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7. A, B, and C engage in trade. A puts in $ 6000, B $ 9000, and C $ 5000. They gain $ 1680. What is each partner's share of the gain ?

357. To find the gain or loss of partners,

Rule. · Take for each partner such a part of the gain or loss as his stock is of the entire capital.

NOTE. — The rule applies to the distribution of the assets of bankrupts and other like apportionments.

8. A, B, and C engage in trade, investing capital to the amount of $ 1280, $ 1760, and $ 1920, respectively. Their profits were $ 2790. How were they divided ?

9. A bankrupt owes three creditors, A, B, and C, $ 1750, $ 2100, and $ 2650, respectively. His assets are $ 4225. What should they each receive ?

10. Hall and Bishop gain by trade $728. Hall put in $ 1200, and Bishop $ 1600. What is the gain of each ?

11. B puts on board of a ship 400 barrels of flour, C 600, and D 400; but when at sea it was found necessary to throw 360 barrels overboard. How much of the loss should fall to each ?

12. A, B, and C hire a pasture for $ 300. A puts in 8 horses, B 6, and C 10. How many dollars should each pay ?

13. A, B, and C engage in trade. A puts in $1400, B $ 600, and C 125 barrels of flour. They gained $ 180; of which C took $60 as his part. What will A and B receive, and what was the value of C's flour a barrel ?

358. When the capital of the partners is employed for unequal times,

Find the product of each partner's stock multiplied by the time it was invested, and divide the gain or loss in proportion to the products.

14. A, B, and C engage in trade. A puts in $300 for 7 months, B $500 for 8 months, and C $ 200 for 12 months. They gain $ 170. What is each man's share of the gain ?

Solution.

A's $300 for 7 months = $ 2100 for 1 month.
B’s $ 500 for 8 months = 4000 for 1 month.

C's $ 200 for 12 months = 2400 for 1 month.
The entire stock is the same as $ 8500 for 1 month.

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16. A, B, and C were in partnership. A had in the business $ 5000 for 8 months, B $ 4000 for 12 months, and C $ 3000 for 15 months. The profits were $ 1330. How much is each partner's part of the profits ?

17. A and B rent a pasture for $ 46.80. A puts in 30 horses for 33 days, and B 21 horses for 42 days. How much ought each to pay of the rent ? • 18. A, B, and C form a partnership. A furnishes $ 500 for

9 months, B $ 700 for 1 year, and C $ 400 for 15 months. They lose $300. What is each man's share of the loss ?

19. A and B are in partnership. A put in $6000, and at the end of 6 months put in $ 4000 more; B put in $ 12000, and at the end of 8 months took out $ 6000. They trade 1 year, and gain $ 2160. What is each man's share of the gain ?

20. A and B enter into partnership for 1 year. A had $ 500 in the business during the first 4 months, and $300 more during the remainder of the year; whereas B had only $ 400 during the first 6 months, but $ 900 during the last 6 months. They gained $ 2400. What was each man's share of the gain ?

21. Jan. 1, 1881, Wood goes into business with a capital of $ 6000. March 1, Furbush joins him with $ 5000. July 1, they take Davis into the partnership with $ 4000 capital, agreeing to pay him 6% interest for his money, and give him an annual salary of $ 1200. The profits were $ 3160, out of which Davis was paid and the balance divided between the other partners. Find each man's share at the end of the year.

22. Jan. 1, 1882, A and B form a partnership for a year. A furnishes $ 2000, and B $3000. May 1, they take C into the firm with a capital of $ 5000. August 1, A receives a legacy of $ 4000, which he adds to the capital. Oct. 1, B withdraws $ 1000 of his capital. At the end of the year the firm's net gains were $ 5850. Divide it equitably among the partners.

INVOLUTION AND EVOLUTION.

INVOLUTION. 359. 1. What is the product of 5 used twice as a factor ? 2. Of what number are 5 and 5 the factors ? 3. What is the product of 5 used three times as a factor ? 4. Of what number are 5, 5, and 5 the factors ?

5. What is the product of .3 used three times as a factor ? Of used three times as a factor ?

360. A Power of a number is the product arising from taking the number a certain number of times as a factor.

The First Power of a number is the number itself;

The Second Power of a number is the product arising from taking the number twice as a factor;

The Third Power of a number is the product arising from taking the number three times as a factor; and so on. Thus,

The first power of 3 is 3. ..
“ second “ 3 is 3 x 3, or 9.
“ third " 3 is 3 x 3 x 3, or 27.
“ fourth " 3 is 3 X 3 X 3 X 3, or 81.

The second power is also called the Square of the number, as the area of a square is the product of two equal factors.

The third power is called the Cube, as the volume of a cube is the product of three equal factors.

361. The Exponent of a power is a small figure placed at the right and above a number. Thus,

3.13 means the third power, or cube, of 3.1; ()* means the fourth power of 4

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362. Involution is the process of finding powers.

WRITTEN EXERCISES. 6. What is the third power of 12 ?

Solution. — 128 = 12 X 12 X 12 = 1728. 7. Find the squares and cubes of the first nine numbers.

363. Rule for Involution. Use the given number as many times as a factor as there are units in the exponent of the required power. Find the powers indicated by the exponents : 8. 23. 11. (3)? 14. 253. 17. 3.52. 9. 163 12. (2}). 15. 115. 18. (141) 10. 134. 13. 3.62. 16. 0.158 19. 0.073

EVOLUTION.

20. What are the factors of 9? Of 25 ? Of 49 ?. 21. What are the two equal factors of 16 ? Of 64 ? Of 81 ? 22. What are the three equal factors of 27 ? Of 64 ?

364. A Root of a number is one of the equal factors which produce it.

The Second, or Square, Root of a number is one of the two equal factors which produce it;

The Third, or Cube, Root of a number is one of the three equal factors which produce it;

The Fourth Root of a number is one of the four equal factors which produce it; and so on.

365. A Perfect Power is a number whose exact root can be found, and an Imperfect Power is a number whose root cannot be exactly found.

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