8. Reduce m n a and to equivalent fractions having the X least common denominator. 9a 76 11a 7(a+8) 9. Reduce and 8m 36m 28m' to equivalent frac 4m tions having the least common denominator. 2 3 2x-3 10. Reduce and x 2x-1 to equivalent fractions 4x2_1 having the least common denominator. Addition of Fractions. 117. The denominator of a fraction shows into how many parts a unit is to be divided, and the numerator shows how many of those parts are to be taken. Fractions can only be added when they are like parts of unity; that is, when they have a common denominator. In that case, the numerator of each fraction will indicate how many times the common fractional unit is repeated in that fraction, and the sum of the numerators will indicate how many times this result is repeated in the sum of the fractions. Hence we have the following RULE. Reduce the fractions to a common denominator; then add the numerators together, and write their sum over the common denominator. If there are mixed quantities, we may add the entire and fractional parts separately. EXAMPLES. 1. What is the sum of and ? Reducing to a common denominator, the fractions become 3x 2ac 5x Adding the numerators, we obtain 6 It is plain that three sixths of x and two sixths of x make five sixths of x. a с and ? a and a +227 2. What is the sum of to a n adn+bon + bdm Ans. bdn 3. What is the sum of and .? ato 6 2a 4. What is the sum of 5x, 3x2 4x 2ąc 5. What is the sum of 2a, 3a+ 7 , and at 583 Ans. 6a+ 45 6. What is the sum of a+a, ama x, and a—? X2 a' - axi a7. What is the sum of and a=b; Ans. a. 2 2 a a-2m + 8. What is the sum of 2 4 4 A-X a a+b and a+2m ? ma-6 na+b? 7y2 +3? 9. What is the sum of and mtn mtn X-n •10. What is the sum of Y-> 2z+n and ? a+y+z' x+y+z +y+2 3y2-2 11. What is the sum of and 772-5 4y2-1 13a--296 7-21a 12. What is the sum of and 5a-3) 5(a−b) 5(a0) Ans. 9. 1+« 1-2 1-x+22 1+x+x2 13. What is the sum of 1-'17 1+x2 1-X and -1? Subtraction of Fractions. 118. Fractions can only be subtracted when they are like parts of unity; that is, when they have a common denominator. In that case, the difference of the numerators will indicate how many times the common fractional unit is repeated in the difference of the fractions. Hence we have the following RULE. Reduce the fractions to a common denominator; then subtract the numerator of the subtrahend from the numerator of the minuend, and write the result over the common denominator. EXAMPLES. 2. Зх 1. From subtract 3 5. Reducing to a common denominator, the fractions become 10. 9.3 and 10x 9cc Hence we have 15 15-15 and it is plain that ten fifteenths of x, diminished by nine fifteenths of x, equals one fifteenth of x. 12 3x 2. From subtract 7 50 9a-4x Ба— Зах 3. From subtract 7 3 It must be remembered that a minus sign before the dividing line of a fraction affects the quotient (Art. 111); and since a quantity is subtracted by changing its sign, the result of the subtraction in this case is 9a-4x 5a - 3x t 7 3 ; which fractions may be reduced to a common denominator, and the like terms united as in addition. 2acx 4. From subtract Ans. 0+ 0 622+73 50-6 3552-6 5. From 2+ subtract Ans. 8 21 168 ax AX b-c 13a-56 25a-116 8. From subtract Ans. 12 ба 64ab-15a__6372 9. From subtract Ans. 27ab-1862 10. From (a+b) subtract unity. 4ab 2x 11. From subtract lly 7x— Бу ax 12. From subtract a a2-22 3x-87 a ac Multiplication of Fractions. 119. Let it be required to multiply s by a First let us multiply s by c. According to the first principle of Art. 109, the product must be . But the proposed multiplier was ä; that is, we have used a multiplier d times too great. We must therefore divide the result by d; and, according to the second principle of Art. 109, we obtain bd; that is, 6 xarici bd. Hence we have the following ас a с Х b RULE. Multiply the numerators together for a new numerator, and the denominators for a new denominator. Entire and mixed quantities should first be reduced to fractional forms. Also, if there are any factors common to the numerator and denominator of the product, they should be canceled. EXAMPLES. 2.c 22 1. Multiply by Ans. 27 xta 2. Multiply by a+7 812 bc ab X 4x 21 X-1 a mnx bac a 24-34 4. Multiply by Ans. b+co 72c+bc2 a> +62 a> +62 5. Multiply Ans. (a+b)2 10x 6. Multiply together and 2' 5' 2x 3ab Зас 7. Multiply together and a 26 +1 aces 8. Multiply together x, and Ans. a? +abi 4ax 9. Multiply by Oman. (18m2nx) (81many) 10. Multiply by a2b2c4. (a2b3c") (a+b3c") (a?b3c4) 5a3712 14aRm 5n1lm6 6am 11. Multiply together and 7man 25n11' 6a15 03n 2mon Ans. a-b3 13(a-b) 5(x-y) 21(m,n) 12. Multiply together and 7(m-n' 39(a—)' 3dn Bbm 5mn 11abc 13. Multiply + by 3ax a2-02 bc+bx 14. Multiply together and 4by' c? — x2' a2 + ax' x 3x Ans. 4y 1-22 1-ya 15. Multiply together and 1+ 1+y' +2' 1-X Ans. 16. Multiply a(a + x) by a+62 by Ans. 55 (36-Y 1+ C 17. Multiply až – 2ab +62 a-6 al |