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EXAMPLE.
Reduce 18 to an improper fraction with denominators 3, 12.

18X3 54
18=

3 3

18X12 216 18=

12 12

IV. TO REDUCE FRACTIONS TO THEIR LEAST COMMON DENOMINATOR.

Rule-1. Find the L.C.M. of the denominators; this will be the L.C.D.

2. Multiply numerator and denominator of each fraction by the quotient of the L. C.D., divided by its denominator.

EXAMPLE. Reduce to their L. C. D. , \, zi, 39, n.

L. C. D. = 3X3X7X11 - 693.

4X77 308

=

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693 495

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11 11X63 693 N.B. It is always better to let the factors of the L. C. D. appear, as then the several multipliers are readily found by casting out the factors of each denominator, and multiplying the rest.

B.-Addition of Fractions.

1. TO ADD FRACTIONS. Rule--1. Reduce the fractions to their L. C. D.

2. Add the numerators for the numerator of the sum; which will have the L. C. D. for denominator.

3. Reduce the result to its lowest terms. 4. If any of the fractions be mixed numbers, add the integers separately.

EXAMPLES. 1. Add together 2, 3, 4, 4, 6.

L.C.D= 2X2X3X5 = 60.

I 1X30 30

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II. TO REDUCE A MIXED NUMBER, TO AN IMPROPER FRACTION.

Rule. Multiply the integer by the denominator, add the numerator to the product. The sum is the numerator of the fraction, which has the same denominator as before.

EXAMPLES. 1. Reduce 3} to an improper fraction.

5 3X7+5 26 3-=

7 7 7 2. Reduce 891821 to an improper fraction.

1817 89X1824+1817 891824

1824

162336+1817

1824 164153

1824

C.-Subtraction of Fractions.

1. TO SUBTRACT ONE SIMPLE FRACTION FROM ANOTHER. Rule-1. Reduce the fractions to their L. C. D. 2. Subtract the numerator of the subtrahend from that of the diminuend.

3. The difference is the numerator of remainder, and the L. C. D. the denominator.

4. Reduce the result to its lowest terms.

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II.

TO SUBTRACT ONE MIXED NUMBER FROM ANOTHER. Rule-1. Reduce the fractions to their L. C. D.

2. If the numerator of the fractional subtrahend be less than that of the diminuend, subtract integer and fraction separately.

3. If the numerator of the subtrahend be greater than that of the diminuend, increase the latter by the C. D., and the integer of the subtrahend by one, and proceed as before.

EXAMPLES. 1. Subtract 37 from 71.

L. C. D. = 7X4 = 28.
1 1 7 4

7 4
7-- 3-=7 - 3-=7-3+
7 28 28

28 28
3
=4
28

C2

2. Subtract 3} from 87.

L. C. D. = 7X8 = 56.

1 7 8 49
..8-- 3-= 8 - 3-
7 8 56 56

64-49
=8-47

56 15 15 = 4+- =4

56 56

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TO SUBTRACT SEVERAL FRACTIONS FROM THE SUM OF SEVERAL

OTHERS.

Rule-1. Add together those which are to be subtracted, and the others in two separate sums. 2. Proceed to subtract by the previous rule.

EXAMPLES. 1. Find the value of f-} + - .

L. C. D. =5X7X8=280.

3 3x56 168

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2. Find the value of 7 -3 - 3} +98 + 8 - 15.

L. C. D.=7X8X3 = 168.

1 1X24 24

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4X42

168

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24 +

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2 2X84 168

3 7 5 3-- 3- +9.

+8 - 15— 4 8 6

2

3 7 1
+9 +8 3- +3m + 15
6
4

2
24+140

126+147+84

21+
168

168
164 357
= 24-

21
168 168

164 21
= 24- 23
168

168
143
=l-

168 D.-Multiplication of a Fraction by an integer. Rule-1. If the fraction be a mixed number, reduce it to an improper fraction.

2. Form a fraction, whose numerator is the product of the numerator and the multiplier; and the denominator that of the multiplicand.

3. Cancel any factors common to numerator and denominator; the resulting fraction is the product required.

EXAMPLES. 1. Multiply a4 by 9.

23 23X9 23 2
--X9=

=7-
27

27 3 3 2. Multiply }} by 25.

13 13X25 13X5

-X25E35

35 7 65 2

7 3. Multiply 21% by 3.

17 55
2- X 3 = X3
19

19

13 = 819 19

165

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