are added together, their sum is equal to the product of the multiplicand and multiplier. Consequently the product of a mixed number by a whole number or a fraction is obtained, without reduction of the factors to improper fractions, by multiplying the integral and fractional parts of the multiplicand separately by the multiplier, and taking the sum of the partial products for that required. Examples of the application of this method: 1st. Let it be required to multiply 8547 by 2. The product of a whole number by a fraction is obtained by multiplying the whole number by the numerator of the fraction, and dividing the product by the denominator (Art. 188). Now 8547 x3=25641, and 25641÷4=64101. .*.8547x=64101. The calculation may be made as follows: 8547 4)25641 64104 Since the product of a fraction by an integer is equal to the product of the integer by the fraction (Art. 191), it follows that, the fraction being made the multiplicand and the integer the multiplier, the product is still formed in the same manner. 2d. Let it be required to multiply 6835 by 7. 68354 6835+}, and 6835×7=6835x7+x7. .*.68353×7=478499. 3d. Let it be required to multiply 2857 by The product of 28573 by & is equal to 5 repetitions of the 6th part of 2857 plus 5 repetitions of the 6th part of. The 6th part of 2857-27-476. 4th. Let it be required to multiply 2463 by 86 The product must be equal to 86 repetitions of 2463 plus 7 repetitions of the 10th part of 2463. The product of 2463 by 86 is found as in Example 2 of this Article, and the product of 2463 by is found as in Example 3, thus, 2463 × 86= 2468x7 {246 x 86=21156 ..246867= 513 246×7 172} × = 21 z = z From the principles and examples of this Article it appears that the product of a mixed number by a whole number or a fraction is composed of two partial products; the one being the product of the integral part of the mixed number by the multiplier; and the other, the product of the fractional part of the mixed number by the multiplier. When the multiplier also is a mixed number, the product of the multiplicand by the integral part of the multiplier is composed of two partial products, and the product of the multiplicand by the fractional part of the multiplier is also composed of two partial products. To obtain the product of two mixed numbers without reducing them to improper fractions, it is, hence, necessary to form four partial products. These are, 1st. The product of the integral parts of the factors. 2d. The product of the fractional part of the multiplicand by the integral part of the multiplier. 3d. The product of the integral part of the multiplicand by the fractional part of the multiplier. 4th. The product of the fractional parts of the two factors. 197. Let a, a denote the integral parts, and b, b′ the fractional parts, of two mixed numbers. Then a+b, a+b' denote the mixed numbers. Multiplying the first of these expressions by the second (Art. 76) (a+b) (a+b)=aa'+ab+ab+bb'. This formula expresses, generally, the conclusion of Article 196. To find the product of a large mixed number by a whole number, a fraction, or another mixed number, Rule. Consider each mixed number to be decomposed into an integral part and a fractional part; then 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. The result is the product of the given factors. 198. Additional exercises in the multiplication of fractions: .... Ans. 770201. .Ans. 36474247}. ...Ans.97301523. ..Ans. 14398721. ..Ans. 156690+%. .....Ans. 47468218. ......Ans. 479640791. .Ans.118168333. Ans. 69089731. .Ans. 7010946181. ....Ans. 353455). .Ans. 1955153,5. ......Ans. 747363. ..Ans. 612679713. ....... Ans. 40174718. ....Ans. 132010423. 199. Division of fractions differs from division of integers in this respect, that the dividend or the divisor, or both the dividend and the divisor, are fractional expressions instead of whole numbers. The result which is to be found, namely, the quotient, being a number whose product by the divisor shall be equal to the dividend (Art. 79), it follows from Article 186 that the dividend must contain some part or parts of the quotient as often as the divisor contains the same part or parts of unity; or, conversely, that the divisor must have to unity the same numerical relation as the dividend has to the quotient. In division of fractions, it may be required, 1st. To divide a fractional expression by a whole number. 3d. To divide a fractional expression by a fractional expression. 200. Let it be required to divide by 4, by 8, and 5 by 9. 1st. To divide by 4: Since unity is the 4th part of the divisor, the quotient is the 4th part of g. Now, by Article 148 the 4th part of is obtained by either dividing the numerator 8 by 4, or multiplying the denominator 9 by 4. As 8 is a multiple of 4, the first method is to be preferred on account of its giving the result in the lowest terms. 