NOTE. It has been shown 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. 2. Reduce 99 to a whole or mixed number. Ans. 12. 3. In of yards of cloth, how many yards? Ans. 4. In 5 of bushels, how many bushels? Ans. 58bu. 5. If I give of an apple to each one of 15 children, how many apples do I give? Ans. 5. 6. Reduce 327 3672 50287 987625, to their whole or mixed numbers. Ans. 25, 24, 71709, 1347317. 7. If I distribute 878 quarter-apples among a number of boys, how many whole apples do I use Ans. 125 153 6941 72301 3114' 8. Reduce 62587, 4927, 2641674, to their whole or mixed numbers. Ans. 9. Reduce 147254149, 145260, 62015735, to their whole or mixed numbers. Ans. 315044452, 1345, 79475293. CASE II. 7803 129. To reduce a mixed number to its equivalent improper fraction. 129. 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 seveneighths? 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. 1. Reduce 4 to its equivalent improper fraction. Here 4x5 20: then 20+4=24; hence, 24 is the equivalent fraction. Ans. 24. This rule is the reverse of Case I. In the example 4 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. 2. Reduce 47 to its equivalent improper fraction. Ans. 287. 3. Reduce 67637, 87433, 69047, 3678, to their equivalent improper fractions. Ans. 34513, 7899, 69047, 38177. 100 4. Reduce 8473, 874876, 67426398, to their equivalent improper fractions. 5. How many 200ths in 675187? 6. How many 151ths in 187? Ans. 7. Reduce 625 to an improper fraction. Ans. 28278. 8. Reduce 15617, to an improper fraction. 407 CASE III. 130. To reduce a fraction to its lowest terms. 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 both terms of the fraction without a remainder. 130. When is a fraction in its lowest terms? (see Art. 127.) 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 sixeighths? Of nine-twelfths? Of sixteen thirty-sixths? Of tentwentieths? Of fifteen twenty-fourths? Of sixteen-eighteenths? Of nine-eighteenths? 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 to its lowest terms. 5) 70 7)14 2 5)1757)35-5' 1ST METHOD. which are the lowest terms of the fraction, since no number greater than 1 will divide the numerator and denominator without a remainder. Ans. J. 8. Reduce 1 to its lowest terms by the 2d method. 9. Reduce 1157 to its lowest terms by the 2d method. 623 Ans. 10. Reduce 792 to its lowest terms by the 2d method. Ans. 4. 1386 CASE IV. 131. To reduce a whole number to an equivalent fraction having a given denominator. Since the denominator of a fraction shows into how many equal parts unity has been divided, it is plain that if we multiply it by the number of units so divided, the product will be equal to the entire number of parts taken. Hence, 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. Since each unit is to be divided into 4 parts, it follows that the number of parts in 6 units will be expressed by 6×4=24; hence the required fraction is 24. 2. Reduce 15 to a fraction whose denominator shall be 9. Ans. 135. 3. Reduce 139 to a fraction whose denominator shall be 175. Ans. 4. Reduce 1837 to a fraction whose denominator shall be 181. Ans. 5. If the denominator be 837, what fractions will be formed from 327 ? From 889? From 575? 6. If the denominator be 216, what fractions will be formed from 876? From 306? From 5047 ? CASE V. 132. To reduce a compound fraction to its equivalent simple one. 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. 131. 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? 132. What is a compound fraction? How do you reduce a compound fraction to a simple one? When you find like factors 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? Onehalf of one-third? One-third of one-fourth? One-sixth of oneseventh? Three-halves of one-eighth? Six-thirds of two-ones? EXAMPLES. 1. Let us take the fraction of §. First, 3× hence the fractions may be written 3× of; that is, three times one-fourth of $. of: hence we have, of But a result which is obtained by multiplying together the numerators and denominators of the given fractions. 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. 2. Reduce of of to a simple fraction. Here, 3. Reduce of of to a simple fraction. Here, 90 Ans. by dividing the numerator and denominator first by 9 and then by 2, as shown in Case III. 4. Reduce of of to a simple fraction. Ans. f. 5. Reduce 24 of 6 of 7 to a simple fraction. Ans. 12-102. 6. Reduce 5 of of of 6 to a simple fraction. 7. Reduce 6 of 71 of 63 to a simple fraction. Ans. 106343 324 METHOD BY CANCELLING 133. The work may often be abridged by striking out or cancelling common factors in the numerator and denominator, which is merely dividing both terms of the fraction by the same number. In every operation in fractions, let this be done whenever it is possible. EXAMPLES. 1. Reduce of of to a simple fraction. Here, 5885 8x8x7=7 |