CASE I. 180. To reduce whole or mixed,numbers to improper fractions. MENTAL EXERCISES. 1. How many fifths in 4? SOLUTION.-In one there are 5 fifths, and in 4 there are 4 times 5 fifths, or 20 fifths, which added to 3 fifths, equal 23 fifths; therefore 43 = 4. 2. How many fourths in 7? in 54? in 94? in 12&? 3. How many fifths in 63? in 7? in 8? in 10? 4. How many sixths in 43? in 83? in 91? in 11? 5. How many sevenths in 74? in 94? in 84? in 12§? 6. Describe the operation we perform in reducing a mixed number to a fraction. WRITTEN EXERCISES. 1. Reduce 27 to fourths. SOLUTION.-In one there are 4 fourths, and in 27 there are 27 times 4 fourths, or 108, which added to the, equals 1. Therefore, etc. Rule.-Multiply the whole number by the denominator of the fraction, add the numerator to the product, and write the denominator under the sum. 5. 11. 6. 27. Ans. 11. Ans. 181. Ans. 319; 970. Ans. 104. 10. 8211; 100%. Ans. 1744; 19882. CASE II. 161. To reduce improper fractions to whole or mixed numbers. MENTAL EXERCISES. 1. How many units in 2? SOLUTION.-In one there are 1, hence in 23 fourths there are as many ones as 4 is contained times in 23, which are 54. Therefore 254. 2. How many units in 3. How many units in 4. How many units in ? in 36? in 35? ? in 41? in 25? 30? in t? in t? 5. How many units in ? in ? in ? 6. Describe the process of reducing an improper fraction to a mixed number. WRITTEN EXERCISES. 1. How many units in ? SOLUTION. Since dividing both terms of a fraction by the same number does not change its value (Prin. 6), by dividing both terms by 5, we have 15, OPERATION. =151. or 154. Rule.-Divide the numerator by the denominator, and the quotient will be the whole or mixed number. Reduce to whole or mixed numbers, 162. To reduce fractions to higher terms. 163. Reducing a Fraction to higher terms is the process of reducing it to an equivalent fraction, having a greater numerator and denominator. MENTAL EXERCISES. 1. Change to twelfths. SOLUTION.-In one there are 14, and in there are 1 of 11, which are 1 and in there are 2 times, which are ; therefore. 2. Change and to twentieths; and to fifteenths. 3. Change and § to thirtieths; } and to sixteenths. 4. Change ‡ and § to seventieths; § and † to eighteenths. 5. Change and to seventy-seconds; and to 120ths. 6. Describe the process of reducing a fraction to higher terms. WRITTEN EXERCISES. 1. How many twentieths in ? SOLUTION. Since multiplying both terms of a fraction by the same number does not change its value (Prin. 5), we multiply both terms by the number which will give the required denominator, which we see is 4; hence, . Rule.-Multiply both numerator and denominator by the number which will give the required denominator. 164. To reduce fractions to lower terms. 165. Reducing a Fraction to lower terms is the process of reducing it to an equivalent fraction having a smaller aumerator and denominator. Principle. A fraction is in its lowest terms when the numerator and denominator are prime to each other. 8. Describe the process of reducing a fraction to lower terms. WRITTEN EXERCISES. 1. Reduce to fifths. SOLUTION. Since dividing both terms of a fraction by the same number does not change its value (Prin. 5), we may reduce to lower terms by dividing both numerator and denominator by 6; dividing, we have equal to ; and since the terms 4 and 5 are prime to each other, the fraction is in its lowest terms. Therefore, etc. Rule I.-Divide both terms successively by their common factors. Rule II.-Divide both terms by their greatest common divisor. Ans., 18856 288. 4. 18, 18. 21 5. H, H. 9. 840 1178. 4812 1512. 10. 726, 9549 6161 H, 72 6.7, H. Ans. 1, 1. 11. 41, 42524. 288 CASE V. Ans. &, f. Ans. H. Ans. 1, f. Ans. H, 4. Ans. H, H. 166. To reduce compound fractions to simple ones. 1. What is of ? SOLUTION. divided; MENTAL EXERCISES. ofis one of the three equal parts into which may be if each fourth is divided into 3 equal parts, 4 fourths, or the unit, will be divided into 4 times 3, or 12 equal parts; hence each part is of unit. Therefore, of is 1. 7. In reducing a compound fraction to a simple one, what do we do with the numerators, and what with the denominators? WRITTEN EXERCISES. 1. What is the value of of? SOLUTION.- of equals(Prin. 3), and since of equals of equals 3 times, which by Prin. 1, equals or f. Rule.-Multiply the numerators together and the denomi nators 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 oncs, see Art. 183. MENTAL EXERCISES. 1. A man gave his son of} of a dollar; how much did the boy receive? 13 2. A man owning of the stock of a bank, sold of his stock how much did he sell? 3. A boy earned of a dollar and gave his mother of it; how much did he give his mother? 4. Having of a bushel of apples I gave of them to my sister what part of a bushel did I give her? 5. Having lost of my money I found of what I lost, and then had $140; how much had I at first? 6. At of a dollar a gallon, what will § of a gallon of syrup cost? 7. At of a dollar a yard, what will of a yard of cloth cost? 8. At $6 a barrel, what will 9. At $8 a case, what will of a barrel of flour cost? of a case of slates cost? WRITTEN EXERCISES. 1. A had of a ton of hay, which is as much as B has; how much has B? of a 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. 2. A has of a certain sum of money, which is of what B has; how much has B? Ans. 1. 3. A barrel of flour cost $83, and a barrel of fish cost as much; what was the cost of the fish? Ans. $68. 4. A lady bought 3 of a yard of velvet, at $15% a yard: what did it cost? Ans. $138. 5. Henry having of a quart of nuts, divided them among 8 of his schoolmates; what did each receive? Ans. To 6. A owns of the stock of a railroad, and 4 of this is 34 times what B owns; how much does B own? Ans.. 7. Mary shared 18 of a bushel of berries with 8 of hel schoolmates; what did each receive? Ans. 8. I drew $580 from bank, which was of what still reained 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 |