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19. What is the quotient of ji = ?

First Solution. ji divided by 1 = is, and by š must equal 9 times ti; if it contains š so many times, it must contain only š as many times. Hence,

2 3 16 8 16 x 9

6 21 9 21 X 8

7

17 Second Solution. tf divided by 1 equals js, and divided by $ must equal 9 times li; if the quotient by g equals so much, the quotient by must equal of this result.* Hence,

3
16 8 16 X 9

6

21

as before.
9
21 x 8

7 7

7 What is the quotient 20. Of = 7z?

25. Of 475 • 11? 21. Oft • 14?

26. Of 824:? 22. Of 8ft ; } ?

27. Of 617 = ? 23. Of 235 • 11?

28. Of 675 = ?
24. Of • 243?
29. What is the quotient of 675 • .9?
The work may be written thus:-

a = 675 = dividend
10 X a=b= 6750 = quotient by .1

Šof b= 750 quotient by .9
What is the quotient
30. Of 8.47 ; .07 ?

33. Of .75 = .05 ? 31. Of 75. • .05 ?

34. Of .075 • .005 ? 32. Of 75. • .005 ?

35. Of .00084 ; .0012?

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* In accordance with the principle, that if a number contains another a certain number of times, it will contain 8 times that number only s as many times ; 12 times the number only i' as many times, &c. Thus, 72 = 2 = 36, and 72 : 6 times 2, or 12, = 5 of 36, or 6.

48 = 3= 16, and 48 - 8 times 3, or 24, = of 16, or 2.
7= = 56, and 7 ;- 5 times g, or = } of 56, or 5 = 113.

To = 59°, and ģ :- 7 times Ty, or To, = 1 of, or %. + Reduce to an improper fraction.

148. Process of Division generalized. (a.) Since the quotient of any number divided by }= 5 times the number; by = 9 times the number; by 1' = 13 times the number, &c.; and since the quotient of a number divided by f= f its quotient divided by t; divided by g= of its quotient divided by $; divided by 1} = of its quotient divided by 1', &c., it follows that the quotient of a number divided by –

= { of 5 times the number, = of the number;

of 9 times the number, of the number; 1}= of 13 times the number, = 11 of the number;

.07 = 7 of 100 times the number = 142 of the number; and, universally, that

The quotient of any number divided by a fraction, is equal to the product of that number multiplied by the fraction inverted.

Illustrations. — 1. To divide by S, we have only to multiply by 9 and divide by 8.

2. To divide by it, we have only to multiply by 17 and divide by 3.

3. To divide by .03, we have only to multiply by: 100 and divide by 3, or, which is the same thing, to divide by 3 and remove the point 2 places towards the right.

4. To divide by .000037, we have only to multiply by 1000000, and divide by 37, or, which is the same thing, to divide by 37, and remove the point 6 places towards the right. 1. What is the quotient of Vi ; 1: ? 5 10 5 x 13

13
Ans.
11 13 11 x 1Ø 22

2
2. What is the quotient of 520.6 = .011?

Ans. The quotient of 520.6 • .011 may be obtained by dividing 520.6 by 11 and removing the point three places towards the right thus :First Form.

Second Form. .011 ) 520.6

011 ) 520.600

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NOTE. — The second form of writing the work differs from the frst only in this, – that in it as many zeros are annexed to the dividend as would be necessary were the point actually changed before performing the division. Great care is necessary, by either of these forms, to insure that the point is placed correctly in the quotient; and if in any case there is a doubt as to its true position, the work should be written in full, as in the model given after example 29th, 147.

3. What is the quotient of of } of i of j} of 31 ? Solution. 4 7 8 8 12 21 4 x 7 x 8 of of of

: 5 9 11 9 25 22 5 X 9 X 11

5 2 8 x 12 x 21 4 X 7 X 8 X 9 X 25 X 22 9 x 25 x 22 5 X 9 X 11 X 8 X 12 X 21

3 3 10 1 9

1g

of

=l

Note. — The more full and analytical explanation would be the following: ii is the number to be operated upon. 5 of ii may be expressed by making 7 a factor of the numerator, and 9 a factor of the denominator. of this may be expressed by making 4 a factor of the numerator, and 5 a factor of the denominator. The quotient of this quantity divided by 31 will be 21 of it, and may be expressed by making 22 a factor of the numerator, and 21 a factor of the denominator. If 24 is contained so many times, i' of {} must be contained 22 h as many times, expressed by making 25 a factor of the numerator, and 12 a factor of the denominator. If this divisor (3} of **) is contained so many times, is of it must be contained § as many times, expressed by making 9 a factor of the numerator, and 8 a factor of the denominator. Hence, 4

8 8 12 21
of of : of of
5 9 11

9

25 22

2 5
8 X 7 X 4 X 22 X 25 X 9 10 1
11 x 9 x 8 x 21 x 12 x 8 9

3 3

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The equation in the parenthesis may be omitted in practical operations

What is the quotient4. Of 134 • 813 ?

7. Of .004 ; 25 ? 5. Of 114 = 41?

8. Of .067 • .02 ? 6. Of 163 = 287 ?

9. Of 3287 = .0004 ? 10. Of of 11 of iti of Boog of 385? 11. Of g of 2 of 2 • of 1% of 61? 12. Of 1 of ļof of į of of; off of it of 11 ? 13. Of of 4 of 6 of j = 1o of iof 14?

149. Complex Fractions. (a.) A COMPLEX FRACTION is one having a fraction in

7 2 either numerator or denominator, or in both ; as,

37° 477 NOTE. — Complex fractions are usually considered as expressions of unexecuted divisions, and are read accordingly. Thus,

7

;

45 (6.) To show their similarity to other fractions, we may explain them thus:7

7 parts of such kind that 3 of them would equal a unit.

3}

36 *

4

259

(c.) Complex fractions can be reduced to simple fractions by the ordinary process of division.

57 1. Reduce to a simple fraction.

91 57

37 36 x 4 144 Solution.

91
7

7 X 37
Reduce each of the following to simple fractions.
74
86

74
2.
5.

8.
11}
121

8
2
3.

97
6.

9. 47

11 4.

41 7.

10. 111

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44 47

134

7

* By reducing 54 to sevenths, and 9 to fourths.

3

(d.) Complex fractions may often be reduced to simple ones, by reducing them to their lowest terms,

3 Thus : Dividing both terms of by 11 gives . Dividing

4 44

41

13 both terms of 1 gives

163

Reduce each of the following in the same manner :-
21
8

63
11.
14.

17.
5

23
14

8
12.
15.

18.
64

99
33

43
13.
16.

19.

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77. 27

64

7 7호

(e.) Complex fractions may also be reduced to simple ones, by multiplying both numerator and denominator by such a number as will give a whole number in place of each. 20. Reduce 4

to a simple fraction.

104 Solution. If 4 be multiplied by 3, or some multiple of 3, and 101 be multiplied by 2, or some multiple of 2, the result will in each case be

43

be multia whole number. Hence, if both terms of the fraction

104 plied by some multiple of both 2 and 3, the resulting fraction will be a simple one. Multiplying by 6 gives

101 In the same way reduce each of the following complex fractions to simple ones :

31 21.

104
24.

27.
56
134

837
21

163

4354
22.
25.

28.
71
627

847
43

481
23.

47
26.
8}
644

286

28

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63

9

29.

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