ART. 137. To reduce improper fractions to whole or mixed numbers. Ex. 1. How many dollars in 37 dollars? OPERATION. 16) 37 (216 32 5 Ans. $25. This question may be analyzed by saying, As 16 sixteenths make one dollar, there will be as many dollars in 37 sixteenths as 37 contains 16, which is 2 times $216. = RULE. - Divide the numerator by the denominator, and the quotient will be the whole or mixed number. 876 EXAMPLES FOR PRACTICE. 2. Reduce 96 to a whole number. 3. Change 178 to a mixed number. 4. Change to a mixed number. 5. Change 1735 to a mixed number. 6. Reduce 1000 to a mixed number. 7. Reduce 378 to a whole number. 8. Change 567 to a whole number. 9. Reduce 43 to a mixed number. 10. Reduce 1848 to a mixed number. 459 ART. 138. To reduce a compound fraction to a simple fraction. Ex. 1. Reduce of to a simple fraction. OPERATION. Ans.. To show the reason of the operation, this question may be analyzed by saying, that, if of an apple be divided into 5 equal parts, one of these parts is of an apple; and, if of be, it is evident that of will be 7 times as much. 7 times is ; and, if of 71 be 5, of 71 will be 4 times as much. 4 times are Or, by multiplying the denominator of by 5, the denominator of, it is evident we obtain of 75, since the parts into which the number or thing is divided are 5 times as many, and consequently only as large as before. Again, since QUESTIONS. Art. 137. What is the rule for reducing improper fractions to whole or mixed numbers? Give a reason for the rule.— - Art. 138. How do you reduce a compound fraction to a simple one? Give the reason for the operation. of 5, of will be 4 times as much; and 4 times 5. This process will be seen to be precisely like the operation. Ex. 2. Reduce of of of § of to a simple fraction. Ans. 3 RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. NOTE 1. All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying the numerators and denominators together. NOTE 2. When there are factors common to both numerator and denominator, they may be cancelled in the operation. Ans. 220. 7. Required the value of of of 14 of 13 of 5. Ans.. 8. Reduce of of of of to a simple fraction. Ans. 77 9. Reduce of of 7 off of 43 to a simple fraction. 10. Reduce 1 Ans.. of 8 of to a simple fraction. Ans. 35 of 22 of 15 of 95 to a whole number. Ans. 3. 12. Reduce of of of 8 of 11 to a simple fraction. Ans. 1. QUESTIONS.- When there are common factors in the numerator and denominator, how may the operation be shortened? What is the rule? What must be done with all whole and mixed numbers in the compound fraction? How may the operation be shortened by cancelling? A COMMON DENOMINATOR. ART. 139. A common denominator of two or more fractions is a common multiple of their denominators. The least common denominator is the least common multiple. NOTE.Fractions have a common denominator, when all their denominators are alike. ART. 140. To reduce fractions to a common denominator. Ex. 1. Reduce 2, §, and 7, to a common denominator. Ans. 11, 192, 199. 160 168 We first multiply the numerator of by the denominators 6 and 8, and obtain 144 for its numerator. We next multiply the numerator of by the denominators 4 and 8, and obtain 160 for its numerator; and then we multiply the numerator of by the denominators 4 and 6, and obtain 168 for its numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation. By this process, since the numerator and denominator of each fraction are multiplied by the same numbers, and consequently both being increased in the same ratio, their relation to each other is not changed, and the value of the fraction remains the same. (Art. 133.) Therefore, Multiplying the numerator and denominator of a fraction by the same number does not alter the value of the fraction. RULE.-Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together for a common denominator. NOTE 1. Compound fractions, if any, must first be reduced to simple ones, and whole or mixed numbers to improper fractions. NOTE 2. Fractions may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor. QUESTIONS. Art. 139. What is a common denominator of two or more fractions? What is the least common denominator? When have fractions a common denominator?