is no number greater than 1 that will divide the numerator and denominator without a remainder. Q. What is reduction? When is a fraction said to be in its lowest terms? Is one-half in its lowest terms? Is two-fourths ? ls three. fourths ? CASE I. $ 89. To reduce an improper fraction to its equivalent whole or mixed number. RULE. Divide the numerator by the denominator, the quotient will be the whole number; and the remainder, if there be one, placed over the given denominator will form the fractional part. EXAMPLES, 1. Reduce and to their equivalent whole or mixed numbers. OPERATION. OPERATION. 4)84 9)67 Ans. 21 Ans. It was shown in $ 44, that the value of every fraction is equal to the quotient arising from dividing the numerator by the denominator: hence the value of the fraction is not changed by the reduction. Q. How do you reduce a fraction to its equivalent whole or mixed number? Does this reduction alter its value? Why not? What is four-halves equal to ? Eight-fourths? Sixteen-eighths ? Twenty-fifths ? Thirty-six-sixths ? Four-thirds ? What is nine-fourths equal to ? Four-fifths? Seventeen-sixths ? Eighteen-sevenths ? 2. Reduce to a whole or mixed number? Ans. 123. 3. In of yards of cloth, how many yards? Ans. yd. 4. In 5 of bushels, how many bushels ? 5. If I give } of an apple to each one of 15 children, how many apples do I give? Ans. 5. 6. Reduce 33, 3672, 50287, 98762,5, to their whole or mixed numbers. 7. If I distribute 878 quarter apples among a number of boys, how many whole apples do I use? Ans. Ans. 5 bu. 125 CASE II. § 90. To reduce a mixed number to its equivalent improper fraction. RULE. Multiply the whole number by the denominator of the fraction ; to the product add the numerator, and place the sum over the given denominator. EXAMPLES. Ans. 2 1. Reduce 44 to its equivalent improper fraction. Here 4x5=20 : then 20+4=24; hence, 24 is the equivalent fraction. This rule is the reverse of Case I. In the example 43 we have the integer number 4 and the fraction. Now 1 whole thing is equal to 5 fifths, and 4 whole things are equal to 20 fifths; to which, add the 4 fifths, and we obtain the 24 fifths. Q. How do you reduce a mixed number to its equivalent improper fraction? How many fourths are there in one? In two ? In three? How many sixths in four and one-sixth ? In eight and two-sixths ? In seven and three-sixths ? In nine and five-sixths? In ten and five. sixths? How many eighths in two and one-eighth? In three and three-eighths? In four and four-eighths? In five and six-eighths ? In seven and seven-eighths ? In eight and seven-eighths ? 2. Reduce 47 to its equivalent improper fraction? Ans. 287. 3. Reduce 67637, 87433, 690 176, 367784, to their equivalent improper fractions. Ans. 4. Reduce 84736, 874178, 67426348, to their equivalent improper fractions. 5. How many 200ths in 675287 Ans. 135187. 6. How many 151ths in 187151 Ans. 28278. CASE III. § 91. To reduce a fraction to its lowest terms. RULE. I. Divide the numerator and denominator by any number that will divide them both without a remainder, and then divide the quotients arising in the same way until there is no number greater than 1 that will divide them without a remainder. II. Or, find the greatest common divisor of the numerator and denominator and divide them by it. The value of the fraction will not be altered by the reduction. EXAMPLES. 1. Reduce 70 to its lowest terms. 175 1st METHOD. 5) 70 7)14 2 which are the lowest terms of the 5)1757)355 fraction, since no number greater than 1 will divide the numerator and denominator without a remainder. 2nd METHOD, BY THE COMMON DIVISOR. 