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a

sense.

Looked at from an Arithmetical point of view, the preceding definitions, although ingenious and plausible, are by no means unexceptionable. The first assumes that a multiplier is derivable from unity in only one way. The second definition assumes ihat unity is derivable from a divisor in only one way; also, that a “ remainder" never occurs in an exercise in Division.

Now, everybody knows that a multiplier is derivable from unity, and that unity is derivable from a divisor, in more ways than one ; so that the definitions hold good only in a particular For instance :

(I.) (a) We obtain 8 from 1 by we do not obtain 152 from 19 adding 7;

by adding 7. (1) We obtain 7 from 1 by we do not obtain 16'1 from 23 subtracting '3;

by subtracting 3. (c) We obtain 9 from i by, we do not obtain £2 is. 3d. adding 8.

from 4s. 7d. by adding 8. (d) We obtain from 1 by we do not obtain 29 from 5 subtracting

by subtracting 3.

(II.) (e) We obtain 1 rom 8

we do not obtain 19 from 152 subtracting 7;

by subtracting 7. (f) We obtain i from •7 by we do not obtain 23 from 16'1 adding *3 ;

1

by adding 3. (g) We obtain i from 9 by we do not obtain 4s. 7d. from

1 subtracting 8;

£2 18. 3d. by subtracting 8. (h) We obtain i from by we do not obtain from z by adding };

adding:

by |

by

COMPOUND PROPORTION.* 169. When two or more ratios are given, and we multiply the antecedents together for a new antecedent, and the consequents together for a new consequent, the ratio which the new antecedent bears to the new consequent is said to be COMPOUNDED of the two or more given ratios.

*

Compound Proportion is sometimes called the “DOUBLE Rule of Three:" a name which, although long in use, can hardly be considered appropriate-particularly when there are more than two ratios to be compounded.

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Thus, the ratio 3x6:4X5 is “compounded” of the two ratios 3:4 and 6:5;

the ratio 9X2X11:8X3X7

9

8 is “compounded " of the 3 4

3 three ratios 9:8, 2:3, and 6

5

7 II:7; &c.

3x6:4X5 9X2X11:8X3X7 170. A Proportion is called COMPOUND when one of the ratios-instead of being expressed in the ordinary way-is represented by two or more other ratios, from the "compounding of which it is ultimately obtained. The following are " Compound" Proportions :(I.)

(II.) 3:4

1} 6 : 5

:: 9:10 2:3 :: 33: 28.

9:8

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II:7

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The first of these Proportions indicates that the ratio 9:10 is equal—not to either 3: 4 or 6: 5, but to the ratio (18:20) compounded of 3 : 4 and 6:5. The second Proportion indicates that the ratio 33: 28 is equal—not to 9:8, or 2: 3, or 11: 7, but-to the ratio (198:168) compounded of 9:8, 2:3, and II:7.

171. When the ratio represented by two or more others is obtained from the “compounding” of those others, and substituted for them, a COMPOUND Proportion becomes a SIMPLE Proportion. Thus, the Compound Proportion

3:41 6:

::9:10 :5

18:20 ::9:10 becomes a Sim

ple Proportion
9:8)

when written
2:3
:: 33 : 28

198 : 168 :: 33: 28 II:7

PRACTICAL EXERCISES IN COMPOUND PROPORTION. EXAMPLE I.-If 8 horses can plough 20 acres in 7 days, how many acres could 5 horses plough in 11 days ?

8

7

5 ? Here we have materials for two exercises in Simple Propor. tion:

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H.

A.

DYS.

20

II

H.

H.

A.

acres.

DYS.

DYS.

:

II

acres.

8X7

(1) If 8 horses can plough 20 acres in a certain time [7 days], how many acres could 5 horses plough in the same time?

20X5 8: 5 :: 20:a; a=

8

20 X 5 (2) If a certain number of horses [5] can plough

8 acres in 7 days, how many acres could the same number of horses plough in 11 days?

A. 2015

20 X 5 XII 7

8

a; a= By means of these Proportions we find (1) that 5 horses, in

5 7 days, could plough 20x5

i

and (2) that 5 horses, in II days, could plough

8x7

20X5X1 It will be seen that the ratio 20 acres :

8x7

acres is equal -not to either 8 horses : 5 horses, or 7 days : 11 days, but-to the ratio (8x7:5X11) compounded of 8: 5 and 7:11. It will likewise be seen (1) that, when writing 8 horses in the first, and 5 horses in the second place, we assumed the number of days to be the same in both cases ; and (2) that, when writing 7 days in the first, and 11 days in the second place, we assumed the number of horses to be the same in both cases. In practice, the work would stand thus :

8 5

} 7

acres
8
20 X 5 XII

acres.

