RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum

(a) To reduce an integer to a fraction having any given denominator:

Multiply the integer by the proposed denominator, and under the product write the denominator (Art. 84, c).

27. Reduce 12 to a fraction whose denominator is 7.

Ans. 8.

28. Reduce 9 to a fraction whose denominator is 8. 29. Reduce 9 to a fraction whose denominator is 5. 30. Reduce 7 to a fraction whose denominator is 1.

Ans. 1.

31. Reduce 87 to a fraction whose denominator is 87. 32. Reduce 16 to a fraction whose denominator is 1. 33. Reduce 16 to a fraction whose denominator is 4. 34. Reduce 20 to a fraction whose denominator is 4. 35. Reduce 14 to five different fractional forms.

139. Rule for reducing a mixed number to an improper fraction? Reason? An integer, how reduced to a fractional form?

PROBLEM 2.

140. To reduce an improper fraction to a whole or

mixed number.

Ex. 1. How many units in 13?

=13÷4=3, Ans."

Ans. 31.

Since the numerator is a dividend and the denominator a divisor (Art. 131), the

fraction is reduced to an equivalent whole or mixed number by the following

RULE. Divide the numerator by the denominator; if there is any remainder, place it over the divisor, and annex the fraction so formed to the quotient.

2. Reduce 3. Reduce

4. Reduce

to a whole or mixed number.

to a whole or mixed number.

to a whole or mixed number.

5. Reduce 756 to a whole or mixed number.

29

Ans. 3.

Ans. 3.

Ans. 249.

Ans. 26.

11. Reduce 1328.

PROBLEM 3.

141. To reduce a fraction to its lowest terms.

Ex. 1. Reduce 3 to its lowest terms.

FIRST OPERATION.

2=18=1, Ans.

Ans. 3.

Dividing both terms of a frac tion by any number does not alter the value of the fraction (Art. 84, b, and 131); .. dividing each term of 3 by 3 gives the equal fraction ; then dividing each term of this result by 4 gives, and as 3 and 4 are mutually prime (Art. 112), 39, in its lowest terms, equals 3.

SECOND OPERATION.

12)=1, Ans.

In this operation both terms of the fraction are divided by theit greatest common divisor, 12 (Art. 119), and thus the fraction is re

duced at once to its lowest terms. Hence,

140. Rule for reducing an improper fraction to a whole or mixed number? Reason?

RULE 1. Divide each term by any factor common to them, then divide these quotients by any factor common to THEM, and so proceed till the quotients are mutually prime. Or,

RULE 2. Divide each term by their greatest common divisor.

142. To multiply a fraction by a whole number.

Ex. 1. Multiply by 3.

FIRST OPERATION.

6

fs X 3 = fs, Ans.

Ans. or .

It is just as evident that 3 times are as that 3 times 2 cents are 6 cents, or that 3 times 2 are ; i. e. when the numerator

is multiplied by 3 the fraction represents 3 times as many parts continues of the same size; .. the frac

as before, and each part tion is multiplied by 3.

SECOND OPERATION.

X3, Ans.

If the denominator is divided by 3, the fraction represents just as many parts as before, but each part is three times as great, and

.. the whole fraction is three times as great. Hence,

RULE 1. Multiply the numerator by the whole number.
RULE 2. Divide the denominator by the whole number.

Or,

NOTE 1. The correctness of Rule 1 is also evident from Art. 83 (a), and Art. 131. Rule 2 also depends on Art. 83 (d).

141. First rule for reducing a fraction to its lowest terms? Second rule? Reason? 142. First rule for multiplying a fraction by a whole number? Why? Second rule? Why? Another reason?

2. Multiply

by 3.

Ans. or.

NOTE 2. The second rule is preferable in this and all similar examples, because it gives the fraction in smaller terms.

NOTE 3. The first rule is preferable for this and all similar examples, bo

cause the second gives a complex fraction.

3

Ans. 1 or 43

14. Multiply by 15.

15 5 X 3.

X5; and § X 334, Ans.

NOTE 4. We may here, as in whole numbers (Art. 61), use the factors of the multiplier, and in using these factors we may apply the 1st or the 24. rule, or both.

(a) If we multiply a fraction by its denominator, the

product will be the numerator.

19. Multiply by 8.

Ans. X8}=7, by Rule 2.

20. Multiply

by 44.

142. May the factors of the multiplier be used? What is the product if a frac tion is multiplied by its denominator?

(b) To multiply a mixed number by an integer:

Multiply the fractional part and the entire part separately, and add the products together; or, reduce the mixed number to an improper fraction (Art. 139), and then multiply.

21. Multiply 3 by 5.

First multiply by 5 and

Ans. 19.

the product is 4; then multiply 3 by 5 and the product is 15. These partial products added give 15+4 19 for the true product. Or, first reduce 3 to and then multiply by 5 and the product is 19, as before.

22. Multiply 83 by 9.

× 9=34; 8 × 9=72; and 72 +39 75§., Ans.

23. Multiply 9 by 12.

Ans. 113.

24. Multiply 18 by 20.

25. Multiply 234 by 7.

PROBLEM 5.

143. To divide a fraction by a whole number. Ex. 1. Divide § by 4.

FIRST OPERATION.

843, Ans.

Ans. or 36

It is just as evident that one fourth of is as that one fourth of 8 cents is 2 cents, or that one fourth of 8 is 2; i. e. when the numerator is divided by 4 the fraction represents only one fourth as many parts as before, and each part continues of the same size; .. the fraction is divided by 4.

If the denominator is multipled by 4, the fraction represents just as many parts as before, but each part is only one fourth as great, and :,

Or,

the whole fraction is only one fourth as great. Hence,
RULE 1. Divide the numerator by the whole number.
RULE 2. Multiply the denominator by the whole number.
NOTE 1. These rules may also be explained by Art. 83 (b) and (c).

143. How is a mixed number multiplied by an integer? 143. First rule for dividing a fraction by a whole number? rule? Why? Another explanation?

Another way?
Why? Second