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2. Two men purchase a ticket for 4 dollars, of which one pays 1 dollar, and the other 3 dollars; the ticket draws 40 dollars; what is each man's share of the money?

3. A and B bought a quantity of cotton; A paid 100 dollars, and B 200 dollars; they sold it so as to gain 30 dellars; what were their respective shares of the gain?

The process of ascertaining the respective gains or losses of individuals, engaged in joint trade, is called the Rule of Fellowship.

The money, or value of the articles employed in trade, is called the Capital, or Stock; the gain or loss to be shared is called the Dividend.

It is plain, that each man's gain or loss ought to have the same relation to the whole gain or loss, as his share of the stock does to the whole stock.

Hence we have this RULE:-As the whole stock: to each man's share of the stock :: the whole gain or loss: his share of the gain or loss.

4. Two persons have a joint stock in trade; A put in $250, and B $350; they gain $400; what is each man's share of the profit?

A's stock,
B's stock,

OPERATION.

$250

$350

Then,

600: 250 :: 400: 166'666 dolls. A's gain. 600: 350 :: 400: 233′3334 dolls. B's gain.

Whole stock, $600

The pupil will perceive, that the process may be contracted by cutting off an equal number of ciphers from the first and second, or first and third terms; thus, 6: 250 :: 4: 166'6664, &c.

It is obvious, the correctness of the work may be ascertained by finding whether the sums of the shares of the gains are equal to the whole gain; thus, $166'6664+ $233′3334

$400, whole gain.

5. A, B and C trade in company; A's capital was $175, B's $200, and C's $500; by misfortune they lose $250; what loss must each sustain ? $ 50', A's loss. Ane. $ 57'1424, B's loss. ($142'8574, C's loss.

6. Divide $600 among 3 persons, so that their shares may be to each other as 1, 2, 3, respectively.

Ans. 100, $200, and $800.

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7. Two merchants, A and B, loaded a ship with 500 hhds. of rum; A loaded 350 hhds., and B the rest; in a storm, the seamen were obliged to throw overboard 100 hhds.; how much must each sustain of the loss?

Ans. A 70, and B 30 hhds.. 8. A and B companied; A put in $45, and took out of the gain; how much did B put in? Ans. $30. Note. They took out in the same proportion as they put in; if 3 fifths of the stock is $45, how much is 2 fifths of it?

9. A and B companied, and trade with a joint capital of 400; A receives, for his share of the gain, as much as B; what was the stock of each?

Ans. {

10. A bankrupt is indebted to A $780, to B $460, and to C $760; his estate is worth only $600; how must it be divided?

S$133'3334, A's stock. $266'6663, B's stock.

Note. The question evidently involves the principles of fellowship, and may be wrought by it.

Ans. A $234, B $138, and C $228. 11. A and B venture equal stocks in trade, and clear $164; by agreement, A was to have 5 per cent. of the profits, because he managed the concerns; B was to have but 2 per cent.; what was eath one's gain? and how much did A receive for his trouble?

Ans. A's gain was $117'142, and B's $46,8574, and A received $70'2854 for his trouble.

12. A cotton factory, valued at $12000, is divided into 100 shares; if the profits amount to 15 per cent. yearly, what will be the profit accruing to 1 share? to 2 shares ?

to 5 shares?

to 25 shares?

Ans. to the last. $450. 13. In the above-mentioned factory, repairs are to be made which will cost $340; what will be the tax, on each share, necessary to raise the sum ? on 2 shares ?

on 3

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on 10 shares?

shares? Ans. to the last, $34. 14. If a town raise a tax of $1850, and the whole town be valued at $37000, what will that be on $1? What will be the tax of a man whose property is valued at $1780? Ans. $'05 on a dollar, and $89 on $1780.

¶ 99. In assessing taxes, it is necessary to have an inventory of the property, both real and personal, of the whole town, and also of the whole number of polls; and, as the polls are rated at so much each, we must first take out from the whole tax what the polls amount to, and the remainder is to be assessed on the property. We may then find the tax upon 1 dollar, and make a table containing the taxes on 1, 2, 3, &c., to 10 dollars; then on 20, 30, &c., to 100 dollars; and then on 100, 200, &c., to 1000 dollars. Then, knowing the inventory of any individual, it is easy to find the tax upon his property.

15. A certain town, valued at $64530, raises a tax of $2259'90; there are 540 polls, which are taxed $'60 each; what is the tax on a dollar, and what will be A's tax, whose real estate is valued at $1340, his personal property at $874, and who pays for 2 polls?

