CASE. VI. To add fractional quantities together. RULE. Reduce the fractions, if necessary, to a common denominator; then add all the numerators together, and under their sum put the common denominator, and it will give the fractions required.* EXAMPLES. 1. It is required to find the sum of - and 2 3 Зх the numerators. Зах 28 50 the sum required. 6 6 6' + e 2. It is required to find the sum of and d f Here a xdxf=adf cxbxf=cbf the numerators. e xbxd=ebd adf , cbf", ebd adf + cbf + ebd Whence the sum. bdfbdfbdf bdf 3.22 2 ax 3. It is required to find the sum of a and b + b Here, taking only the fractional parts, = 3cx =2abx 2aba Зcar2 Whence a +6+ =a+b+ the sum. be bic be 20 4. It is required to find the sum of and we shall have { 5X 5 39x Ans. 35 3х 5. It is required to find the sum of and 2a 5. 15x + 2ax Ans. 10a * In the adding or subtracting of mixed quantities, it is best to bring the fractional parts only to a common denominator, and then to affix their sum or difference to the sum or difference of the integral parts, interposing the proper sign. C C a 6. It is required to find the sum of and 2' 3' 4 X 4x 2 7. It is required to find the sum of and 7 5 272 - 14 Ans. 35 8x 8. Required the sum of 2a, 3a + 5: 9 22x Ans. 6a 45 2x and a a Зх 9. Required the sum of 2a +5a and 3a-x - 3ax: + 5* Ans. 2a +2+ 5a? 5ax a 2 2x -3 10. Required the sum of 5x + and 4x 3 5x 522 - 16x + 9 Ans. 9x + 153 2α 11. It is required to find the sum of 5%, and 3х2) 4x 8a + 3ax + 62: Ans. 5x + 12.x2 a + 2x CASE VII. To subtract one fractional quantity from another. RULE.-Reduce the fractions to a common denominator, if necessary, as in addition; then subtract the less numerator from the greater, and under the difference write the common denominator, and it will give the difference of the fractions required. And 3 x 5= 15 the common denominator. And 26 x 3c = 6be the common denominator. 3cx Зас 4ab 8bx 3cx - 3ac-4ab-t 8bx Whence the 6bc 6bc 6bc difference required. 123 3x 42 3. Required the difference of and Ans. Cat 7 5 35 1 + 2y 4. Required the difference of 15y and 8 118y - 1 Ans. 8 a and a a. and a atx 7. Required the difference of a + ata ac 2a + 2x2 Ans. a2 2x2 2x + 7 5x 6 8. Required the difference of ax to 8 21 86x 99 Ans. as 168 3х 5 9 Required the difference of 2 + and 3x + 7 118 - 10 320 + 5 Ans. x + a to 20 10. Required the difference of a + and a(a + 2) a(a - 2) 4x Ans. a a2 а CASE VIII. To multiply fractional quantities together. Rule.-Multiply the numerators together for a new nunerator and the denominators for a new denominator; and the former of these being placed over the latter, will give the product of the fractions, as required.* EXAMPLES. 2x 1. It is required to find the product of a and 6 9 the product required. 54 27 4x 2' 59 a a ox2 + ax 2 2. It is required to find the continued product of 10x and 21. x 4x x 10x 4023 423 Here the product. 2 X 5 X 21 210 21 atX 3. It is required to find the product of and. * X (a + x) Here the product. a X (a - x) 3x 52 4. It is required to find the product of and 5x2 Ans. 26 2x 3х2 5. It is required to find the product of and 5 323 Ans. 5a. * When the numerator of one of the fractions to be multiplied, and the denominator of the other, can be divided by some quantity, which is common to each of them, the quotients may be used instead of the fractions themselves. Also, when a fraction is to be multiplied by an integer, it is the same thing whether the numerator be multiplied by it, or the denominator divided by it. Or if an integer is to be multiplied by a fraction, or a fraction by an integer, the integer may be considered as having unity for its denominator, and the two be then multiplied together as usual. 2x 402 6. It is required to find the continued product of 3373 8αα3 ? and Ans. a ta 218 +21x 22. 3ab 7. It is required to find the continued product of a 5ac and Ans. 15ax. 26. bx b 8. It is required to find the product of 2a + and 3a с a 26 Ans. 60% + 3bx a? 9. It is required to find the continued product of 3x, X + 1 XC 1 3203 3х and Ans. 2a at6 202 + 2ab a" x2 10. It is required to find the continued product of ad - 72 a25 and at Ans. 3 a ax tona CASE IX. To divide one fractional quantity by another. RULE.-Multiply the denominator of the divisor by the numerator of the dividend, for the numerator; and the numerator of the divisor by the denominator of the dividend, for the denominator. Or, which is more convenient in practice, multiply the dividend by the reciprocal of the divisor, and the product will be the quotient required.* * When a fraction is to be divided by an integer, it is the same thing whether the numerator be divided by it, or the denominator multiplied Also, when the two numerators, or the two denominators, can be divided by some common quantity, that quantity may be thrown out of each, and the quotients used instead of the fractions first proposed. by it. |