greater than 1, is subtracted from + 1 or 21 7 21 is carried to the whole number 1, as in the Subtraction of Integers. Examples for Practice. (1) Find the differences of and; of and ; of and, and of and. (2) What are the respective differences of 19 and 13; of 8 and 17, and of 1000 and 3847? Answers: 6, 95 and 6158. (3) Required the difference of 13 of 3 and 23 of 16; also, of of of and of of 25. 2 Answers: 39 and 1799. 924 鮮 (4) Find the difference of of and 2 of 15 33 6 also, of and 47 12 Answers: 4 and 645. : (5) Prove that the sum of 5 and 3, is equal to four times their difference. III. MULTIPLICATION OF FRACTIONS. RULE. Multiply together the respective numerators and denominators of the proposed quantities, reduced to fractional forms if necessary; and the fraction thence arising will be the product, which may generally be simplified by means of the preceding articles. For, let the fractions be and; then if be multiplied by 7, the product will be by article (75): but 14 7 being 8 times as great as, the multiplier above used is 8 times too large, and the product will therefore be 8 times too large also: whence the product required must be 14 9 ÷ 8 that is, 718 14 7 72 = or, 213 and 314 1 2 3 1 × 2 × 3 = 1/2 6 6 = 2 3 6 1 = 12 24 2 3 4 2 × 3 × 4 24 4 and thus the rule may be proved to be general: also, in cases like this, the reduction is much shortened by cancelling from the products of the numerators and denominators, any factor or factors common to them both, and effecting the multiplications of what are left; as, (1) Required the respective products of and ; of and; of 23 and 7%, and of 8 and 101⁄2. (3) Required the continued products of,, and of,,and, and of, 2, 1 and 1. (4) Multiply 2 by of of, and 18 of 7 by of of 121. (6) Find the continued product of the fractions, IV. DIVISION OF FRACTIONS. RULE. Multiply the dividend by the divisor inverted, and this result reduced when possible, will be the quotient: or, which is the same thing, invert the divisor, and then proceed according to the rule for the Multiplication of Fractions. For, let be to be divided by ; then it is 3 3 manifest that ÷÷4= is 5 times too small, because 7 28 the divisor has been taken 5 times too great: whence the quotient required will be and the operation may be expressed in this form ; 88. To denote the division of one integer by another, as for instance, that of 4 by 5, we shall have, according to the principles already established, or, in words, a simple fraction may be considered as an adequate expression of the implied division of its numerator by its denominator. Examples for Practice. (1) Find the respective quotients of by ; of (2) What are the respective quotients of 2 by 33; of 10 by 13%, and of 17 by 7? (3) Divide 3 by of of; and 15% of 8; by of of 15%. 7 Answers: 14 and 19%. 10 (4) Compare the product and quotient of by 19. 89. What has been proved in the adaptation of the four fundamental operations to fractional quantities, will furnish the means of simplifying arithmetical expressions formed by any of their combinations: thus, (2) Reduce to its simplest form, the expression, (5) Simplify as much as possible, the arithmetical expression (×3+2×) - († X- 5 1 X X 9 (6) Determine the simple fraction which expresses 5 2 54 the value of 13)+(x+5). 9 7 (7) What is the value of the expression, (8) Required the value of the expression, REDUCTION OF FRACTIONS. 90. Our attention has hitherto been confined to fractions considered generally, without regard to the particular species of their units; and it remains to apply what has been said to such concrete quantities as constitute the principal subjects of practical computation. 91. A fraction may always be transformed into another, so that the value of the unit in the latter may have a specified relation to that of the unit in the former. |