329. 1. A Compound Proportion is a proportion in which there is a compound ratio.

2. The following problem involves a compound proportion:

If a man, working 8 hours a day, build 60 feet of fence in 2 days, how many feet of fence can a man build in 6 days, working 10 hours a day?

NOTE. What is assumed about the two fences? About the men?

ANALYSIS. If the days were of the same length, the proportion would read, 2 days: days 60 feet: x feet. Since the 2 days are as long as the 6 days, the work done in 2 days will not be to the work done in 6 days as 2 is to 6, but will be of that ratio, which is found by multiplying 2:6 by 8: 10.

2.61 The form:

8.10

Because there are two elements involved in determining the time that each man worked (the number of days and their length), the relation of the times is expressed by a compound ratio.

RULE.

To solve a problem involving a compound proportion, take for the third term the antecedent of the ratio of which the required term is the consequent. With one pair of the remaining terms for the first ratio, state the proportion according to the conditions of the problem.

Proceed with the remaining pairs in the same way

until all of the conditions are stated. Find the product of the third term and all of the second terms, and divide it by the product of the first terms.

PROBLEMS.

1. If 4 men, in 5 days of 8 hours each, can dig a ditch 120 yards long, 3 feet wide, and 4 feet deep, in how many days of 10 hours each can 12 men dig a ditch 300 yards long, 4 feet wide, and 4 feet deep?

Why is 12 put first in the fourth ratio? Why 10 in the last? What cancellation can be employed? Why?

2. If $180 be paid for the work of 5 men for 24 days, what should be paid for the work of 17 men for 36 days?

3. If 15 men, in 16 days of 9 hours each, can do a piece of work, how many men will be needed to do the same piece of work in 8 days of 6 hours each?

4. If 15 men, in 16 days of 9 hours each, can do a certain piece of work, in how many days of 6 hours each can 45 men do the same work?

5. If 45 men can do a piece of work in 8 days of 6 hours each, how many hours a day must 15 men work to do the same work in 16 days?

6. If 50 tons of coal are required to run 4 engines 15 hours a day for 6 days, how many tons will be required to run 7 engines 18 hours a day for 11 days, with 3 times as heavy a load?

7. If it cost $50 to make a walk 8 feet wide and 60 feet long, what will it cost to build a walk 73 feet wide and 72 feet long?

8. If it cost $50 to make a walk 8 feet wide and 60 feet long, what is the width of a walk that is 72 feet long, and costing $57.50?

9. If a walk that is 72 feet long and 73 feet wide cost $57.50, how long a walk that is 8 feet wide can be built for $50?

10. If 83 horses eat 933 bushels 3 pecks of oats in 30 days, how many bushels will 125 horses eat in 45 days?

11. If 44,640 bricks, 4 inches by 8 inches, will pave a court-yard, how many tiles 8 inches wide and 15 inches long will pave it?

NOTE. Solve these problems by straight-line analysis. Additional problems may be formed from those given, as illustrated in 7 and 8. Remark. No problem can be solved by proportion that cannot be more easily solved by straight-line analysis.

1. A and B went into business together, A investing $5,000, and B $7,000. They agreed to share gains and losses in proportion to their investments. $2400. What was each one's share?

(a) Solve by analysis.

The net gain was

What was the whole investment? What A invest? What part of it did B invest? share? B's?

(b) Solve by proportion.

Explain these proportions.

(1) $12000: $5000 :: $2400 : A's share. (2) $12000: $7000 :: $2400 : B's share.

part of it did

What was A's

2. The action of A and B is called "the formation of a partnership." The amount invested is called the Capital. A and B are called Partners. The agreement into which they enter is called the Conditions of Partnership.

3. A Partnership is an association of persons for the prosecution of business on joint account.

PROBLEMS.

1. A, B, and C formed a two-year partnership, agreeing to share gains and losses in proportion to their investments. A put in $5,000; B, $6,000; C, $7,000. Their net gain. was $8,000. Find the share of each by analysis and by proportion.

2. If their net loss had been $2,160, what would have been the loss of each?

3. A, B, and C formed a partnership, C being a silent partner. A invested $6,000; B, $8,500; C, $10,000. By the conditions of partnership, A was to receive a salary of $1,000, and B, $700. The net profits were to be divided in proportion to investments. At the end of the first year the profits, exclusive of all expenses but salaries, were $4,150. What was the share of each?

4. A, B, and C formed a partnership for 3 years. They were to draw equal amounts as salaries, and were to share the net profits equitably. A invested $5,000 at the beginning, added $3,000 to it at the end of the first year, and $2,000 more at the end of the second. B invested $6,500 at the beginning, withdrew $2,000 at the end of the first year, and $1,500 at the end of the second. C invested $5,000 at the beginning, and did not change it. At the end of the time their net profits were $4,850. What was the share of each?

5. The investments of three partners are in the ratio of 3, 4, and 5. If they gain $3,600, what is the share of each?

6. A, B, and C owned a mill valued at $18,000. A owned of it; B, % of it; and C, the rest. It was insured for of its value. If it should be destroyed by fire what would each partner lose?

7. A, B, C, and D constructed a street railroad costing $135,000. A furnished of the capital; B, of it; end C and D each furnished of the remainder. The company

sold to E of the road for $9,000; what part of this amount should each receive? What part of the stock would each of the original partners own after the sale?

8. A, B, and C took a contract for excavating a railroad cut. A furnished 60 men for 25 days; B, 50 men for 48 days; C, 75 men for 56 days. They received $20,250 for the work. What was the share of each?

9. A, B, and C engaged in business for one year, agreeing to share the profits in proportion to their investments. On January 1, A put in $3,000; B, $3,500; and C, $2,500. On March 1, A. increased his share $500, B diminished his $500, and C increased his $250. On July 1, A withdrew $1,000, B put in $800, and C increased his $1,000. On October 1,

A put in $600, B withdrew $400, and C withdrew $750. They gained $3,000. What was the share of each?

10. A, B, and C engaged in business for 2 years, with a capital of $16,000. A furnished and B of the capital. C conducted the business for one half the net profits. The gross earnings were $4,800. The expenses were 12%. What was A's share? B's?

11. What would have been A's share, if at the end of the first year he had transferred to B one third of his interest?