or value of the fraction, for the words dividend, divisor, and quotient, respectively, and we shall have the following GENERAL PRINCIPLES OF FRACTIONS. 124. PRIN. I. Multiplying the numerator multiplies the fraction, and dividing the numerator divides the fraction. PRIN. II. Multiplying the denominator divides the frac tron, and dividing the denominator multiplies the fraction. PRIN. III. Multiplying or dividing both terms of the frac tion by the same number does not alter the value of the fraction. These three principles may be embraced in one GENERAL LAW. 125. A change in the NUMERATOR produces a LIKE change in the value of the fraction; but a change in the DENOMINATOR produces an OPPOSITE change in the value of the fraction. REDUCTION. 126. To reduce fractions to their lowest terms. A fraction is in its lowest terms when its numerator and denominator are prime to each other; that is, when both terms have no common divisor. 1. Reduce the fraction 48 to its lowest terms. FIRST OPERATION. #=#=#=t, Ans. ANALYSIS. Dividing both terms of a fraction by the same number does not alter the value of the fraction or quotient, (124, III;) hence, we divide both terms of 48, by 2, both terms of the result, 14, by 2, and both terms of this result by 3. As the terms of are prime to each other, the lowest terms of 18 are . We have, in effect, canceled all the factors common to the numerator and denominator. First general principle? Second? Third? General law? What is meant by reduction of fractions? Case I is what? What is meant by lowest terms? Give analysis. SECOND OPERATION. 12) 48=t, Ans. In this operation we have divided both terms of the fraction by their greatest common divisor, (97,) and thus performed the reduction at a single division. RULE. Cancel or reject all factors common to both numera tor and denominator. Or, Divide both terms by their greatest common divisor. EXAMPLES FOR PRACTICE. 2. Reduce 1 to its lowest terms. Ans. Ans. 1. 10% 11. Reduce 4680 to its lowest terms. 12. Express in its simplest form the quotient of 441 dı vided by 462. Ans. 13. Express in its simplest form the quotient of 189 divided by 273. Ans. s. 14. Express in its simplest form the quotient of 1344 divided by 1536. CASE II. Ans. J. 127. To reduce an improper fraction to a whole or mixed number. 1. Reduce 324 to a whole or mixed number. OPERATION. = 324 324 15 21 213, Ans. ANALYSIS. Since 15 fifteenths equal 1, 324 fifteenths are equal to as many times 1 as 15 is contained times in 324, which is 21 times. Or, since the numerator is a dividend and the denom. Rule. Case II is what? Give explanation. inator a divisor (118), we reduce the fraction to an equivalent whole or mixed number, by dividing the numerator, 324, by the denominator, 15. RULE. Divide the numerator by the denominator. 1. When the denominator is an exact divisor of the numerator, the result will be a whole number. 2. In all answers containing fractions reduce the fractions to their lowest terms EXAMPLES FOR PRACTICE. 2. In 13 of a week, how many weeks? Ans. 14. Ans. 234. 4. In 4§1 of a dollar, how many dollars? 5. In 2 of a pound, how many pounds? Ans. 541. 6. Reduce 1 to a mixed number. 7. Reduce & to a whole number. 18 9. Change 131 to a mixed number. 10. Change 237040 to a mixed number. 225 11. Change 2531820 to a whole number. CASE III. Ans. 18%. Ans. 1053. Ans. 7032. 128. To reduce a whole number to a fraction having a given denominator. 1. Reduce 46 yards to fourths. OPERATION. 46 4 184, Ans. = ANALYSIS. Since in 1 yard there are 4 fourths, in 46 yards there are 46 times 4 fourths, which are 184 fourths 184. In practice we multiply 46, the number of yards, by 4, the given denominator, and taking the product, 184, for the numerator of a fraction, and the given denominator, 4, for the denominator, we have 184. RULE. Multiply the whole number by the given denominator; take the product for a numerator, under which write the given denominator. Rule. Case III is what? Give explanation. Rule. A whole number is reduced to a fractional form by writing 1 under it for a de nominator; thus, 9= = 1. EXAMPLES FOR PRACTICE. 2. Reduce 25 bushels to eighths of a bushel. 3. Reduce 63 gallons to fourths of a gallon. Ans. fe 4. Reduce 140 pounds to sixteenths of a pound. 5. In 56 dollars, how many tenths of a dollar? Ans. 560. 6. Reduce 94 to a fraction whose denominator is 9. 7. Reduce 180 to seventy-fifths. 8. Change 42 to the form of a fraction. 9. Change 247 to the form of a fraction. Ans. 4. 10. Change 347 to a fraction whose denominator shall be 14. CASE IV. Ans. 1858. 129. To reduce a mixed number to an improper fraction. 1. In 5 dollars, how many eighths of a dollar? OPERATION. 5 8 43, Ans. ANALYSIS. Since in 1 dollar there are 8 eighths, in 5 dollars there are 5 times 8 eighths, or 40 eighths, and 40 eighths + 3 eighths = 43 eighths, or 48. RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum write the denominator. EXAMPLES FOR PRACTICE. 2. In 4 dollars, how many half dollars? Ans. . 3. In 714 weeks, how many sevenths of a week? 8. Reduce 225 to an improper fraction. Case IV is what? Give explanation. Rule. Ans. 532 9. In 96, how many one hundred twentieths? 10. In 1297, how many eighty-fourths? Ans. 108951. 11. What improper fraction will express 400f? CASE V. 130. To reduce a fraction to a given denominator. As fractions may be reduced to lower terms by division, they may also be reduced to higher terms by multiplication; and all higher terms must be multiples of the lowest terms. (103.) 1. Reduce to a fraction whose denominator is 20. OPERATION. 20÷4=5 3 x 5 4 x 5 =18, Ans. ANALYSIS. First divide 20, the required denominator, by 4, the denominator of the given fraction, to ascertain if it be a multiple of this term, 4. The division shows that it is a multiple, and that 5 is the factor which must be employed to produce this multiple of 4. We therefore multiply both terms of & by 5, (124,) and obtain 15, the desired result. RULE. Divide the required denominator by the denominator of the given fraction, and multiply both terms of the fraction by the quotient. EXAMPLES FOR PRACTICE. 2. Reduce to a fraction whose denominator is 15. Ans. 3. Reduce to a fraction whose denominator is 35. to a fraction whose denominator is 51. Ans. If 4. Reduce 5. Reduce 6. Reduce to a fraction whose denominator is 150. 88 7. Reduce to a fraction whose denominator is 1000. Case V is what? How are fractions reduced to higher terms? What are all higher terms? Give analysis. Rule. |