Two ways have been shown to multiply fractions, and two ways to divide them. C Reducing Fractions to Lower Terms. XVII. If both numerator and denominator be 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 b a certain number of times, if a is as large or larger than b; or a part of one time, 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. to its lowest terms. 108 a x2+81 x — 90 m2 x3 8. Divide 35 a bmx by 7 a3n m3 x. Write the divisor under the dividend in the form of a fraction, and reduce it to its lowest terms. 21. Divide 18 a* m3 — 54 a5 m2 + 42 a3 m2 by 30 am3 d-12 a2 cm3. 22. Divide (a+b) (13 ac+be) by (m3 — c) (a+b). 23. Divide 3 c2 (a-2c)3 by 2b c3 (a — 2 c)3. 24. Divide 36 b3 c2 (2a + d)2 (7 b—d)3 This addition may be expressed by writing the fractions one 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. 1. Add together and 3 5 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. c d Multiply both terms of the first by d, and of the second by a d b c b, they become and b d bd 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 bcfh bdeh bdfg bd fh bdfh bdfh b d fh come and 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 eg, the second by c2g, and the third by bce, each becomes be g. 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 cdg and aeg eb cf beeg beeg The answer is aeg + c d g + b c e f 10. Add together 11. Add together beeg 2 ac be cb2 eg and h 17. Add together and 3m n3r ъс and 2 h3 n2 and 13 cd. 2 m3 r h2 n3 x and 2 a c-5 b. |