The preceding rule may be derived in another manner, thus: To divide a number by 2, is to take of it, or to multiply it by ; to divide by 3, is to take of it, or to multiply it by. In the 1

same manner, to divide a quantity by m, is to tako of it, or to

1

m

multiply it by. Hence, to divide a fraction by an entire quan

m

m

tity, we write the divisor in the form of a fraction (thus, m=—), and invert it, and then proceed as in multiplication of fractions.

REVIEW.-141. How do you divide a fraction by an entire quantity? Explain the reason of the rule, by analyzing an example. How may the work be abbreviated, when the numerators of the fraction and the entire quantity contain common factors?

ART. 142.-To divide an integral or fractional quantity by a fraction.

1. How often is contained in 4, or what is the quotient of 4

divided by ?

4 is equal to

(2), as often as 2

2. How often is

12, and 2 thirds (3), is contained in 12 thirds

is contained in 12, that is, 6 times.

contained in na, that is

times.

Or,

is contained in a, na

ՊՆ

n

Here,

8

12,

9

and 3=12, and 8 twelfths (1) is contained in 9 twelfths (1), as often as 8 is contained in 9, that is, g==1 times

4. How often is

m
n

contained in

Reducing these fractions to a common denominator,

is contained in as often as mc is contained in

na

nc

times. This is the same result as that produced by

multiplying - by inverted, that is -X

с

n

с m тс

An examination of each of these examples, will show that the process consists in reducing the quantities to a common denominator, and then dividing the numerator of the dividend, by the numerator of the divisor. But, as the common denominator of the fraction is not used in performing the division, the result will be the same as if we invert the divisor, and proceed as in multiplication. Hence, the

RULE,

FOR DIVIDING AN INTEGRAL OR FRACTIONAL QUANTITY BY A FRACTION.

Reduce both dividend and divisor to the form of a fraction; then invert the terms of the divisor, and multiply the numerators together for a new numerator, and the denominators together for a new, denominator.

NOTE. After inverting the divisor, the work may be abbreviated, by canceling all the factors common to both terms of the result.

REVIEW.

142. How do you divide an integral or fractional quantity

by a fraction? Explain the reason of this rule, by analyzing an example. Wher, and how, can the work be abbreviated?

ART. 143. To reduce a complex fraction to a simple one. This may be regarded as a case of division, in which the divi dend and the divisor are either fractions or mixed quantities.

Thus,

21

31

is the same as to divide 21 by 31.

b

с

Also, is the same as to divide a+ by m+2.

n

m+

In the same manner, let the following examples be solved.

A complex fraction may also be reduced to a simple one, by multiplying both terms by the least common multiple of the denominators of the fractional parts of each term. Thus, we may

41

reduce to a simple fraction, by multiplying both terms by 6, 51

the least common multiple of 2 and 3; the result is §. In some cases this is a shorter method, than by division. Either method may be used.

ART. 144.-Resolution of fractions into series.

An infinite series consists of an unlimited number of terms, which observe the same law.

The law of a series is a relation existing between its terms, so that when some of them are known, the succeeding terms may be easily derived.

REVIEW.-143. How do you reduce a complex fraction to a simple one, by division? How, by multiplication?