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::20:a

: II

M.

HRS.

YDS.

DYS.

IO

-15

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8X7:5X11 :: 20:0 a=20 X 5x11+8X7=1100:-56–1988 or 19, acres. EXAMPLE II.-If 6 men, working 10 hours a day, can build a wall 360 yards long in 15 days, in what time could 4 men, working 12 hours a day, build a length of 6

-360480 yards ?

4

-480 ? Here we have materials for three exercises in Simple Proportion:

(1) If 6 men can do a certain quantity of work [360 yards] in 15 days, in how many days of the same length [10 hrs.) coulă 4 men do the same quantity of work ?

DYS.
4 : 6 :: 15 :a; a=

15x6

days. 4

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4

(2) If a number of men [4], engaged 10 hours a day, can do a certain quantity of work [360 yds.] in

15x6

days, in what time could the same number of men, engaged 12 hours a day, do the same quantity of work ?

dys. 15x6

days. 4

4X12

HRS.

HRS.
I2

:

IO

:

: a; a=15X6X10

a

4X 12

(3) If a certain number of men [4], working a certain num.

15x6x 10 ber of hours (12) daily, can build 360 yards of a wall in days, in what time could the same number of men, working the same number of hours daily, build 480 yards ?

dys. 360 : 480 :: 15х6х10

days. 4X12

4X 12 X 360

YDS.

YDS.

:a; a=15X6X10X 480

days;

4

in 15x6x10

4X12

By means of these Proportions we find (1) that 4 men,

15x6 working 10 hours a day, could build 360 yards in (2) that 4 men, working 12 hours a day, could build 360 yards

days; and (3) that 4 men, working 12 hours a day, could build 480 yards in 15X6 XIo X 480

4X12X360

days. It will be observed that the ratio 15 days :

15X6 X 10 X 480
4X12X360

days is equal—not to 4 men : 6 men, or 12 hours: 10 hours, or 360 yards : 480 yards, but-to the ratio (4 X 12 X360:6X10X480) compounded of 4:6, 12:10, and 360 : 480. It will also bé observed (1) that, when writing 4 men in the first, and 6 men in the second place, we supposed the length of the day, as well as the quantity of work, to be the same in both cases; (2) that, when writing 12 hours in the first, and 10 hours in the second place, we supposed the number of men, as well as the quantity of work, to be the same in both cases; and (3) that, when writing 360 yards in the first, and 480 yards in the second place, we supposed the number of men, as well as the length of the day, to be the same in both cases. Here is the work, as it would appear in practice :

4:
6

15 : a.
360 : 480

12:

IO

::

4x 12 x 360 : 6 x 10 x 480 :: 15 : a. a==15x6x 10 x 480- 4x 12 x 360=432000: 17280 = 25 days.

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ExamPLE III.-If 10 weavers, in 5 days, can make 30 pieces

. of cloth, each 16 feet long and 2 feet broad, how dys.

long.

broad. many pieces, each 12 feet

5 -30

16 ft.long and 3 feet broad, could 9- -8. ?

-3 ,, be made by 9 weavers in 8 days?

Leaving the fourth place for a (the "answer"), and making 30 pieces the third term, we know, after what has just been explained, that the ratio 30: a is equal to one "compounded" of four other ratios-namely:

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5: 8

8: 5

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2:

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3: 2

(3)
16:12

12:16
(4)

3 In determining, in the case of each of the four ratios referred to, which term should be made the antecedent, and which the consequent, we have merely to take, in turn, every two numbers of the same kind, and (disregarding, for the moment, the remaining three pairs) consider how they should be placed if they and the third term were the only numbers upon which the answer depended :

(1) If 10 weavers can make 30 pieces of cloth under certain circumstances, how many such pieces could 9 weavers make under the same circumstances ? " The Proportion being 10:9:: 30:0, we write to in the first, and 9 in the second place.

(2) If 30 pieces of cloth can, under certain circumstances, be made in 5 days, how many such pieces could, under the same circumstances, be made in 8 days ? The Proportion being 5:8:: 30 :Q, we write 5 in the first, and 8 in the second place.

(3) If 30 pieces of cloth, each 16 feet long, can be made under certain circumstances, how many pieces (of the same breadth), each 12 feet long, could be made under the same circumstances ? The Proportion being 12 : 16:: 30: , we write 12 in the first, and 16 in the second place.

(4) If 30 pieces of cloth, each 2 feet broad, can be made under certain circumstances, how many pieces (of the same length), each 3 feet broad, could be made under the same circumstances ? The proportion being 3:2 :: 30 : a, we write 3 in the first, and 2 in the second place.

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