33. From } of a pound take # of a shilling. First, + of a shilling reduced to the fraction of a pound is # then I re duce + and # to a common denominator; then subtracting 7 from 108, the remainder is 101. Lastly, I reduce # L. to its proper quantity, which is - o 3 34. From II of a hlid. of wine take 3. gallons. 25. From # of a pound take # of a shilling. Diff. 13s. 2 8 37. From II of a barrel take 9 of a gallon. Diff, 5gal. 24t. 198. To multiply fractions together. RULE I. Multiply the numerators together for a numerator, and the denominators together for a denominator. II. If tile new fraction be a proper fraction, reduce it to its lowest terms; if an improper one, reduce it to its equivalent whole or mixed number". ExAMPLEs. 1. Multiply #, #. and # together. OPERATIon. Thus, 3. X 2. X 7. – 42 ~ 7. (1718wer. 4 3 8 96 16 Earplanation. I first multiply the numerators 3, 2, and 7 together, and the product is 42. Then I multiply the denominators 4, 3, and 8 together, and the product is 96. * To multiply a fraction by a whole number, we must evidently multiply the numerator by the whole number; but to divide it, we must multiply the denominator by the whole number: thus, if -- be multiplied by 2, the pro4. d in be # 1 × 2 B !-air is evid w! l uct will be 4. (or 4 ). But 4 ivided by 2 is evidently g, (or I-7) for two fourths are double of one fourth, and one eighth is the half of one fourth : this being premised, let it be required to multiply + by +. now 2 multiplied by 4 equals 8; but it is not 2, but a third part of 2, which is to be multiplied, and therefore the product will be the third part of 8 only, or +, but the multiplier is not 4, but the fifth part of 4 only, wherefore the product - 8 ... 8 2 4 8 will be the fifth part only of T. that is, T5 3 wherefore T x-E = Tā; or the numerators multiplied together give the numerators of the product, and the denominators multiplied give the denominators; which is the rule. 5. Multiply 3. 3, and 12 continually together. Pro4 8 13 45 duct 199. When mixed numbers or complex fractions are to be RULE. Reduce the mixed numbers to improper fractions, and the complex fractions to simple ones, and proceed as before". * The reason of this process evidently follows from the preceding note. Earplanation. 7 First, I reduce 3; to the improper fraction T. Secondly, I reduce the 13 7 complex fraction to its equal T5 . Thirdly, I multiply the two fractions T2 13 - 91 and 15 together, and reduce the product 30 to its equivalent mixed num 1 * 3. 13. Multiply 2. and # together. Prod. 525 T ‘T 200. To multiply a whole number and fraction together. Rule. Multiply the numerator of the fraction by the whole number, and under the product set the denominator; then reduce this fraction to its lowest or proper terms, as the case requires, and it will be the answer". * Whatever parts a fraction consists of, its product when multiplied by any whole number will consist of like parts; thus two sevenths multiplied by three - 2 2 × 3 6 - - will produce six sevenths, that is, 7 X 3 = -- (= +), which is the rule. 14. Multiply 2 by #. OPERATION. Erplanation. 3. 6 3 Here I multiply the numerator 3 by 2 x — = − = - product. the whole number 2, and under the pro8 8 4 duct 6 I place the denominator 8; then 4 16. Multiply #. #, and 12 together. Prod. ++. 8 I 17. Multiply 7 by # of To Prod. 53. . . 4 ... 1 .. 2 ,..., 8 18. Multiply 7 of 5 of # by 3. Prod. 63' 201. When the whole number will divide the denominator of the fraction without remainder, divide by it, and set the quotient under the given numerator; this fraction reduced as before will give the answer". 19. Multiply; by 19. OPERATIon. Earplanation. 4 l Here I divide the denominator 48 x 12 = — = -- prod. by the whole number 12, and set 48 -i- 12 4 the quotient 4 under the numerator l for the answer. * To divide the denominator by any number is the same as to multiply the numerator by it; for let + be multiplied by 2, the product by the last rule will be +. which reduced to its lowest terms is +. Let now the denomina tor 4 of the faction- be divided by 2, thus, 4 1 2’ and the quotient will be 2, making the resulting fraction __ , the same as by the former method; and 2 the same holds true in every other instance: wherefore the truth of the rule is manifest. |