« AnteriorContinuar »
133. The value of a fraction is the quotient arising from the division of the numerator by the denominator. Thus, the value of , or 6 = 2, is 3; and the value of , or 3 • 4, is .
134. Reduction of Fractions is the process of changing their form without altering their value.
A fraction is in its lowest terms, when its terms are prime to each other. (Art. 112.)
135. To reduce a fraction to its lowest terms. Ex. 1. Reduce 188 to its lowest terms.
We divide the terms of the fraction by 2, a factor 2) est common to them both, and obtain f. We divide, 3) }
again, both terms of g by 3, a factor common to
them, and obtain . Now, as 1 and 3 are nunibers prime to each other, the fraction is in its lowest terins.
The same result would have been produced, if we had divided the terms by 6, the greatest common divisor.
Since the numerator and denominator of a fraction correspond to the dividend and divisor in division (Art. 132), dividing both by the same number, or canceling equal factors in both (Art. 115), changes only the form of the fraction, while the value expressed remains the
Therefore, Dividing the numerator and denominator of a fraction by the same number does not alter the value of the fraction.
RULE. Divide the numerat and denominator by any number greater than 1, that will divide them both without a remainder, and thus proceed until they are prime to each other. Or, Divide both terms of the fraction by their greatest common divisor.
EXAMPLES FOR PRACTICE.
2. Reduces to its lowest terms. 3. Reduce to its lowest terms. 4. Reduce je to its lowest terms. 5. Reduce que to its lowest terms. 6. Reduce it to its lowest terms. 7. Reduce it to its lowest terms. 8. Reduce 367 to its lowest terms.
Ans. * Ans. Ans. š. Ans. ß.
Ans. : Ans. 336.
133. What is the value of a fraction ? - 134. What is reduction of fractions ? When is a fraction in its lowest terms ? - 135. Why docs dividing both ternis of a fraction by the same number not alter the valuc ? Has $ the same value as 198? Why? Repeat the rule.
Ans. 11t 9. Reduce iit to its lowest terms.
Ans. 335. 10. What is the lowest expression of 448?
136. To reduce a mixed number to an improper fraction.
Ans. 3. Ex. 1. In 72 how many fifths ?
Since there are 5 fifths in 1 whole one, there will be 5 times as many fifths as whole ones; therefore, in 7 there are 35 fifths, and the 3 fifths being added make 38 fifths, which are expressed thus, Be.
38 fifths RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place the sum over the given denominator.
Note. — To reduce a whole number to a fraction of the same value, having a given denominator, we multiply the whole number by' the given denoninator, and make the product the numerator ; thus 5, reduced to a fraction, having 3 for a denominator, becomes the
EXAMPLES FOR PRACTICE. 2. In 8 dollars how many sevenths ?
Ans. 3. In 34 oranges how many fourths ? 4. In 9 gallons how many elevenths ? 5. Reduce 8 to an improper fraction. 6. Reduce 157 to an improper fraction.
Ans. 182 7. In 187 how many ninths ?
Ans. 192. 8. In 161114 how many one hundred and seventeenths ?
Ans. 18848. 9. Change 43111 to an improper fraction. Ans. 5147 10. What improper fraction will express 2713? Ans. 36. 11. Change 1111tt to an improper fraction. Ans. 12112 12. Change 125 to an improper fraction. Ans. 12.
13. Change 25 to an improper fraction, having 6 for a denominator.
Ans. 14. 14. Reduce 75 to ninths.
Ans. zS. 15. Change 343 to the form of a fraction. Ans. 343. 16. Reduce 84 to fifteenths.
Ans. Ans. 103 Ans. 11
136. What is the rule for reducing a mixed number to an improper frar. tion? The reason ? How do you reduce a whole number to a fraction of the same value, having a given denominator ?
137. To reduce improper fractions to whole or mixed numbers. Ex. 1. How many dollars in f dollars ?
16 ) 37 (216
Since 16 sixteenths make one dollar, there will be as many dollars in 37 sixteenths of a dollar as 37 contains times 16, or $ 216
RULE. — Divide the numerator by the denominator, and the quotient will be the whole or mixed number.
