166. To reduce compound fractions to simple ones 1. What is the value of of? SOLUTION. of equals(Prin. 3), and since OPERATION. of equals of equals 3 times, which by of = Prin. 1, equals 15 or §. 3X5 4X6 , Ans. Rule.-Multiply the numerators together and the denominators together, cancelling the factors common to both terms. NOTE.-Reduce whole or mixed numbers to fractions before commencing the reduction to a simple fraction. To reduce complex fractions to simple ones, see Art. 183. 1. A had of a ton of hay, which is as much as B has; how much has B? SOLUTION.If of a ton of hay is & of what B has, of what B has is of, which is of a ton, and of what B has is 4 times ton, which is of a ton. Therefore, etc. of a 2. A has of a certain sum of money, which is of what B has; how much has B? Ans. 3. A barrel of flour cost $84, and a barrel of fish cost as much; what was the cost of the fish? Ans. $6. 4. A lady bought of a yard of velvet, at $15% a yard: what did it cost? Ans. $131. 5. Henry having of a quart cf nuts, divided them among 8 of his schoolmates; what did each receive? Ans. To of this is 34 Ans.. 6. A owns 4 of the stock of a railroad, and times what B owns; how much does B own? 7. Mary shared 18 of a bushel of berries with 8 of her schoolmates; what did each receive? Ans. f. 8. I drew $580 from bank, which was of what still remained in bank; what was my bank deposit? Ans. $1276. 9. William lost of of his money, and found that $132 was of of of what remained; how much had he at first? Ans. $486. 'COMMON DENOMINATOR. 167. A Common Denominator is a denominator common to several fractions, or a denominator to which all may be reduced. 168. Similar Fractional Units are those which are of the same kind; as 3 fifths and 2 fifths. 169. Dissimilar Fractional Units are those which are of different kinds; as, 3 fourths and 3 fifths. Principle. A common denominator of several fractions must be a common multiple of their denominators. OPERATION. 105 188, 118, 198 1. Reduce,, and to a common denominator. SOLUTION. Since the product of the denominators of the fractions is a common multiple of their denominators, 4×5×7, which equals 140, will be the common denominator. Then multiplying both terms of by 5×7 we have 3195 (Prin. 5). Multiplying both terms of by 4×7, we have 13, etc. Hence the following Rule. Multiply both terms of each fraction by the de nominators of the other fractions. Reduce to a common denominator, 2 1536 1536' 1536 768 864 960 8.,,,, and . Ans. 57, 182, 112, 112, 118. 7. 1152, 9. Show that the common denominator of several fractions is a common multiple of the denominators of those fractions. LEAST COMMON DENOMINATOR. 170. The Least Common Denominator of several fractions is the smallest denominator to which all may be reduced. Principle. The least common denominator of several fractions is the least common multiple of their denominators. 1. Reduce, §, and to their least common denominator SOLUTION. We find the least common multiple of the denominators to be 24, hence 24 is the least common denominator. Dividing 24 by 3, the denominator of, we find we must multiply 3 by 8 to produce 24; hence multiplying both terms of by 8, we have (Prin. 5). Dividing 24 by 6, the denominator of 3, we find we must multiply 6 by 4 to produce 24; hence, multiplying both terms by 4, we laave 4, etc. Rule.-I. Find the least common multiple of the denomi nators, for the least common denominator. II. Divide the least common denominator by the denominator of each fraction, and multiply both terms by the quotient. NOTE.-Reduce compound fractions to simple ones, mixed numbers to improper fractions, and all to their lowest terms, before finding the least tommon denominator. To their least common denominator, 2. Reduce, 4, 7. Ans. 18, 18, 1. 42 55 Ans. 15, 15, 18. 68 Ans. H, fi, 13. 72 95 Ans. 7, 128, 120. Ans. II, II, 14, 117. 2521 216 Ans. 182, 490, 45, 48, 48. 70 ADDITION OF FRACTIONS. 171. Addition of Fractions is the process of fiuding the sum of two or more fractions. PRINCIPLES. 1. T add two or more fractions, they must express similar fractional units. 2. To add two or more fractions they must be reduced to a common denominator. 1. What is the sum of 4, §, and ? SOLUTION.-Reducing the fractions to a common denominator that they may express similar fractional units, we have 1, 3=34, 7=31; 18 twenty-fourths plus 20 twenty-fourths plus 21 twentyfourths equals 59 twenty-fourths. Hence the following OPERATION. $+8+= Rule. Reduce the fractions to a common denominator, then add the numerators and write the sum over the common denominator. NOTES.-1. Reduce compound fractions to simple ones, and reduce each fraction and the sum to lowest terms. 2. To add mixed numbers, add the integers and fractions separately, and then unite their sums. 8. Find the sum of 3,, Ans. 33. Ans. 3 9. Find the sum of 3, 1, 8, 10. Find the sum of §, }, %, H. Ans. 32. Ans. 1218. Ans. 943. Ans. 27388. Ans. 83474. Ans. 21. SUBTRACTION OF FRACTIONS. 172. Subtraction of Fractions is the process of finding the difference between two fractions. PRINCIPLES. 1. To subtract two fractions they must express similar fractional units. 2. To subtract two fractions they must be reduced to a common denominator. 1. What is the difference between § and 7? SOLUTION. Reducing the fractions to a common denominator that they may express similar fractional units, we have 39 and 45: 56 seventy-seconds minus 45 seventy-seconds equals 11 seventy-seconds. Hence the following OPERATION. Rule. Reduce the fractions to a common denominator, take the difference of the numerators, and write it over the common denominator. NOTE.-Reduce compound fractions to simple ones, and reduce each fraction and the difference to its lowest terms. |