10. Multiply together, 3, 4, 4, 3, and §. Ans. T DIVISION OF FRACTIONS. 26. LET it be required to divide by §. We know that can be divided by 5, by multiplying the denominator by 5, (see Prop. II., Art. 16,) which Now, since is but one-eighth of 5, it follows that 4, divided by, must be eight times as great as divided ..4, divided by, must be 4x8 From this, by 5. we see that has been multiplied by, when inverted. Hence, to divide one fraction by another, we have this RULE. Reduce the fractions to their simplest form. Invert the divisor, and then proceed as in multiplication. 1. Divide by 1. EXAMPLES. Inverting the divisor, and then multiplying, we obtain 2×2; which, by canceling, becomes 1 X 27. SOMETIMES fractions occur, in which the nu merator, or denominator, or both, are already fractional. REDUCTION OF COMPLEX FRACTIONS. 28. SINCE the value of a fraction is the quotient arising from dividing the numerator by the denominator, it follows that the complex fraction is the same as 2÷4=44=43. Again, 2 Hence, to reduce a complex fraction to a simple one, we have this RULE. Divide the numerator of the complex fraction by the denominator, according to Rule under Art. 26. EXAMPLES. 1. Reduce to a simple fraction 41 Dividing 4= by 3=0, we get 7=17%. 2. Reduce to a simple fraction. Ans. 3. Reduce to a simple fraction. Ans. =3}} 29. SUPPOSE We wish to change the fraction to an equivalent one, having 6 for its denominator. It is obvious that if we first multiply by 6, and then divide the product by 6, its value will not be altered. 4 × 624_41 By this means, we find that = 6 6 6 52 Hence, to reduce a fraction to an equivalent one having a given denominator, we have this RULE. Multiply the fraction by the number which is to be the given denominator, (see Rule under Art. 25,) under which place the given denominator, and it will be the fraction required. EXAMPLES. 1. Reduce to an equivalent fraction having 8 for its denominator. In this example, we first multiply by 8, which gives 24 3 24; therefore, placing 8 under 24, we get 8 8. the fraction required. for 2. Reduce to an equivalent fraction having 12 for its denominator. Ans. TT 12 3. Reduce to an equivalent fraction having 7 for its denominator. Ans. 7 4. Reduce 1, 3, 4, and 13 to fractions having 12 or 10 their common denominators. 10 Ans.,,, and 5. Reduce,,, and, to fractions having 100 for their common denominator. 6. Reduce,, 1, 1, and, to fractions having 30 for their common denominator. 30. A denominate fraction is a fraction of a number of a particular denomination. Thus, of a foot, of a yard, of a dollar, and 1⁄2 of a shilling are denominate fractions. Reduction of denominate fractions is the changing of them from one denomination to another, without altering their values. 31. Suppose we wish to reduce of a pound sterling to an equivalent fraction of a farthing, we proceed as follows: Since there are 20 shillings in 1 pound, it follows that of a pound is the same as 20 times of a shilling, and this is also the same as 12 times 20 times of a penny; which, in turn, is 4 times 12 times 20 times of a farthing. Hence, of a bound sterling is equivalent to π of 2o of 12 of † of a farthing. Again, let us reduce of a farthing to an equivalent fraction of a pound sterling. In this case, we must use the reciprocals of 20, 2, we thus find that of a farthing is equivalent to sterling. ; of of, of of a pound |