30 a' m3 d- 12 a'c m3. - a2c 21. Divide 18 a m3-54 a3 m2 + 42 a3 m* by 22. Divide (a+b) (13 ac+bc) by (m2 — c) (a + b). 23. Divide 3 c (a-2c) by 2 be3 (a—2c). 24. Divide 36 b3 c2 (2 a + d)2 (7 b — 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. 1. Add together and Ans. 3 + 2 = 5. 2. Add together and—. Ans. a‡c 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 g. 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 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 befh bdeh bdfg bdfh bdfh bd ƒ h bdfh come and The answer is a dƒh + b c f h +b de h+b d f g 8. Add together fh bdfh 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 c❜g, and the third by bce, each becomes b c e 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 deg cdg bceg beeg' a e g + c d g+bce f The answer is and eb cf bceg 3bf4aed q + 3 = be 2dg 26d5 5 am ec За 2m n ατ eg. and en saka an and 3mp 2 c 5b m2 5bn' 3 m2 s and 2 ar 3mn'r 3 sign 31. From was changed to +. See Art. VI example 6th Subtract XIX. Division of whole numbers by Fractions, and Fractions by Fractions. How many times is contained in 7 ? Ans. is contained in 7, 35 times, and is contained as many times; that is, 35 or 11 times. 2. How many times is Ans. contained in a? is contained in a, 8 a times, and is contained as many times; that is, . 8 @ |