15. Divide by 13 a cr. 16. Divide 28 a* cy by 14 a® y. 17. Divide 36 amoy by 54 army. 18. Divide 75 a' by by 35 a* c* yo x. 19. Divide a +b by 2c-d. 20. Divide 2 a' c-7abc + 15 a' cd by 13 a'cd. 21. Divide 18 a mi 18 am - 54 am + 42 am by 30 amd12 a'cm. 22. Divide (a + b) (13 a c + bc) by (m -c)(a + b). 23. Divide 3 co (a---2c) by 2bco (a - 2 c)'. 24. Divide 36 63 c (2a + d): (76_d) by 126* (2 a + d)* (76 -- d)' (a---d). Addition and Subtraction of Fractions. e e XVIII. Add together Ở and į and 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. 3 2 1. Add together 3+2 and Ans. 7 ng 7 2. Add together and Ans. a toc 5 and 2a 2 a 3 a 5a 3. Add together 3a + 2a Ans. cd cd cd cd 5 a 6 4. Add together and Ans. 2a+ 5 ab 3 cd 3 cd 3 cd 5. Add together and 4. 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 it. 6. Add together and á Multiply both terms of the first by d, and of the second by ad b, they become The denominators are now alike bd 6d and the numerators may be added. The answer is ad + b c bd and b c ş, and h come 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 and bd f1bdfh' bdfh' bd fh The answer is adfh+bcfh+bdeh +b df&. bdfh 2 C 8. Add together and Ans. 15 ad +46 2bc 5d 10 b cd 3 a a and of 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. 9. Add together d bo be cg 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 cg, and the third by bce, each becomes b c eg. 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 beg' boeg The answer is a e g +cRdg + bcef cdg and eb of cbeg 3 тр 3 a 36 12. Add together and 2mn 3 cd 13. Add together and 2 mon 2 ar 14. Add together and bc 2 mar 15. Add together and 2 ac 16. Add together 37, and 13 cd. 17. Add together 2 a m and 2 ac-56. 4 a no 18. Add together 13 an 40 and 11 ac—5n. 3 bm and 7 ab + 80 4 a b 26 + 16 ab 19. Add together and 2a.c 20. Add together e 3 a e 21. Subtract from bc 260 3a ca But if they are reduced to a common denominator, the numerators may be subtracted. 3ac-2e Ans. 2 b c 2 ab 5 m n 22. Subtract from 3d 2 cm 3cd c de 23. Subtract from 3 rx 2m yo from 27 a d 2 abd - 3 mc 4 62 c? Solution. 27 a d 2 abd 3 cm? (27 a d) 26 (2 b c*) 26 4 6 4 62 ca 4 62 c? which is the answer. When the fraction 2 a b d 3 c m was subtracted, the 46° c sign - was changed to + See Art. VI, example 6th. XIX. Division of whole numbers by Fractions, and Fractions by Fractions. 1. How many times is contained in 7? Ans. } is contained in 7, 35 times, and is contained į as many times; that is, vor 119 times. 2. How many times is contained in a ? Ans. is contained in a, 8 a times, and is contained } as many times; that is, a. |