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THEOREM. The least common multiple of two or more fractions has for its numerator the least common multiple of all the numerators, and for its denominator the greatest common divisor of all the denominators.
The least common multiple of all the numerators will evidently be a common multiple of all the fractions; for each numerator regarded as an integer is a multiple of its denominator regarded as · a fraction; and when the denominators are prime to each other, this common multiple will be the least common multiple of the fractions; but when all the denominators have a common factor, then the galue of each fraction is reduced by it, and the common multiple may be reduced by the same; hence, the greatest common divisor of all the denominators is the denominator of the least common multiple.
EXA MPLES. 1. Find the L. C. M. of 1 and 1.
Evidently 1. 2. Find the L. C. M. of I and t. Evidently .
THEOREM. The greatest common divisor of two or more fractions has for its numerator the greatest common divisor of all the numerators, and for its denominator the least common multiple of all the denominators.
The greatest common divisor of all the numerators is evidently the numerator of the greatest common divisor of the fractions; and as multiplying the denominators divides the fraction and the less the denominator the greater the value of the fraction, hence the least common multiple of all the denominators is the denominator of the greatest common divisor of the fractions.
EXAMPLES. 1. Find the greatest common divisor of and .
Evidently to 2. Find the greatest common divisor of and .
ADDITION AND SUBTRACTION OF FRACTIONS.
1. Add and subtract 1 and 1.
4 is L. C. D. 2. Add and subtract and 1. 6 is L. C. D.
Ans. Sum, 7; difference, f. 3. Add and subtract f and . 20 is L. C. D.
Ans. Sum, 7; difference, po 4. Add and subtract and . 30 is L. C. D.
Ans. Sum, 8; difference, o . 5. Add and subtract fand . 42 is L. C. D.
Ans. Sum, 45; difference, s. CoR.- When the denominators have no common face tor, then their product is the L. C. D., and each numerator is multiplied by all the denominators except its own 6. Add and t. L. C. D., 12. 2 x 4 8
3 x 3 9
4 x 3 12
Sum = 1 = 18.
12 7. Add , , and H.
Sum = 133 = 21%. As no two numbers have a common factor, the L. C. D. is the product of all the denominators; and then as each denominator is multiplied by the other two denominators, so each numerator must be multiplied by the product of all the denominators except its own. 8. Add 4, $, and to 4) 5 8 12
5 2 3 4 x 5 x 2 x3 = 120.
120 = 24. 192 = 15. 120 = 10. :: 24, 15, and 10 are the multipliers of the fractions.
ADDITION OF FRACTIONS.
EXAMPLES. 1. Add į and 1.
4, and +4= = Sum. 2. Add 1 and 1. 10 is the least common denominator Exeo Po and = ko ; and t + t = %= Sum.
to 3. Add 1, fand 1. 12 = least common denominator,
1 4 5 3 .. 3 X 4 X 5 X 3 = 180.
45 36 20 REM.—The terms of the 1st fraction must be multiplied by 60, the 2d by 45, the 3d by 36, and the 4th by 20.
COR.-Fractions are reduced to a common denominator by multiplying both terms of the fraction by the quotient, obtained by dividing the common denominator by the denominator of each given fraction.
5. Add 5, f and 11 5 x 9 x 11 = 495 ny xp x 11 = 539 9 x x 9= 567
= 7 x 9 x 11 = 693
693 ) 1601 ( 2411 = Sum.
REM. When the denominators are prime to each other, as no two have a common factor, the least common denominator is the product of all the denominators, and each numerator is multiplied by all the denominators except its own. 6. Add ki, 11 and 14.
187% = 118!! . 17. Add 16, 17 and 12%
Here 120 is the common denominator.
15% = 11 . 8. Add Par to, 16 and 17.
1% = Sum.
SUBTRACTION OF FRACTIONS.
EXAMPLES. 1. Subtract fe from H. C. D. = 60
Difference = 18. 2. Subtract from 1 C. D. = 176.
Difference = 1 3. Subtract 4 from 1. C. D. = 84.
Difference = ft. 4. Subtract from .
C. D. = 2793. Difference = 5. Subtract from . Difference = 44
REM.—The above examples should be repeated, or similar ones given, until the class is familiar with addition and subtraction of fractions.
6. Subtract from A. Difference 17. 7. Subtract fo from H.
Difference = 1960 8. Subtract 5from 8f. Difference = 31. 9. Subtract 3f from 57. Difference = 11€. REM.--Reduce Ex. 8 and 9 to improper fractions.
5 6 8 2793
MULTIPLICATION OF FRACTIONS.
The product of two proper fractions is less than either fraction.
For, if a number is multiplied by one, the product is the same as the number multiplied. If the multiplier is greater than one, the product is greater than the number; and if the multiplier is less than one, the product is less than the number.
In the multiplication of two proper fractions, each factor is less than one; hence the product is less than either fraction.
1. Multiply k by 1.
Ans. f x1 = . 2. Multiply by 2.
Ans. x 2 =$. 3. Multiply by :
Ans. x = $. It is evident that f multiplied by 1 = j, that ff x 2 is $, and f x for 1 time is }; and, as alternating the factors does not change the product, therefore, 1xf=}, 2 x = = $, and txt = f. 4. Multiply } by .
Ans. } * = . Cor. 1.-In multiplying by a fraction the numerator is a multiplier and the denominator a divisor.
COR. 2.-In the multiplication of fractions, the product of all the numerators is the numerator of the product; and the product of all the denominators is the denominator of the product.