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Thus, taking the expression (2} - 3) .

We must remember that the sign of 2; when within the bracket is, according to a remark made in Ex. 3, understood to be plus. In fact, though it is unusual, we might write the expression thus: -(+2} 3). Now, remove the bracket, and it becomes - 2 + 37.

} Again + (4} – 14) =

= + 4} - lt.
And – (2130 + 37'3) =
Hence our given expression-
:7} - 2} + 3} + 47 - 1} – 27 – 375
14

} + 1 + 1 - } -1-
: 6 +
1X24 + 1X60

1X10
6 15 – 24+6 0 +- 2 0 - 40 – 3 6 -10 6 + 95 -110
= 6 - 1to = 6 - 5 = 5 + (1 - 1) = 53.

216

31's:

8

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8 + s

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1X15
1 20

+ 1 x 20 -1X40

120

3X12
1 2 0

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120

1 2 0

120

+

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Ex. V. 1. Add together (1.) }, }, }; (2.) , I'o, 1}; (3.) *, I's, I'g. 2. Find the sum of 3, 1747, 276, 11'5, 6, 33. 3. Add together 23,

,

2 of 13,5} of 175.

27' 4. Find the difference between (1.) 7 and ; (2.) é and 4; (3.) 4 and 7.

5. Subtract (1.) 6 from 875; (2.) 37 from 41'; (3.) 210 from 61.

1 6. Take

1 %
from

116
13

2 – 11 7. Find the value of 1.7 2 + 3} - 11. 8. Simplify the expression (24 - 17) - (3 - 78).

}– % 9. By how much does 34 14 exceed 21% - 1o? 10. Take the difference of 617 and 1341 from their sum. 11. Add the difference of the same two fractions to their

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Multiplication of Fractions. 22. RULE.—Multiply together the numerators of the fractions for a new numerator, and the denominators for a new denominator.

The reason of this rule is easily seen. Let it be required to find the product of } and , or the value of 3 x .

Now, what is the meaning of multiplying the ratio 3:5 by the ratio 7:8? It means evidently that the ratio 3:5 is to be multiplied by 7, and the result divided by 8.

Now (Art. 8) the ratio 3: 5, when multiplied by 7, becomes 3 ~ 7:5; and (Art. 9) the ratio 3 x 7:5, when divided by 8, becomes 3 * 7:5 x 8; and we have

x ]

3x7. And so on for any number of fractions. Hence the above rule. Ex 1.-Multiply together the fractions , &, 13. 6 x Š x 13

5X3X12 Before actually performing the operation of multiplication, it is advisable to strike out any factor common to both numerator and denominator. We see that 5 is common to 5 and 25, 3 common to 3 and 6, 4 common to 12 and 8, and we then have

x x The whole operation is sometimes written thus6 x š x 33

XSX1X

70

5 X 8

3 63

6 X 8 X 2 5

1X1 X3
2 X 2 X 5

3

& X9 XXX
2 2 5

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Ex. 2.-Multiply together 23, 313, 170, $7.
2 x 3.13 x 130 $= x x $ 7

x

1200 = 13X100 X Y x 8 g

3

3

i = 9.

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Division of Fractions. 23. RULE.-Invert the divisor, and proceed as in multiplication. To explain this rule, let us endeavour to divide z by

몹 We may evidently consider the required quotient as nothing

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9 X 5

else than the ratio 7: , and this, by the reasoning of Art. 11, is equivalent to the ratio 7 ~ 8:9 x 5, and we hence get

공중

3+ f = Now, is the divisor when inverted, and hence the above rule. Ex. 1.--Divide I'd by Io = Io x

10

3 4

X8 JOX3 5

=

= = 23.

57

4.9 467.

2

Ex. 2.—Divide 14 of 7 by 3} of 3%.
13 of 7 - 3 of 3% = { x: (19 x V)

}

3 x 3 x 6 x 2 ន៍ ៖ We have introduced a bracket on the right side of the first equality, for otherwise the sign ; affects only the first fraction .

On the other side a bracket is unnecessary, for the sign - standing before a compound fraction (not two fractions) affects the whole. Ex. 3.—Simplify the expression

115 ; 62 x 41 : 2 of 2473.
The given expression = 1% x + ( x 2)

} =
is x x x x ਲੈ ਲ•

3

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3. Simplify the expression {(37%)*- (114)} = {3147 – 17}.

Ex. VI.
1. Find the sum, difference, and product of 23 and 14.

2. Multiply the sum of the fractions 33, 21 by their difference.

14. Reduce to a simple fraction each of the following expressions

(1.) 11 - 7 x 8 = 2 of 201.

(2.) 11 : 7 of 83 - 24 - 2015 5. What is the difference between (81 -37) and (5:- 473)?

4

of 13 6, Divide

3 +

2.4 7급

by 90*

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X

2

(2.)
6

{3} - (24 – 1})}. of 14 of 2} 9. Find the quotient of 10338 by 30} of 18 s. 10. The cost of 7} articles is £653, what is the cost of each article? 11. Find the cost of 89 articles, when one cost .£4,5.

12. The sum of two quantities is 341., and their differonco is 611 ; required the greater.

Reduction of Fractions to Decimals.

24. If we place a decimal point to the right of an integer, and add as many ciphers as we please, it is clear, from Art. 1, that we do not alter its value. And hence a given ratio, as 3:8, is not altered in value by writing it 3.000 : 8; and further, dividing each of its terms by 8, according to the rule for division of decimals, it becomes •375 :1. It therefore follows, putting each of these ratios in a fractional form, that

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We get, therefore, the following rule :

RULE.—Place a decimal point to the right of the numerator, and add as many ciphers as may be thought necessary. Divide the new numerator by the given denominator, according to the rule for division of decimals, and, if necessary, add ciphers to the successive remainders until the division terminates, or until we have obtained as many decimal figures as required.

Ex. 1.- Reduce to

a decimal.
32)5.0(-15625

32
180
160
200
192
80
64
160
160

Hence n'a

•15625.

5

Ex. 2.--Reduce to a decimal.

296)5.00(-016891

296
2040
1776
2640
2368
2720
2664

560
296

264 It will be seen that we have arrived at a remainder, 264, exactly the same as the second remainder; and that, therefore, the quotient figures 891 will continually repeat, and that the division will never terminate. We call 891 the recurring period of the decimal, and it is usual to indicate the fact of its recurrence by placing dots over its first and last figures, as above. We have, therefore, as a result,

·016891.

2

NorE.-It is easy to see that no fraction, reduced to its lowest terms, whose denominator contains any prime factor, other than 2 or 5, can be expressed as a terminating decimal. For every terminating deci. mal is an exact number of tenths, hundredths, &c., and may, there. fore, be transformed into a fraction, having some power of 10 as its

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