(a) To multiply a mixed number by a whole number.

RULE. Reduce the mixed number to an improper fraction, (Art. 140,) and then multiply. Or, multiply the fraction and whole number separately and add the products together.

NOTE. To multiply a whole number by a fraction is just the same as multiplying a fraction by a whole number. e. g. & X 4 = 4 ק.

CASE 5.

144. To divide a fraction by a whole number.

143. How is a mixed number multiplied by a whole number? Another method?

Ex. 1. Divide by 3.

1st OPERATION.

2d OPERATION.

Ans. .

One third of 6 apples is 2 apples; it is equally clear that one third of 6 sevenths (9) is 2 sevenths (7.)

If I divide by 1, the quotient will be . Now if I divide it by 3 instead of 1, I

9/ x3 = obtain a quotient only one third as great, or of. In this instance the number of parts remains the same, while the size of the parts is diminished.

RULE 1.

Divide the numerator by the whole number.

Or,

RULE 2. Multiply the denominator by the whole number.

NOTE. If the dividend be a mixed number first reduce it to an improper fraction, or divide the whole number and fraction separately and add the results.

11. Divide 26 by 6.

12. Divide 16% by 7.

Ans. 218.

13. Divide 287 by 7.

144. First rule for dividing a fraction by a whole number? Second rule? A mixed number how divided by a whole number?

14. Divide 697 by 13.

15. Divide 211 by 12.

CASE 6.

Ans. 5%. Ans. 17%.

145. To multiply a fraction by a fraction. Ex. 1. Multiply by .

Ans. 1.

We first multiply the fraction by 7, (Art. 143, Rule 1.) and obtain 21. Now, as 7 is eight times the true multiplier 3, the product is 8 times too large; and we obtain the true product by dividing 21 by 8 (Art. 144, Rule 2.) 4×7=4 and 2181. Hence,

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

2

6 2

2. Multiply X

7 3

Ans. .

In the above example, we have the factor 3 in the numerator of the, and also in the denominator of the . These we reject in the operation, since this is equivalent to dividing both terms of the product by 3, which (Art. 139, 5th) does not alter the value of the fraction, and obtain the answer in its lowest terms. This process of cancellation may be employed advantageously in many cases, as the principle is the same, as when applied in division. (See Art. 88.)

7. Multiply by #2.

145. Rule for multiplying a fraction by a fraction?

8. Multiply 269 by 185.

9. Multiply 63 by .

Ans. 1.

NOTE 1. Reduce the mixed numbers to improper fractions.

A compound fraction may be reduced to a simple one by the rule for multiplying one fraction by another.

OPERATION.

by 3.

Ans.

= 13.

We first divide by 2, and obtain 3, Art. (144, Rule 2,) but the divisor

1÷=1× used is 3 times too great, and consequently the quotient is only of the required quotient, and hence must be multiplied by 3 to obtain the correct result.

From the above we have the following

RULE. Invert the divisor, and then proceed as in multiplication (Art. 145).

Ans..

2. Divide

by 14.

NOTE. If either of the quantities is a mixed number it must be reduced to an improper fraction.

CASE 8.

147. To reduce fractions that have different denominators to equivalent fractions having a common denominator.

Ex. 1. Reduce and to equivalent fractions having a common denominator.

314

20100

OPERATION.

X

X

9

616

9

4,4

=

=

27

36

2828

20

36

We multiply both terms of each fraction by the denominator of the other fraction; this (Art. 139, 5th) does not alter the value of either fraction, and of necessity it makes the denominators alike as they are both the product of the two denominators, 4 and 9.

RULE. Multiply both terms of each fraction by the continued product of the denominators of all the other fractions. Ex. 2. Reduce 3, 3, and 4, to equivalent fractions having a common denominator.

63

3. Reduce,, to equivalent fractions having a common denominator. Ans. 3, 113, 144. 4. Reduce,, to equivalent fractions having a common denominator. Ans. 493, 450, 48.

7709

5. Reduce, §, to equivalent fractions having a common denominator. Ans. 388, 388, 318.

200

6. Reduce, &, to equivalent fractions having a common denominator.

147. Rule for reducing fractions to equivalent fractions having a common denominator?