- Art. 140. How do you find a common denominator of two or more fractions? Give the reason of the operation. What inference is drawn from it? What is the rule for finding a common denominator? How may fractions having a common denominator be reduced to lower terms? EXAMPLES FOR PRACTICE. 2. Reduce and to common denominators. Ans. 1,, or, 1. 3. Reduce 3,, and, to a common denominator. Ans. 18, 33, 45. 4. Reduce,, and, to a common denominator. Ans. 18, 21, 218. 5. Reduce,, and, to a common denominator. Ans. 3, 1, 316, or 38, 18, 38. 288 135 324 6. Reduce, 2, 3, and 4, to a common denominator. 160 384 30 Ans. 68, 88, 848, 648, or 120, 120, 125, 120. ART. 141. To reduce fractions to their least common denominator. Ex. 1. Reduce,, and, to the least common denomi nator. OPERATION. 1 2 common denominator. 33 6 12 21 2 4 3 4 1 1 2 = 8 numerator for 62×5= 10 numerator for : 12 1x7 = 7 numerator for 3 × 2 × 2 = 12, the least common denominator. = = Having first obtained a common multiple, or denominator of the given fractions, we take the part of it expressed by each of these fractions separately for their new numerators. Thus, to get a new numerator for, we take of 12, the common denominator, by dividing it by 3, and multiplying the quotient 4 by 2. We proceed in this manner with each of the fractions, and write the numerators thus obtained over the common denominator. NOTE. The change in the terms of the fractions, in reducing them to the least common denominator by this process, depends upon the same principle as explained in the preceding article. RULE.-1. Find the least common multiple of the denominators for the least common denominator. 2. Divide the least common denominator by the denominator of each of the given fractions, and multiply the quotients by their respective numerators, for the new numerators. NOTE.Compound fractions must be reduced to simple ones, whole QUESTIONS. Art. 141. How do you find the least common denominator of two or more fractions? Upon what principle does this process depend? What is the rule for reducing fractions to their least common denominator? What must be done with compound fractions, whole numbers, and mixed numbers? and mixed numbers to improper fractions, and all to their lowest terms before finding the least common denominator EXAMPLES FOR PRACTICE. 2. Reduce, 4, 5, and 7, to the least common denominator. 90 96 Ans. 12, 126, 128, 125. 1010, 3. Reduce,,, and, to the least common denominator. Ans. 1485, 1980' 1980' 1980° 792 880 360 4. Reduce, o, and 72, to the least common denominator. 35 Ans. 28, 36, 310. 5. Reduce, 14, 1, and 5%, to the least common denomi Ans. 18, 18, 11, 152. §,, and, to the least common denomiAns. 12, 14, 24, 24, 24, 14. 9 6 3 7. Reduce,, 1, 1, †, and, to the least common denomiAns. 18, 34, 18, 36, 36, 36. 8. Reduce,, and 72, to the least common denominator. nator. 9. Reduce 74, 5, 7, and 8, to nator. 10. Reduce 2, 4, 5, 7, and 9, to nator. 44.9 Ans. 38, 18, 3. the least common denomiAns. 341, 244, 308, 352. 449 44 the least common denomiAns. 2, 16, 20, 28, 36 ADDITION OF COMMON FRACTIONS. ART. 142. ADDITION of Fractions is the process of finding the value of two or more fractions in one sum. ART. 143. To add fractions that have a common denominator. Ex. 1. Add 4, 4, 4, 4, and §, together. OPERATION. 1 2 4 5 6 Ans. 24. These fractions all being sevenths, that is, having 7 for 7 +7 +7 +7 +7=18-24. a common denominator, we ++++ add their numerators together, and write their sum, 18, over the common denominator, 7. Thus we obtain 18 Hence, to add fractions having a common denominator, Write their sum over the common denominator, and reduce the fraction, if necessary. QUESTIONS. Art. 142. What is addition of fractions? Art. 143. How are fractions having a common denominator added? Give the reason. |
