This proportion will give

7x=4y.

From the two Equations x will be found equal to $327, and y= $572,

6. Three persons engaged in a speculation, towards which they contributed, respectively, $300, $400, and $500. The profit amounted to $550. What are the respective shares of profit?

Let x, y, and z, represent the respective shares of profit; then we shall have

x+y+2=550;

xy: 300 400; and

yz: 400: 500.

Simplifying these proportions (158), and multiplying together the extremes and means,

4x-3y; and
5y=42.

The values of x, y, and z, are now easily found equal to $137, $183, $2291, respectively.

7. A, B, and C, in a joint mercantile adventure, lost $742. A's part of the capital employed was to B's as 4 to 3, and B's was to C's as 5 to 6. What amount of loss should be borne by each?

Let x represent A's amount of loss; then

from which we shall find x=$280, A's loss; hence of 280-210, B's loss; and of 280-$252, C's loss.

8. Four persons rented a pasture, in which the first kept 8 oxen, the second 6, the third 10, and the fourth 12. The sum paid was $40. What amount should have been paid by each person?

Let x, y, z, and w, represent the several shares of pay.

which, with the first Equation, will give a $83, y= $63, z=11}, and x= y=$63, w=$131.

9. A testator bequeathed his estate, amounting to $7830, to his three children, in such a manner that the share of the first was to that of the second as 2 is to 2, and the share of the second to that of the third as 3 to 3. What were the shares ?

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

from which two Equations, with the first, we shall find x=$3150, y= $2520, and z=2160.

10. A, B, C, and D, together have $3000; A's part is to B's as 2 to 3. B and C together have $1500, and C's part is to D's as 3 to 4 What is the sum possessed by each person?

Let x, y, z, and w, represent the respective shares; then

x+y+z+w=3000;
x:y:2:3;

y+z=1500; and

zw:: 3:4.

Converting the two proportions into Equations, we shall have four

Equations, from which we shall find x=

=$500, y=$750, z=$750, and

11. Three persons contributed funds in a joint speculation as follows: A $200 for 5 months, B $400 for 3 months, and C $500 for 4 months. The profit amounted to $600. What are the several shares of profit?

Let x, y, and z, represent the several shares of profit.

Now, each dollar contributed produced a profit proportional to the time it was in the business. Each person's share of profit is therefore proportional to his amount of capital X its time; in other words, the respective shares are to each other in the compound ratio of capital and time, (131).

Then we shall have

xy:: 200×5:400×3;

x: z:: 200 ×5:500×4; and
x+y+z600.

Simplifying the two proportions, we shall have

xy::5:6; and

:::1:2, (158).

From these proportions and the Equation, we shall find x=$142%, y=$171, and z=$2854.

12. Two persons rented a pasture for $43. The first put into it 100 sheep for 15 days, and the second 120 sheep for 9 days. What amount of rent should be paid by each person?

Let x and y represent the respective shares of rent to be paid by cach; then

xy: 100 x 15: 120 x 9; and
x+y=43.

The proportion simplified becomes

x:y:25:18;

from which and the Equation, we shall find x=$25, and y=$18.

13. A, B, and C, trade together; A ventures $1000 for 5 months, B $1200 for 4 months, and C $800 for 7 months. The profits of the partnership amount to $2310. What share of profit should be assigned to each?

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

xy: 1000×5: 1200 x 4;
y:z: 1200 x 4: 800 x 7; and
x+y+2=2310.

which, with the Equation, will give x=$750, y=$720, and z=$840.

14. An estate consisting of 1000 acres of land is to be divided between three persons, so that the first share shall be to the second as 2 to 3, and the first to the third as 1 to 2. What are the shares?

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

From this Equation, and the two proportions, we shall find x= 2223, y=333, and z=4444 acres.

In ac

15. Two men contracted to do a certain work for $5000. complishing the work, the first employed 100 laborers for 50 days; and the second 125 laborers for 60 days. To what shares of the stipulated sum are the two men respectively entitled?

Let x and y represent the respective shares; then

xy:: 100 x 50: 125 × 60; and
x+y=5000.

The proportion simplified becomes

xy::2:3;

from which and the Equation we shall find x-$2000, and

y=$3000.

16. A gentleman bequeathed $18000 to his widow and his three sons, in the proportions of 2, 2, 3, and 3, respectively. His widow dying before the division was effected, the whole is to be divided proportionably among the three sons. What are their several shares ?

Let x, y,

and

z, represent the several shares; then, as the widow died, the whole sum must be divided among the three sons in the proportions of 21, 3, and 34, respectively. Hence we shall have

x:y :: 21:3;
yz3:31; and

x+y+z=18000.

From the three Equations, a will be found = $5000, y=$6000, and z=$7000.

17. Find the Formulas for dividing between two partners the profits s of a joint adventure, in which the first had the capital a for the time b, and the second the capital c for the time d.

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

xy::ab: cd; and

x+y=s.

From this proportion and Equation, we shall find x=

cds

y == ab+cd

Problems in Percentage.

18. A merchant finds that his capital, which is now $12000, has increased in one year at the rate of 20 per cent. What was his capital at the beginning of the year?

Let x represent his capital at the beginning of the

100 120 x 12000;

which will give x= $10000

year; then

The solution of this problem will perhaps be more intelligible by forming an Equation directly from the conditions of the problem.

Let x represent his capital; then since

20

100

20x

100

is the ratio of percentage (171),

represents the amount of percentage on his capital, (172).

Then, since the capital plus the percentage on the capital is equal to $12000, we shall have the Equation