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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;

2×3, or

7x
2'

is the value of the rye, in shillings;

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

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

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

4x 21'

x+y=505.

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

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

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

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+600 represents the number wounded;

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

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

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from which we shall find x=24000 men.

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?

Let x 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.

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

X

14

х

1

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

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X- -14

28x

,

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

is the part A and B did in 1 day.

Hence, we have the Equation.

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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

1

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У

represents the part which one cask would occupy.

The first Equation will therefore be

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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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which will give y=3x.

2y-6x=0;

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

which gives x=24.

30+33=3x;

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

46. A gentleman left a sum of money to be divided among four servants, so that the share of the first was the sum of the shares of the other three; the share of the second was of the sum of the other three; and the share of the third of the sum of the other three; and it was also found that the share of the first exceeded that of the last by $14. What was the whole sum? 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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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::3:5.

By converting this proportion into an Equation, we find

5x=3y, (149).

From this and the first Equation we shall find x=$356, 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).

as

From the two Equations we shall find x=

bs

and y= a+b'

a+b

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

tions of 1, 2, and 3.

Let x represent A's share; then

12 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 × 500 $1500, C's share.

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

to

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

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The three Equations will give x=4000, y=2000, and z=1600.

5. A bankrupt is indebted to A $400, and to B $700. He is able 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
xy: 400 700, or
xy4:7, (158).

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