Two ways have been shown to multiply fractions, and two multiply the denominator. ways to divide them. To multiply a fraction, S the numerator To divide a fraction, divide S the numerator. To multiply a fraction, the denominator. Arith. Art. XVIII. XVII. Reducing Fractions to Lower Terms. If both numerator and denominator he multiplied by the same number, the value of the fraction will not be altered. Arith. Art. XIX. For multiplying the numerator multiplies the fraction, and multiplying the denominator divides it; hence it will be multiplied and the product divided by the multiplier, which reproduces the multiplicand. α In other words, signifies that a contains o a certain num ber of times, if a is as large or larger than b; or a part of ontime, ifb is larger than a. Now it is evident that 2 a will contain 2 b just as often, since both numbers are twice as large as before. So dividing both numerator and denominator, both divides and multiplies by the same number. Hence, if a fraction contain the same factor both in the numerator and denominator, it may be rejected in both, that is, both may be divided by it. This is called reducing fractions to lower terms. 8. Divide 35 a b m* x3 by 7 a3n m3 x. Write the divisor under the dividend in the form of a frac tion, and reduce it to its lowest terms. a3 by 2 13 a' c d. 21. Divide 18 am-54 a m2 + 42 a3 m* 22. Divide (a+b) (13 ac4-be) by (m2-c) (a+b). 23. Divide 3 c2 (a-2c) by 2bc3 (a — 2 c)3. 24. Divide 36 b3 c2 (2 a + d)2 (7 b — d) 3 by 12 b' (2a + d)3 (7 b — d)3 (α — d). This addition may be expressed by writing the fractions ore after the other with the sign of addition between them; thus N. B. When fractions are connected by the signs and -, the sign should stand directly in a line with the line of the fraction. It is frequently necessary to add the numerators together, in which case, the fractions, if they are not of the same denomination, must first be reduced to a common denominator, as in Arithmetic, Art. XIX. These must be reduced to a common denominator. It has been shown above that if both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered. If both the numerator and denominator of the first fraction be multiplied by 7, and those of the second by 5, the fractions become and. They are now both of the same denomination, and their numerators may be added. The answer is 3. Multiply both terms of the first by d, and of the second by a d b, they become and be b d b d The denominators are now alike In all cases the denominators will be alike if both terms of each fraction be multiplied by the denominators of all the others. For then they will all consist of the same factors. Applying this rule to the above example, the fractions bead fh b c f h b d e h bdfh bdfh bdfh come fh and bdfg bd fk The answer is a dƒh + b c f h + b de h + b d f g bdfh and 2bc 5 d It was shown in Arithmetic, Art. XXII, that a common denominator may frequently be found much smaller than that produced by the above rule. This is much more easily done in algebra than in arithmetic. Here the denominators will be alike, if each be multiplied by all the factors in the others not common to itself. If the first be multiplied by e g, the second by c2g, and the third by bce, each becomes bceg. Then each numerator must be multiplied by the same quantity by which its denominator was multiplied, that the value of the fractions may not be altered. The fractions then become aeg_ c2 d g bceg bceg The answer is a e g + c2 d g + b c e f 10. Add together 11. Add together 12. Add together 13. Add together bceg 2 ac be and 5 am ес 3bf 2dg 2r' 36 h 3 a 2m n and "" 14. Add together and 15. Add together and h2 n3 x e b c f bceg |