70)175(2 35) 70 2 Ans. Greatest common div. 35)70(2 35)1755 70 Q. When is a fraction in its lowest terms? (see 88.) How do you reduce a fraction to its lowest terms by the first method ? By the second? What are the lowest terms of two-fourths ? Of six-eighths ? Of nine-twelfths ? Of sixteen-thirty-sixths ? Of ten-twentieths ? Of fifteen-twenty-fourths? Of sixteen-eighteenths ? Of nine-eighteenths ? 2. Reduce 114 to its lowest terms. Ans. * 3. Reduce 1949 to its lowest terms. Ans. 4. Reduce 275 to its lowest terms. Ans. 5. Reduce 351 to its lowest terms. 6. Reduce 17? to its lowest terms. 7. Reduce si to its lowest terms by the 2nd method. Ans. 117 795 265 Ans. 13 Ans. 8. Reduce 215 to its lowest terms by the 2nd method. Ans. į 9. Reduce 101 to its lowest terms by the 2nd method. Ans. 10. Reduce 792 to its lowest terms by the 2nd method. Ans. CASE IV. § 92. To reduce a whole number to an equivalent fraction having a given denominator. RULE, Multiply the whole number by the given denominator, and set the product over the said denominator. EXAMPLES. 1. Reduce 6 to a fraction whose denominator, shall be 4. Here 6x4=24; therefore 24 is the required fraction. It is plain that the fraction will in all cases be equal to the whole number, since it may be reduced to the whole number by Case I. Q. How do you reduce a whole number to an equivalent fraction having a given denominator ? How many thirds in 1? In 2? In 3? In 4? If the denominator be 5, what fraction will you form of 5? Of 4? Of 9?. Of 7? Of 8? With the denominator 6, what fraction will you form of 3? Of 4? Of 5? Of 6? Of 7? Of 9? 2. Reduce 15 to a fraction whose denominator shall Ans. 136. 3. Reduce 139 to a fraction whose denominator shall be 175. Ans. 4. Reduce 1837 to a fraction whose denominator shall be 181. 5. If the denominator be 837, what fractions will be formed from 327 ? From 889 ? From 575? be 9. CASE V. $ 93. To reduce a compound fraction to its equivalent simple one. EXAMPLES. 1. Let us take the fraction of First, =3x1: hence the fractions may be written of =3x} of; that is, three times one-fourth of q. But of s: hence we have, of=3x2=; a result which is obtained by multiplying together the numerators and denominators of the given fractions. 5 Ans.az When the compound fraction consists of more than two simple ones, two of them can be reduced to a simple fraction as above, and then this fraction may be reduced with the next, and so on. We therefore have the following RULE, I. Reduce all mixed numbers to their equivalent improper fractions by Case II. II. Then multiply all the numerators together for a numerator and all the denominators together for a denominator: their products will form the fraction sought. 2. Reduce } of } of to a simple fraction. Here {x}x= 3. Reduce sof of to a simple fraction. Here, şxx==1= by dividing the numerator and denominator of 19.o, first by 9 and then by 2, as shown in Case III. Or, sxx =, by cancelling or striking out the 3's and 6's in the numerator and denominator. By cancelling or striking out the 3's we only divide the numerator and denominator of the fraction by 3; and in cancelling the 6's we divide by 6. Hence, the value of the fraction is not affected by striking out like figures, which should always be done when they multiply the numerator and denominator. 4. Reduce i of, of is to a simple fraction. Here XX15=1032=*= Ans. Or, sx$*15=*= Ans. Q. What is a compound fraction? How do you reduce a compound fraction to a simple one? When you find like figures in the numerator and denominator, what do you do with them? Does this alter the value of the fraction? What is one-half of one-half ? One-half of one-third ? One-third of one-fourth? One-sixth of one-seventh? Three-halves of one-eighth? Six-thirds of two-ones ? 5. Reduce 21 of 6 of 7 to a simple fraction. Ans. 819=1025 6. Reduce 5 of } of 4 of 6 to a simple fraction. Ans. 7. Reduce 6 of 71 of 631 to a simple fraction. Ans. 1063 43 3 24 |