. 2.-Multiplying the numerator or dividing the denominator by any number, multiplies the fraction by the same number.

EXAMPLES.

6
> 89

=

5.6

=

1. Multiply 1, 5, 1, 5, 1, 4, 3, 4, 3. Product

to. 2. Multiply 4, &, 4, }, f.

Product = $.
3. Multiply ), it is and 1. Product = 18
4. Multiply , }, ir, 1 and 11. Product = 語.
5. Multiply in x $ *£* . Product = d.
6. Multiply 451 by 15.

73
451 x 15 = 225 one x 15 = 910
45

455
6827

1365

= 6821. 2

7. Multiply 743 by 12.

3

743 x 12 = 282 x 12 = 897.
8. Multiply 274 by 33}.
111 x 100 = 18,426 = 921% = 921 %.

,

81

11 x 18 = 2413 = 921% = Product.

2

9. Multiply 54% by 37% = 42 x 143 = 24147

} 10. Multiply 674 by 517. 11. Multiply 91} by 561. 12. Multiply 53767 by 8214. 13. Multiply 6274 by 2374.

REM.When one or both factors are mixed numbers, it is generally best to reduce them to improper fractions.

DIVISION OF FRACTIONS.

PROBLEMS.

=

Ans. $x2

1. Divide 10 by 5.

Ans. 10 -5== 2. 2. Divide 6 by 3.

Ans. f = 2. 3. Divide 3 by 3.

Ans. = 1. 4. Divide 1 by 2.

Ans. 5. Divide 1 by 3.

Ans. $. 6. Divide 2 by 3.

Ans. . 7. Divide 3 by 4.

Ans. * 8. Divide 5 by 4.

Ans.

11. 9. Divide 1 by 2, or divide into two equal parts.

Ans. Hx2 = 1 10. Divide f by 2, or divide finto two equal parts.

Ans. +9 = 11. Divide by 2, or divide into two equal parts.

1

= 12. Divide f by 5, or how often is contained in f.

Ans. Evidently twice. 13. Divide 4 by 4, or how often is 4 contained in 1=*.

Ans. Evidently twice. 14. Divide 4 by }, or how often is contained in .

Ans. 1 = $ and j = *;};$= = 11.
f

BE
15. Divide } by 4, or how often is 4 contained in }.
Ans. f = % and 1 = Bi;= $.

ਝੂਠ £ ਝੂਠ ਨੂੰਨ COR. 1.-The numerator of a fraction is a dividend, the denominator a divisor, and the fraction itself the quotient.

COR. 2.-To divide a fraction by a fraction, reduce both to a common denominator and then divide the

}

EXAMPLES.

6

56 195

1. Multiply 1, 1, 1, 4, 5, 6, , 8,1. Product = t.

53$ 2. Multiply $, %, 4, 1, s.

Product $ 3. Multiply }, it, if and 1. Product 4. Multiply 5, 1, 1, 1 and H. Product = 16 5. Multiply it * * * * * * H. Product = t. 6. Multiply 451 by 15.

21 x 15 = 910

451 x 15 = 225

45

455

6821

1365

= 6821 2

3

7. Multiply 744 by 12.

744 x 12 = 242 x 12 = 897. 8. Multiply 27 by 333.

111 x 196 = 18426 = 921% = 921 %.

81

1 x = 4113 = 921% = Product.
111 LRT

2

9. Multiply 544 by 373 42 x 13 = 34742 . 10. Multiply 674 by 51}. 11. Multiply 915 by 56%. 12. Multiply 5376} by 8214. 13. Multiply 6274 by 237%.

REM.-When one or both factors are mixed numbers, it is generally best to reduce them to improper fractions.