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inators are prime to each other, multiply both terms of each fraction by all the other denominators.

CoR. 3.-An integral number is reduced to a fraction by multiplying it by the denominator of the fraction.

EXAMPLES.

24

360
5049

3465

24.0

1. Reduce $, , and to a common denominator.

Ans. 48, 48, and 48. 2. Reduce 4, , and to a common denominator.

Ans. , 44, and 3. Reduce , J, and f to a common denominator.

Ans. 464, 44, and 444. 4. Reduce }, }, }, and it to a common denominator.

Ans. 447%, $175, 276, and 45 5. Reduce 1, 25, too, and Tooo to a common denominator.

Ans. 100%, 40%, 1780, and To EXEMPLIFICATION.—Since == == P, etc., and } =*=*=*=*s, etc., and 15, 8:35}, 4143, etc., and I, 1, 425, etc.; hence, if both terms of a fraction are divided by the same number, the value of the fraction is not changed.

COR.-When both terms of a fraction have a common factor, it may be canceled, and the fraction is thereby reduced to its lowest terms.

REM.—The common factor may be either a prime or a composite number, and it is the greatest common divisor of the terms.

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6=2

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EXAMPLES.

Ans. 4:9=1

3.

1. Reduce as to its lowest terms.
2. Reduce if to its lowest terms.
3. Reduce 1783 to its lowest terms.

Ans. Ans. 11:

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 jalue 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.

EXAMPLES. 1. Find the L. C. M. of 1 and 4. Evidently 1. 2. Find the L. C. M. of į and . Evidently to

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 4 and 3.

Evidently to 2. Find the greatest common divisor of and Z.

Evidently

ADDITION AND SUBTRACTION OF FRACTIONS.

EXAMPLES.

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1. Add and subtract and .

4 is L. C. D. 2. Add and subtract f and 1. 6 is L. C. D.

Ans. Sum, 7; difference, to 3. Add and subtract and 20 is L. C. D.

Ans. Sum, 7; difference, po 4. Add and subtract and . 30 is L. C. D,

Ans. Sum, 48; difference, to 5. Add and subtract f and 4. 42 is L. C. D.

Ans. Sum, 45 ; difference, de 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. L. C. D., 12. 2 x 4 8

3 x 3 9
3 x 4
12

4 x 3 12
8 + 9 = 17.

Sum 1 =18 7. Add , , and . Sum = 133 = 27%.

2용 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 its 4) 5 8

5 2 3 4 x 5 x 2 x3 = 120.
144 = 24.
19= 15.

120 = 10. .: 24, 15, and 10 are the multipliers of the fractions.

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12

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ADDITION OF FRACTIONS.

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EXAMPLES. 1. Add 1 and 1.

을 4, and if +=*= Sum. 2. Add } and 3. 10 is the least common denominator 公号 = Po and x = b; and to t to to; and to + = + = Sum.

. 3. Add }, {and 1. 12 = least common denominator,

*&= , *4 = t and Ixt=
and t + i + = + = 11: = Sum.
4. Add 3, 4, and J.
3) 3 4 5 9

1 4 5 3 3 X 4 X 5 X 3 = 180.
3 ) 180 4)

180 5 ) 180 9 ) 180
60
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 , J and fr. 5 X 9 X 11 - 495 7 x 7 x 11 = 539 9 x 17 x 9= 567

1601

= 693 ) 1601 ( 2413 on X 9 X 11 = 693

1386

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= Sum.

215

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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 fi, 1} and 14.

28344 = 11884. 7. Add H, 17 and . Here 120 'is the common denominator.

18% = 11 . 8. Add Paris, 11 and 47.

1}} = Sum. %

39

120

= 84.

SUBTRACTION OF FRACTIONS.

EXAMPLES. 1. Subtract is from H. C. D. = 60

Difference = - 14. 2. Subtract & from 1

C. D. = 176. Difference = M. 3. Subtract from 11.

4 C. D.

Difference = **. 4. Subtract from H. C. D.

Difference 5. Subtract from 17.

Difference = 14 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 = . %. Subtract is from H.

Difference = 1% 8. Subtract 5} from 8. Difference = 37. 9. Subtract 3 from 57. Difference = 11€. REM.--Reduce Ex. 8 and 9 to improper fractions.

= 2793.

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