Examples. 5 1. &X3=====21. Or, XB==21 =. 2. FX4=34; 11 X6=54; 1: X3=23; 2Xq=p. 3. X 11=95; 134 X 1 =123 13; 1] x204=18219. 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. 2279 8 159 8 33. Excamples. 1. 53 x 53=1 X 48 =2843. Or (138) thus : 53 x 5=265, and 53 x = = 192: then, 265 + 193 = 2845. 3 11 RA 2. 21 X 43 X 33 Х 3. 153 X 53. (138.) 153 53 Product of 15 X 5 75 s x 5 3} or 15 X 13] or X 24 9224 By the general method, 153X51=X35X2209 = 924, as before. 4. 13 x 29 x =1; 23 X 33 X4=39; 331X102=360. 5. X 31 X 23 X 1=6,11; 64 X 326 X 2=393. 6. 949 X 7711 = 7341131; 691 X 39,5 = 270748. 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 313; then introducing it into 131, we have 152, and multiplying the results, instead of the numbers, we havé 468for the product, as follows: 66. 455 1. 13 | X 3311 =152 X 311 = 456 + 123 =4683. 2. 61 X 169 = 50 X 23 = 100 + 5% = 105%. 3. 163 x 18% = 50 x 64 = 3121. 4. 183 x 141=129 X 2=258 + 141=2721. 5. 1153 X 250=578 X 50=28900 +961=289963. 211. A fraction of a fraction is called a compound fraction. Thus of Z is a compound fraction. To take of Jis (144) to divide by 5; that is, (165,) to multiply its denominator 8 by 5, which gives yo; therefore, of z must be 3 times y; and this (165) is found in multiplying the numerator 7 by 3, which gives . Hence, we may substitute the sign X for the word of, and proceed as in mul tiplication of fractions. Wherefore, of=X= Also, it is evident that of of i=of li ; that is, as above, it x8=188. We see, therefore, that to reduce a compound fraction to a simple one, the fractions composing the compound fraction must all be multiplied together. Examples. 1. of į of 9=X ý x =; } of 4 of 11 = 2. { of of =; s of % of 25 = 15. 3. of of of = %; of of 32=16. 4. z of 19 of 51} = 4231 ; of z's of 284 =1. Division of Fractions. 212. To divide a quantity Q by any fraction d' multiply the d dividend by the reciprocal of the divisor ; that is, by Thus, d Qd QX the required quotient. d First, is the quantity divided by the numerator. But (208) n= xd: that is, n is d times too great; conse đ quently, (165) is d times less than it should be; where od fore, (165,) Q must be multiplied by d, which gives for the true quotient. n n n Q n n Q n n Q n n Examples. 1. 24+1=24X=8X8=64, and 24+1=24X4=32. 36 X8 298 x 411, and 36 =- =46%. 2. 36=7 n p pd inis 3. 43 = 1=1573, and 99 = 's=2475. d 213. Let Q be any fraction 2. Then, ? X Х 9 d , (165;) but, as n is d times too ng 2 ng great, is á times less than it should be; wherefore, we mulпq pa tiply p by d, and have for the true quotient. ng To divide by a fraction, therefore, we have this simple rule: invert it, (that is, turn it upside down,) and multiply. Examples. 1. = = * X = 1; 1! = H=11 x il =2. 2. * -f=;&:10=121; = =211. 3. 9+1=1777 13 - 1s = 45; s=200. 4. 11 - 1's=169=234; $i = }=135. pd 214. The ratio of ? to -7, which (188 and 213) is is 9 р d (189) compounded of the ratios and and is, evidently, 2 the product of these ratios. 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 to is ng y pdy pd and is compounded of the two ratios and; or of пqх d ng the three ratios , and ; that is, a ratio compounded q of any number of ratios is the product of those ratios. Escamples. 1. Find the ratio of b adn Answer, ng' n pd beq e to е 3. What is the ratio, compounded of the ratios A: to B, B to C, and C to D? Answer, D 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 ? Answer, the duplicate ratio, or square of the fraction. Thus, the ratio of 2 Q р 9 р po P Х 2 р 9 р 2 2 5. What is the ratio of a fraction to the square of its reciprocal ? Answer, the triplicate ratio or cube of the fraction, 6. What is the ratio compounded of the ratios a to as a to d d and ? af f Answer, be 7. What is the ratio compounded of the ratios toto , to 5, and to 4 ? Answer, 1. P р pd y 215. As Х a d 9 an an y an pdy р p dy Hence, pdy X That is, qnx 2 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==3==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 : 4 =24 X =(24 X 11)=24+4=28 24 : j =24 X=(24 X 11)=24 +12=36 Divide 168, successively, by 1, j, 1, 4, 5, and find the sum of the quotients. Answer, 1200. n d pd .X Х pd ; so n n X 5 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. Examples. 1. 4+5=1=9=1x=*4=45, and +4=ix Thus we see that the divisor, divided by the dividend, gives the reciprocal of the quotient, which serves as proof. 2. 87 63=81: 4 = 87 xas=203 Also, 63 -- 87=1 X 81=2033. 7} = 3=fj=1}}, and 33 = 7*=* 4. of 5 = 4 of 91==1=: Xig=it! 5. of 915 + 4 of of 2=216. 6. 75811 - 833=9346 7. 14351 - 62=23-38. 8. 37296 - 311=95221% 9. 14353 23,7%8=62. 10. 449833 : 2857=15738917 . 218. When we divide a unit by a fraction, the quotient is the reciprocal of that fraction. Thus, 1=j=iX=3. 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 limited 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 diviile 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 : : |