EXAMPLES FOR PRACTICE.
2. Reduce g@ to a whole number.
Ans. 12. 3. Change 18 to a mixed number.
Ans. 104 4. Change Y1 to a mixed number.
Ans. 10T 5. Change 93.5 to a mixed number.
Ans. 1878 6. Reduce 10.00 to a mixed number.
Ans. 1424. 7. Reduce 373 to a whole number.
Ans. 1. 8. Change 567 to a whole number.
Ans. 567. 9. Reduce 7442 to a mixed number.
Ans. 99% 10. Reduce 1849 to a mixed number.
Ans. 43 138. To reduce a compound fraction to a simple fraction. Ex. 1. Reduce of Yr to a simple fraction.
Ans. OPERATION. If It be divided into 5 equal parts, one of # XT1= these parts is 5's; and if of be 3'5, it is
evident that } of 11 will be 7 times as much. 7 times z's is 35; and if } of vi be 35 of 11 will be 4 times as much.
4 times 35 are Or, by multiplying the denominator of 11 by 5, the denominator of , it is evident we obtain } of 11 575, since the parts into which the number or thing is divided are 5 times as many, and consequently only š as large as before. Again, since ś
137. What is the rule for reducing improper fractions to whole or mixed numbers? A reason for the rule. — 138. How do you reduce a compound fraction to a simple one ? The reason for the operation ?
of t1 = 35, $ of It will be 4 times as much; and 4 times 35 = This process will be seen to be precisely like the operation.
Ex. 2. Reduce of of 4 of 5 of 11 to a simple fraction.
OPERATION BY CANCELLATION. 2
3 X 4 X 5 X 6 X 7 4 X 5 X 7 X 9 X 11
The numerators and denominators which are common factors we cancel according to the principles of cancellation. (Art. 117.)
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.
NOTE 2. - When there are factors common to both numerator and denominator, they may be canceled in the operation.
EXAMPLES FOR PRACTICE.
3. What is off of $ ?
Ans. = 36. 4. What is of I of 7 ? 5. What is of it of f of 4 ?
Ans. 375 6. Change 14 of 1 of of zo of 7 to a simple fraction.
7. Required the value of of 1 of 1 of 33 of 54.
8. Reduce of g of ii of of to a simple fraction.
Ans. 3. 9. Reduce 4 of 4 of 7 of 1 of 44 to a simple fraction.
Ans. . Ans. dr
10. Reduce to off of Ir to a simple fraction.
Ans. 3. 12. Reduce £ of is of te of 8 of 4 to a simple fraction.
138. When there are common factors in the numcrator and denominator, how may the operation be shortened? The rule? What must be done with all whole and mixed numbers in the compound fraction? How may the operation be shortened by canceling?
A COMMON DENOMINATOR. 139. A Common Denominator of two or more fractions is a common multiple of their denominators.
The Least Common Denominator of two or more fractions is the least common multiple of their denominators.
Note. — Fractions have a common denominator, when all their denomi nators are alike.
140. To reduce fractions to a common denominator.. Ex. 1. Reduce , , and ; to a common denominator.
Ans. 119, 199, 99.
160 TS2, 168 192•
3 X 6 X 8 1 4 4, new numerator.is
4 X 6 X 8=192, common denominator. We first multiply the numerator off by the denominators 6 and 8, and obtain 144 for a new numerator. We next multiply the numerator of á by the denominators 4 and 8, and obtain 160 for a new numerator ; and then we multiply the numerator of } by the denominators 4 and 6, and obtain 168 for a new 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, only the form of the fraction is changed, while the quotient arising from díviding the numerator by the denominator, or the value of the fraction (Art. 133), remains the
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.
139. What is a common denominator of two or more fractions? What is the least coinmon denominator? When have fractions a common denominator? - 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 frac tions having a common denoininator be reduced to lower terms ?