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and 2. 2 × 4

8

=

3 x 4 12 8+ 9 = 17.

Ans. Sum, f; difference,

L. C. D., 12.

COR.-When the denominators have no common fac tor, then their product is the L. C. D., and each numerator is multiplied by all the denominators except its own

6. Add

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=

7. Add,, and . Sum = 133 133 218. 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 7.

4) 5 8 12

120.

5 2 3 4x5×2×3 120 = 24. 180 15. 120 10. . 5, 8, and 12 are the multipliers of the fractions.

=

ADDITION OF FRACTIONS.
EXAMPLES.

1. Add and 1.

2. Add and .

=

†, and 4 + 1 } = Sum.

=

.

10 is the least common denominator. fo and X ; and + Sum. fo = Fo 3. Add 1, and 12 = least common denominator. 1 × 8 = 12, 14 and 1} = 12, and ++ √ = }} = 1}= Sum. 4. Add 1, 1,

=

29

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

36

60 45 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.

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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 1,17 and 13.

3982 20828

7. Add 18, 17 and . Here 120 is the common denominator.

37 120

158 = 14% = 1.
=1

8. Add f,, 11 and 17.

= 118824.

20825

1. Subtract

C. D.

2. Subtract

C. D.

3. Subtract from 11.

C. D.

84.

4. Subtract

from 44.

C. D.

2793.

5. Subtract

from 7.

SUBTRACTION OF FRACTIONS.

EXAMPLES.

from 1.

60

from 15.

176.

133 Sum.

6. Subtract from

7. Subtract

from 18.

8. Subtract 5 from 8.

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REM.-The above examples should be repeated, or similar ones given, until the class is familiar with addition and subtraction of fractions.

Difference = 17.
Difference =

9. Subtract 3 from 51.

REM.-Reduce Ex. 8 and 9 to improper fractions.

19

Difference= 31.

Difference = 173.

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MULTIPLICATION OF FRACTIONS.

THEOREM I.

The product of any number multiplied by a proper fraction is less than the number itself.

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AXIOM 7.-If any number be both multiplied and divided by the same number, the value of the original number is not changed.

If any number is multiplied by 2 and divided by 3, it is diminished.

.. The product of any number multiplied by a proper fraction is less than the number itself.

COR. 1.-The product of two proper fractions is less than either fraction.

COR. 2.-In multiplying by a fraction, the numerator is a multiplier and the denominator a divisor.

COR. 3. The factors may be alternated.

COR. 4.-In the multiplication of fractions, the product of all the numerators will be the numerator of the product; and the product of all the denominators will be the denominator of the product.

REM.-Cancellation can be applied to the multiplication of fractions, as in the division of integers.

EXAMPLES.

1. Multiply 1, 3, 4, †, k, f, I, §, 1%, H, H. 1××××××××××

H = 1

By analysis, of = 1, of = 1, 1 of 4 = 1, of 1 2 † f=t, f of 4 = 4, + of } = 1, of = t, t of fo=10, 3 1% 1 of 1 = 11, 1 of 1 = 1.

REM.-The expression, of of, etc., is reduced to a simple fraction by the multiplication of their factors.

2. Multiply 3, 1, and ; thus,

2

1 11 1
× × = 1, product.

3

3. Multiply and 2.

f/ × 1 = 1, product.

THEOREM II.

The product of two proper fractions is less than either fraction.

For if a number is multiplied by 1, the product is the same as the number. If a number is multiplied by any number greater than one, the product is greater than the number. If a number is multiplied by any number less than one, the product is less than the number.

4. Multiply 35

by 9.

2 × 9 = 21= 64
35 × 9

= 315

321 Product.

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