If ad+bd+c d be divided by a+b+c, the quotient is d. When a compound quantity is to be divided, let the quantity, if possible, be so arranged that the divisor may appear as one of the factors, and then that factor being struck out, the other factor will be the quotient.

19. Divide 12 ab-9ac by 3 a.

12 a2b-9ac3a (4 ab-3 c)

a2

3c)

Ans. 4ab-3c.

Observe that a is a factor of both terms, and also 3. Hence the quantity 12 a b - 9 a c, can be resolved into factors; thus 3 (4a2 b 3 a c), or a (12 a b 9 c), or 3 a (4 ab 3 c). In the last form the divisor 3 a appears as one factor, and the other factor 4 ab- 3 c is the quotient.

Note. Any simple quantity, which is a factor of all the terms of any compound quantity, is a factor of the whole quantity; and that factor being taken out of all the terms, the terms as they then stand, taken together, will form the other factor.

20. Divide 8 a' b3- 16 a3 bc by 2ab- 4 a2 c.
8 a2 b3 — 16 a3 b3 c4ab (2 ab- 4 a' c.)
a2

21. Divide

Ans. 4 a b3.

3abc-15 abd+9 a3 bd by 3 ab.

22. Divide 15 abc-30 ac2 + 25 a3 c d

by

13

11

5 a c.

23. Divide 36 a13 b2 c— 28 a11 b1 c2 + 40 ao b® c3

24. Divide 42 a"-84 a10 b c by 1-2 a3 b2 c.

Algebraic Fractions.

XV. When the dividend does not contain the same letters as the divisor, or but part of those of the divisor, the division cannot be performed in this way. It can then only be expressed. The usual way of expressing division, as has already been explained, is by writing the divisor under the dividend in the form of a fraction. Thus a divided by b is expressed

a

This gives rise to fractions in the same manner as in arithmetic. It was shown in arithmetic, that a fraction properly expresses a quotient. Algebraic fractions are subject to precisely the same rules as fractions in arithmetic. Many of the operations are more easily performed on algebraic fractions.

In these, as in arithmetic, it must be kept in mind, that the denominator shows into how many parts a unit is divided; and the numerator shows how many of those parts are used; or the denominator shows into how many parts the numerator is divided.

I shall here briefly recapitulate the rules for the operations on fractions, referring the learner to the Arithmetic for a more full developement of their principles.

of 7 is; for
+

of 7 is, and is 3 times as much. of

a is 24; for of a is, and

is 2 times as much. The part

Hence, to multiply a fraction by a whole number, or a whole number by a fraction, multiply the numerator of the fraction and the whole number together, and divide by the denominator.

Arith. Articles XV. & XVI

This cannot be done like the others, but it may be done by multiplying the denominator as in Arith. Art. XVII. For the fraction denotes, that one is divided into as many equal parts as there are units in b, and that as many of these parts are used as there are units in a; or that a is divided into as many equal parts as there are units in b; hence if it be divided into twice as many parts, the parts will be only one half as large, and the fraction will have only one half the value.

Hence, to divide a fraction by a whole number, divide the numerator; or when that cannot be done, multiply the denominator by

the divisor.

Hence, to multiply one fraction by another, multiply the numerators together for a new numerator, and the denominators together

for a new denominator.

Arith. Art. XVII.