8 84 2 Whence+4=-3, the quotient required. If the denominator is multiplied by 4, 8 2d. To divide by 8: Since unity is the 8th part of the divisor, the quotient is the 8th part of the dividend. 3, the numerator of the dividend, is not a multiple of 8; therefore, in order to divide the fraction by 8, the denominator 5 must be multiplied by 8. 3 3 3d. To divide 5 by 9: Unity being the 9th part of the divisor, the quotient must be the 9th part of 17. 17, the numerator of the dividend, is not a multiple of 9; the denominator of the dividend must therefore be multiplied by 9. 17 17 3x9 27' Whence 53+9=+9=3 the quotient required. From the investigations in this Article it appears, that when the dividend is a fractional expression and the divisor a whole number, the quotient is obtained by either dividing the numerator or multiplying the denominator of the dividend by the divisor. 201. Let it be required to divide 8 by 7, and 15 by 34. 1st. To divide 8 by : Unity contains the 7th part of the divisor 9 times; Therefore 8 must contain the 7th part of the quotient 9 times. Now, the 7th part of 8 is, the 9th of the quotient; and 9 times 9-10, the quotient required. 7 Unity contains the 7th part of the divisor twice; Therefore 15 contains the 7th part of the quotient, also, twice. Hence it appears, that to divide a whole number by a fractional expression, it is necessary to multiply the whole number by the denominator of the divisor, and to divide this product by its numerator. Let it be required to divide 1 by : The division of 1 by is effected by multiplying 1 by 5, and dividing the product by 3: Whence 1+3=1x5=1x= 5 The fraction obtained by dividing 1 by g, namely, is termed the reciprocal of the fraction; and, generally, a fraction, whose numerator and denominator are the denominator and numerator of another fraction, is termed the reciprocal of that other fraction. The rule for the division of a whole number by a fraction may now be expressed thus: multiply the dividend by the reciprocal of the divisor; the result is the quotient required. 202. Let it be required to divide by, and 123 by 63. 1st. To divide by: 1 contains the 8th part of the divisor 11 times; Therefore contains the 8th part of the quotient 11 times. 3 The 8th part of is which is the 11th part of the quotient, 33 5 5x89 and 11 times -3x11-33 Whence is the quotient required. 40 2d. To divide 123 by 6: 3 and 6.12 +63 +20. ...12=1, and 6= Now, it may be shown, as in the last example, that 73 To obtain the quotient in the 1st and 2d examples of this Article, it has been found requisite to multiply the dividend by the denominator, and to divide it by the numerator, of the divisor. The reasoning employed is general. Whence, to divide one fractional expression by another, multiply the dividend by the reciprocal of the divisor. When any of the given quantities are mixed numbers, reduce them to improper fractions. 203. The general rule for the division of fractions may be investigated without reference to particular quantity, thus, Let a represent the numerator and 6 the denominator of the dividend; e the numerator, and d the denominator, of the divisor: a a Then expresses the fractional dividend, and the divisor. Since 1 contains the cth part of the divisor d times, therefore the dividend contains the cth part of the quotient d times; but the cth part of is whence ; bxc the whole quotient. is the dth part of the quotient, and a с ad a d a bxc a xd_axd is bxc Wherefore bd-be-bx; that is, the quotient is equal to the product of the dividend multiplied by the reciprocal of the divisor. When d=1, the formula α с a d + C bd b Xis applicable to the case of Article 200; and when b=1, it coincides with that of Article 201. a 204. Since the division of by c can be made by either dividing a by c, or b Whence, if the terms of the dividend are multiples of the terms of the divisor, the quotient can be obtained by dividing the terms of the dividend by the corresponding terms of the divisor. 205. From Articles 200, 201, 202, 203, 204, are deduced the following practical rules for the division of fractional quantities, it being premised that, to render the rules applicable, mixed numbers must first be reduced to improper fractions; 1st. To divide a fraction by a whole number; divide the numerator, or multiply the denominator, of the dividend by the divisor. 2d. To divide a whole number by a fraction; multiply the dividend by the denominator of the divisor, and divide this product by its numerator. 3d. To divide a fraction by a fraction; multiply the dividend by the reciprocal of the divisor, or, if the numerator of the dividend contains the numerator of the divisor, and the denominator of the dividend also contains the denominator of the divisor, divide the terms of the dividend by the corresponding terms of the divisor. The result is in each case the quotient required. |