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Now, 4s. 6d. is equal to 4 shillings;

3s. 6d. is equal to 3 shillings;

86 × 4, or 387, shillings is the value of the wheat;
is the value of the rye, in shillings;

7x

x × 31, or

2

3y is the value of the barley in shillings; and 136 × 4, or 544 shillings, is the value of the mixture. The two Equations will then evidently be

86+x+y=136; and

7.x

387+ +3y=544;

2

from which we shall find x=14, and y=36 bushels.

and

41. A composition of copper and tin, containing 100 cubic inches, weighs 505 ounces. How many ounces of each metal does it contain, supposing the weight of a cubic inch of copper to be 51 ounces, and of a cubic inch of tin 4 ounces?

Let x represent the number of ounces of copper, and y the number of ounces of tin; then we shall have

x+y=505.

4x

Now, x-51, or represents the number of cubic inches of copper;

21'

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The second Equation will then be

4x 4y

+ = =100.
17

21

From these Equations x will be found equal to 420 ounces, and y equal to 85 ounces.

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4y

y÷41, or represents the number of cubic inches of tin.

17'

42. A general, having lost a battle, found that he had only one half of his army plus 3600 men left fit for action; of his men plus 600 being wounded, and the rest, who were of his whole army, either slain, taken prisoners, or missing. Of how many men did the army consist? Let x represent the number; then

+600 represents the number wounded;

represents the number slain, taken prisoners, or missing; and

X

8100

-600, represents the number re

maining.

Now, as only of his men plus 3600 were left, we shall have

27x

x

+3600;

40

-600

1

)。

2

from which we shall find x=24000 men.

14

43. Two pipes, one of them running 5 hours, and the other 4, filled a cistern containing 330 gallons; and the same two pipes, the first running 2 hours, and the second 3, filled another cistern containing 195 gallons. How many gallons did each pipe discharge per hour?

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Let a represent the number of gallons discharged by the first, and y the number discharged by the second, per hour; then

5x is the number discharged by the 1st in 5 hours; and 4y is the number discharged by the 2d in 4 hours. The first Equation will then be

5x+4y=330;

and in a similar manner, we shall have

2x+3y=195;

from which Equations, we shall find x=30, and y=45 gallons.

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27x

40

or

44. After A and B had been employed on a piece of work for 14 days, they called in C, by whose aid it was completed in 28 days. Had C worked with them from the beginning, the work would have been completed in 21 days. In how many days would C alone have accomplished the work?

Let x represent the number of days; then

1

is the part C did in 1 day; and, since he was employed 14 days,

X

14

is the part of the work performed by C;

X

-600: =

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2-14

28x

Hence, we have the Equation

1

x-14 1
+

28x

x 21'

This Equation will give x=42 days.

, is the part performed by A and B in 28 days;

is the part A and B did in 1 day.

the part the three could do in 1 day.

45. Some smugglers discovered a cave which would exactly hold their cargo, viz, 13 bales of cotton, and 33 casks of wine. A revenue cutter coming in sight while they were unloading, they sailed away with 9 casks, and 5 bales, leaving the cave two-thirds full. How many bales or casks would it contain?

Let x represent the number of bales, and y the number of casks,

1

then represents the part of the cave which one bale would occupy,

and

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1

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represents the part which one cask would occupy.

y

The first Equation will therefore be

13

33

=1, the entire cave;

+
x y

and, as there were left in the cave 13-5, or 8 bales, and 33-9, or 24 casks, the second Equation will be

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8 24

+

2

y 3

Clearing these Equations of fractions, we have

13y+33x=xy;

24y+72x=2xy.

Multiplying the third Equation by 2,

x

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26y+66x=2xy;

subtracting the fourth Equation from this, we have

2y-6x=0;

which will give y=3x.

Substituting 3.x for y, in the third Equation, and dividing both sides of the resulting Equation by x, we shall have

30+33=3x;

which gives x=24.

The value of y is now easily found to be 72.

46. A gentleman left a sum of money to be divided among four serthe sum of the shares of the

vants, so that the share of the first was
other three; the share of the second was
and the share of the third
also found that the share of
What was the whole sum?

of the sum of the other three; of the sum of the other three; and it was the first exceeded that of the last by $14. and the share of each?

Let x, y, z, and w, represent the several shares;
By the conditions of the problem, we shall then have

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x=w+14.

These Equations will give x=$40, y = $30, z=24, and w=$26. The whole sum will then be 40+30+24+26=$120.

PROBLEMS

In Proportion, Percentage, Interest, &c.

1. Divide $950 between two persons, so that their shares shall be to each other as 3 to 5.

Let x represent the share of the 1st, and y the share of the 2d; then we shall have

x+y=950; and

xy::35.

By converting this proportion into an Equation, we find

5x=3y, (149).

From this and the first Equation we shall find x= =$3561, and y= $5933.

2. Find the Formulas for dividing any given sum s between two persons so that the shares shall be to each other as any two numbers a and b.

Let x and y represent the respective shares; then

x+y=s; and
xy::a:b.

This proportion converted into an Equation, gives
bx=ay, (149).

From the two Equations we shall find x=

Let x represent A's share; then

3. Divide the sum of $3000 between A, B, and C, in the propor tions of 1, 2, and 3.

1: 2 ::x : 2x, B's share; and
2:3: 2x: 3x, C's share (150).

Then we shall have the Equation

x+2x+3x=3000;

from which we shall find x=$500, A's share; hence 2×500=$1000, B's share; and 3 x 500-$1500, C's share.

x+y+2=7600;

xy; and

y: 2.

The two proportions converted into Equations, will give

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4. Divide the sum of $7600 between three persons, in the propor tions of,, and }.

Let x, y, and z, represent the several shares; then

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.

y

as

and y= a+b'

2

Z

bs

a+b

; and

4

The three Equations will give x=4000, y=2000, and 2=

= 1600.

5. A bankrupt is indebted to A $400, and to B $700. He is able to pay both $900. What sum should each of the two creditors receive?

Let x and y represent the respective shares; then we shall have

x+y=900; and
xy400 700, or
xy::47, (158).

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