Also, both numerator and denominator of a fraction may be divided by the same quantity, without changing the value of the fraction; for, dividing the numerator divides the fraction, and dividing the denominator multiplies the fraction; but if a quantity is divided, and the result is multiplied by the divisor, the value of that quantity remains unchanged.

ART. 62. From the principle last stated, we deduce the following

RULE FOR SIMPLIFYING A FRACTION.

Divide the numerator and denominator by all the factors common to both.

Simplify the following fractions.

In the following examples, some of the common factors are polynomials; but they can be easily discovered by separating the numerators and denominators into factors. 3(x+y) (xy)

14.

3x2-3 y2
6x-6y

This is the same as

3.2(x-y)

ART. 63. When the division of one integral quantity by another cannot be exactly performed, it is expressed in the form of a fraction, the divisor being placed under the dividend. The fraction should then be simplified.

1. Divide 3 a b2c by 6bc2. Expressing the division, we have

3 a b2 c

which reduced,

6 b c2

becomes

a b
2 c'

Ans.

2. Divide 12 am2 3. Divide 13 x4 y 4. Divide 22 m x2 5. Divide 45 a b x3 6. Divide 54 b2 c2 x 7. Divide 3abc3abm 8. Divide 522 +5 y2 9. Divide 3 (a+b) 10. Divide 6 a2 (x+y)

by 15 a2 m3.
by 39 x2 y3.
by 33 m2 x.
by 50 a2 b3 x.
by 33 65 c x3.
by 15 a2b2x.
by 20 (x4y4).

by 15 (a2+2ab+b2).
by 14 a3 (x3+ y3).

SECTION XVIII.

MULTIPLICATION OF FRACTIONS BY FRACTIONS.

ART. 64. Find of; that is, multiply by . According to the rule for the division of fractions by integers, of is, and, according to the rule for the multiplication of fractions by integers, 3 of 7 is 2, Ans. Find theof; that is, find the product of

by

RULE FOR MULTIPLYING FRACTIONS BY FRACTIONS.

Multiply all the numerators together for a new numerator, and all the denominators together for a new denomi

nator.

Remark. As the resulting fractions should be simplified, it is best to represent the operation, then strike out the common factors, previous to the actual performance 2 am 6(a+b) 2.6 am (a + b) of the multiplication. Thus, 3.4 m2 x y

a (a + b)

3xy

4 m2

=

ART. 65. When one quantity is divisible by another, the former is called a multiple of the latter. a multiple of 5; it is also a multiple of 2.

Thus, 10 is

A common multiple of two or more quantities is one which is divisible by them all; and the least common multiple is the least quantity divisible by them all. Thus, 24 is a common multiple of 6 and 4, but 12 is the least common multiple of these numbers.

Let it be required to find the least common multiple of 8 am and 6 a m3.

It is manifest that this multiple must contain all the factors of 8 am and 6 am3. Separating these quantities

into their prime factors, we have 8 am 23 a2m, and 6 am3 2.3 am3. The different prime factors are 2, 3, a, and m, each of which must be contained in the multiple required, as many times as it is found in either of the given quantities; that is, 2 must be contained three times, 3 once, a twice, and m three times, as a factor. The least common multiple is, therefore, 23. 3 a2 m3, or 24 a2 m3

ART. 66. Hence we have the following

RULE FOR FINDING THE LEAST COMMON MULTIPLE OF SEVERAL QUANTITIES.

First, separate the quantities into their prime factors; then unite in one product all these different factors, each raised to the highest power found in either of the given quantities.

Find the least common multiple in each of the following examples.

1. 4a2, 10 a b3.
2. 6m1x, 8my2.

3. 4x3y, 2xy2, 9 m3.

4. 25, 15 m2, 45 x3 m.

Ans. 22.5 a2 b3 = 20 a2 b3

5. 3xy, 15 x3 y2, 3 (a+b).
6. 11 p2q3, 33 py, 22 p q2.
7. 7(a+b), 14 (a2+2ab+b2).
8. 2ab+4b2, and 14 a2 b2.
9. 18(x-y), 9 (x2 — y2).