Ex. 1. How many dollars in 7 dollars? OPERATION. Ans. $216. This question may be analyzed by saying, As 16 16) 37 (2% sixteenths make one dollar, there will be as many dollars in 37 sixteenths as 37 contains 16, which is 2 times, = $25. 32 5 16 16 RULE.-Divide the numerator by the denominator, and, if there be a remainder, place it over the denominator at the right hand of the whole number. ART. 138. To reduce a compound fraction to a simple fraction. Ex. 1. Reduce OPERATION. 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 of an apple; and, if of be, it is evident that will be 7 times as much. 7 times is ; and if be, of will be 4 times as much. 4 times is of of by 5, the denomi 35, since the parts nator of, it is evident we obtain of √1 into which the number or thing is divided are 5 times as many, and consequently only as large, as before. Again, since of 5, of will be 4 times as much; and 4 times 28. This process will be seen to be precisely like tho operation. 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. Ex. 2. Reduce of of of § of to a simple fraction. Ans. RULE.-1. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator, and then reduce the fraction to its lowest terms. 2. If there are factors in the numerator similar to those in the denominator, cancel them in the operation. NOTE. All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying the numerators and denominators together. 7. Required the value of of of 1 of 1 of 52. 8. Reduce of of of of to a simple fraction. 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? 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, §, and to a common denominator. OPERATION. 160 Ans. 11, 182, 189. 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 the numerator and denominator of each fraction are multiplied by the same numbers, and consequently, both being increased an equal number of times, 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 into all the denominators except its own for a new numerator; and all the denominators into each other for a common denominator. NOTE. Fractions of this form 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. 3. Reduce 3, 4, and to a common denominator. to a common denominator. to a common denominator. to a common denominator. ART. 141. To reduce fractions to their least common denominator. Ex. 1. Reduce, %, and to the least common denomi nator. 33 6 12 21 2 4 1 2 OPERATION. 112 common denominator. 3 4X2: = 8.numerator for 6 2 X 510 numerator for &=12. 12 1X7 = 7 numerator for 12. 3 × 2 × 2 = 12, thẹ 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 of the several fractions, and it will be their 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, and their products will be the numerators of the fractions required. 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,,, and to the least common denominator. 3. Reduce,,, and to the least common denominator. 4. Reduce, o, and 7 to the least common denominator. 5. Reduce,,, and 5% to the least common denominator. 6. Reduce 1, 2, §, §, 7, and 1 to the least common denominator. 7. Reduce,, 1, 1, †, and to the least common denominator. 8. Reduce §,, and to the least common denominator. 9. Reduce 7, 5, 7, and 8 to the least common denominator. 10. Reduce †, 4, 5, 7, and 9 to the least common denomi nator. ADDITION OF VULGAR FRACTIONS. ART. 142. ADDITION of Vulgar 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 7. 7 7 7 7 =4824. Ans. 24. These fractions all being sevenths, that is, having 7 for a common denominator, we simply add their numerators together, and write the sum over the common denominator. RULE.-Add together the numerators of the fractions, and place their sum over the common denominator, and reduce the fraction if necessary. QUESTIONS. Art. 142. What is addition of vulgar fractions? - Art. 143. What is the rule for adding fractions having a common denominator? Give the reason for the rule. |