15 6 Examples. 1. §×3=}=}="={=2}. Or, &×8={= 21. 2. 7×4=81; 11×6=5; 13×8=28; 2×q=p. 3. 11-9; 134 × 13=123; 13×204-18210. 12 Multiplication of Mixed Numbers. 209. Reduce the mixed numbers to improper fractions, and multiply them (206) as fractions. Also, when there are whole numbers, express them as fractions; and, in all cases, cancel as much as possible. 4. X28X1=1; 23×33×4=39; 331×10=360. 5. 3×3×23 × 1 = 6; 61 × 3 × 2 = 391. 6. 949 × 7711=7341151; 6939270718. 210. The calculation is often facilitated by striking a factor from one number and introducing it into another. Thus, striking the factor 11 from 3311, we have 3; then introducing it into 13, we have 152, and multiplying the results, instead of the numbers, we have 4683 for the product, as follows: 1. 13, 3311152 × 3456 +123=4683. 2. 6116850 × 21=100 +55=105§. 3. 163 18350 × 613121. 4. 18 × 147=129 × 2 = 258 + 141 = 2721. 5. 115 250-578 501-28900+961289961. 211. A fraction of a fraction is called a compound fraction. Thus of is a compound fraction. To take of is (144) to divide by 5; that is, (165,) to multiply its denominator 8 by 5, which gives; therefore, of must be 3 times; and this (165) is found in multiplying the numerator 7 by 3, which gives 1. Hence, we may substitute the sign for the word of, and proceed as in multiplication of fractions. Wherefore, of = X× 7 = ? }. Also, it is evident that of of of; that is, as 9 2 above, X=18. We see, therefore, that to reduce a compound fraction to a simple one, the fractions composing. the compound fraction must all be multiplied together. 440 Examples. 66 1. of of 9×? × } = 1; 1 of 4 of 11 = √25· = 2. of of 35; of % of 25 15. 23 Division of Fractions. n 455 212. To divide a quantity Q by any fraction d' multiply the dividend by the reciprocal of the divisor; that is, by. Thus, d Qd - n d n First, is the quantity divided by the numerator. But n n (208) n == Xd: that is, n is d times too great; conse quently, (165,) is d times less than it should be; where n fore, (165,) Q must be multiplied by d, which gives the true quotient. 1. 24÷=24X3=8×8=64, and 24÷3=24×3=32. 36×8 2. 36÷3= 7 -288414, and 36+7=46%. 7 11 3. 43 157, and 99252475. 4. 52811576, and 528111152. 213. Let Q be any fraction P. Then, + -pd For P nis, (185;) but, as n is d times too nq ng d Р n Ρ great, Pis d times less than it should be; wherefore, we mul nq tiply p by d, and have pd ng for the true quotient. To divide by a fraction, therefore, we have this simple rule: invert it, (that is, turn it upside down,) and multiply. 2. 7% = 1; } ÷ 7% = 1}; } ÷ 3 = 211. 3. =127; 1818=45; 5200. 10 Р n 9 d' pd 214. The ratio of to which (188 and 213) is is Р (189) compounded of the ratios and the product of these ratios. n Thus, we see that the quotient arising from the division of one fraction by another is a compound ratio, and is the product either of the direct ratio of the numerators and the inverse ratio of the denominators, or that of the dividend and the reciprocal of the divisor. Hence, the ratio of of any number of ratios is the product of those ratios. 3. What is the ratio, compounded of the ratios A to B, B to C, and C to D? A Hence it is plain that, if we have any number of quantities, the ratio of the first to the last is compounded of the ratios, the first to the second, the second to the third, the third to the fourth, and so on to the last. 4. What is the ratio of a fraction to its reciprocal? the ratio of P to is 5. What is the ratio of a fraction to the square of its reci procal ? d Answer, the triplicate ratio or cube of the fraction. 6. What is the ratio compounded of the ratiosto, to d e 2, and to ? e C d af Answer, be 7. What is the ratio compounded of the ratios to to, to 5, and to? Answer, . х pd q d n qn qn y pdy = dy X pdy That is, Չ d qnx y q Nx qnx to divide a quantity, successively, by any number of fractions, we multiply that quantity by the product of their reciprocals. In the same manner we find that 4÷1÷1÷÷÷÷§=252. 216. A quantity divided by a fraction is always increased; because the reciprocal, by which we multiply, is greater than a unit. Also, it is increased in the ratio of the numerator of the reciprocal to its denominator. Wherefore, if this numerator be one unit greater than the denominator, the quantity will be increased by such part of itself as is signified by the denominator. Thus: 24÷=24X7=(24 × 1})=24+4=28 2424 X = (24 × 11)=24+12=36 Divide 168, successively, by 1, 3, 4, 4, 7, and find the sum of the quotients. Answer, 1200. Division of Mixed Numbers. 217. Reduce mixed numbers to improper fractions; express whole numbers as fractions, and reduce compound fractions to simple ones: after which proceed as with ordinary fractions. Thus we see that the divisor, divided by the dividend, gives the reciprocal of the quotient, which serves as proof. 2. 87÷63-87÷1517x7=203-13, 8. Also, 62874587-203 45 15 35 3. 733=8=138, and 33-73=88. 68 218. When we divide a unit by a fraction, the quotient is the reciprocal of that fraction. Thus, 1÷3 1÷== But the dividend is the product of the divisor and quotient; that is, the product of any fraction multiplied by its reciprocal is a unit. Wherefore, to divide or multiply by a fraction, is to multiply or divide by its reciprocal. To divide one whole number by the factors of another: 219. As the object of this method is, not only to expedite, but facilitate calculation, the number of factors is usually li mited to two: because, where strict accuracy is required, and the dividend is prime to the factors, a greater number would render intricate the calculation of the final fraction. Whenever, therefore, the divisor can be resolved into two factors, neither of which exceeds 12, we first divide by one factor, and then divide the quotient by the other. For example, to divide 165369 by 45: As 45 is the product of 9 and 5, we first divide by one of these factors, no matter which, and then by the other, as follows: |