tiplier contains some part or parts of the multiplicand as often as the multiplier contains the same part or parts of unity. с α If the multiplicand is, the product by multiplied by α ac d ac b ac c and divided by d. Now xc= and ÷d= (Art. 70.) b b b bd' But ac is the product of the numerators of the proposed factors, and bd the product of their denominators; also a c = ca, and bd = db. (Art. 35.) Whence the product of two fractions is equal to the product of their numerators divided by the product of their denominators. a. If the continual product of more than two fractional factors is required, by multiplying together two of these factors, this product by a third factor, &c., it may be shown that such product is equal to the product of all the numerators, taken in any order, divided by the product of all the denominators, taken also in any order. b. To render this method general, whole numbers must be reduced to the form of fractions, and mixed numbers to improper fractions. (Arts. 69. and 74.) 3 . 3 4 5 3 x 4 × 5 60 Ex. 3×4×24=x=5x2 -10-6. c. The calculation may be abridged by cancelling factors common to the numerator and denominator of the product. (Art. 70. c.) If all the prime factors common to both terms are cancelled, the product is expressed in the lowest terms. (Art. 62.) Ex. X X 3 x 3 x 2 x 3 x 2 x 5 3x3x3 27 d. Hence, to find the product of any fractional factors, Rule,-Reduce mixed numbers to improper fractions, and whole numbers to the form of fractions: then multiply together all the numerators of the factors for the numerator of the pro duct, and all the denominators for its denominator. If any prime factors are common to the numerators and denominators, these may be cancelled before multiplication. The product of the proposed fractions is thus expressed in the lowest terms. 93. It is required to find the products of the following fractional numbers 94. In some cases the product of fractional numbers may be found by other methods more easily than by the general rule. Thus, if the one factor is a fraction, and the other a whole number, the numerator of the fraction may be multiplied by the whole number, or the denominator divided by it: in either way, the result is the product of the proposed fraction and whole number. (Art. 70. b.) Conversely, the whole number, if large, may be multiplied by the numerator of the fraction, and this product divided by the denominator. a. The product of a mixed number by a whole humber or a fraction may be found by multiplying the integral and fractional parts of the mixed number separately by the whole number or fraction, and then adding the partial products together. b. The product of two mixed numbers may, in like manner, be found by multiplying, 1st. The two integral parts together; 2nd and 3rd. Each integral part by the fractional part of the other factor; 4th. The two fractional parts; And, lastly, adding these partial products into one sum. Ex. 215 × 100=215 × 100+ 215 × 2+ 100 × 3+} × c. Whence the following Rules : To multiply a fraction by a whole number:-Multiply the numerator or divide the denominator of the fraction by the whole number. To multiply a whole number by a fraction :-Multiply the whole number by the numerator of the fraction and divide the product by its denominator. To multiply a mixed number by a whole number, a fraction, or another mixed number :-Decompose each mixed number into an integral part and a fractional part: find the product of each of the parts into which the multiplicand is decomposed by each of the parts into which the multiplier is decomposed, and add together the partial products. Their sum is the product of the given factors. Note. When the multiplier is a proper fraction, the product is less than the multiplicand: for the product is the same part of the multiplicand as the multiplier is of unity. In multiplication of whole numbers, two numbers are given to find a third which shall contain the first as often as the second contains unity. In multiplication of fractions, two numbers are given to find a third which shall have to the first the same relation which the second has to unity. Thus multiplication of fractions involves an extension of the original meaning of multiplication. 95. Additional examples, in multiplication of vulgar fractions. Multiply together, 96. Examples, the factors being literal fractions. (See Arts. 92. and 92. a.) 97. It follows, from Art. 92., that the product of any multiplicand by a fractional multiplier contains some part or parts of the multiplicand as often as the multiplier contains the same part or parts of unity. Whence, by substituting dividend for product, quotient for multiplicand, and divisor for multiplier, (Art. 45.) it also follows that the dividend contains some part or parts of the quotient as often as the divisor contains the same part or parts of unity. Then the divisor contains the dth part of unity c times. .. the dividend contains the dth part of the quotient c times. .. the cth part of the dividend=the dth part of the quotient (or contains it once). ..d times the cth part of the dividend = the quotient. a But the cth part of the dividend = (Art. 70.) bc' Now the numerator of this quotient is the product of the numerator of the dividend and denominator of the divisor; and its denominator is the product of the denominator of the dividend and numerator of the divisor. The same result may be obtained by changing the numerator of the divisor into its denominator, the denominator into its numerator, and multiplying the dividend by this fraction. с 1 d |