540 X '60 = $324, amount of the poll taxes, and $2259'90- $324 = 1935'90, to be assessed on property. $64530: $1935'90 :: $1: '03; or, 18859203, tax on $1.

TABLE.
dolls.

dolls. dolls.
Tax on 1 is '03 | Tax on 10 is
20
30 '90

2.. '06
3 '09

..

4 ..
'12

40

5..

'15

50

'18

60

1..

'21

70

8..
(24

80

9 627

90

..

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6..

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dolls.

'30 | Tax on

'60

$1000

300

40

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Now, to find A's tax, his real estate being $1340, I find, by the table, that

is

The tax on

The tax on

The tax on

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...

dolls. dolls. 100 is 3' 200..

6'

300.. 9'

400.. 12'

500.. 15'

600.. 18'

700.. 21'

800.. 24'

900.. 27' 1000.. 30'

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Tax on his real estate

In like manner I find the tax on his personal

property to be 2 polls at '60 each, are

$30

9'

1620

$40'20

26'22

1'20

Amount, $67'62

16. What will B's tax amount to, whose inventory is 874 dollars real, and 210 dollars personal property, and who pays for 3 polls? Ans. $34'32.

17. What will be the tax of a man, paying for 1 poll, whose property is valued at $3482 ? at $768? at $940? at $4657? Ans. to the last, $140'31. 18. Two men paid 10 dollars for the use of a pasture 1 month; A kept in 24 cows, and B 16 cows; how much should each pay?

19. Two men hired a pasture for $10; A put in 8 cows 3 months, and B put in 4 cows 4 months; how much should each pay?

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100. The pasturage of 8 cows for 3 months is the same as of 24 cows for I month, and the pasturage of 4 cows for 4 months is the same as of 16 cows for 1 month. The shares of A and B, therefore, are 24 to 16, as in the former question. Hence, when time is regarded in fellowship,― Multiply each one's stock by the time he continues it in trade, and use the product for his share. This is called Double Fellowship. Ans. A 6 dollars, and B 4 dollars. 20. A and B enter into partnership; A puts in $100 6 months, and then puts in $50 more; B puts in $200 4 months, and then takes out $80; at the close of the year, they find that they have gained $95; what is the profit of each?

Ans. {$43711, A's share.

$51'288, B's share. 21. A, with a capital of $ 500, began trade Jan. 1, 1826, and, meeting with success, took in B as a partner, with a capital of $600, on the first of March following; four months after, they admit C as a partner, who brought $800 stock; at the close of the year, they find the gain to be $700; how must it be divided among the partners?

Ans.

$250, A's share. $250, B's share. $200, C's share.

QUESTIONS.

1. What is fellowship? 2. What is the rule for operating? 3. When time is regarded in fellowship, what is it called? 4. What is the method of operating in double fellowship? 5. How are taxes assessed? 6. How is fellowship proved?

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ALLIGATION.

101. Alligation is the method of mixing two or more simples, of different qualities, so that the composition may be of a mean, or middle quality.

When the quantities and prices of the simples are given, to find the mean price of the mixture, compounded of them, the process is called Alligation Mediál.

1. A farmer mixed together 4 bushels of wheat, worth 150 cents per bushel, 3 bushels of rye, worth 70 cents per bushel, and 2 bushels of corn, worth 50 cents per bushel; what is a bushel of the mixture worth?

It is plain, that the cost of the whole, divided by the number of bushels, will give the price of one bushel.

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9 bushels cost

910 cents.

2. A grocer mixed 5 lbs. of sugar, worth 10 cents per lb., 8 lbs. worth 12 cents, 20 lbs. worth 14 cents; what is a pound of the mixture worth? Ans. 121

3. A goldsmith melted together 3 ounces of gold 20 carats fine, and 5 ounces 22 carats fine; what is the fineness of the mixture? Ans. 211.

4. A grocer puts 6 gallons of water into a cask containing 40 gallons of rum, worth 42 cents per gallon; what is a gallon of the mixture worth? Ans. 36 cents.

5. On a certain day the mercury was observed to stand in the thermometer as follows: 5 hours of the day, it stood at 64 degrees; 4 hours, at 70 degrees; 2 hours, at 75 degrees, and 3 hours, at 73 degrees: what was the mean temperature for that day?

It is plain this question does not differ, in the mode of its operation, from the former. Ans. 69 degrees.

¶ 102. When the mean price or rate, and the prices or rates of the several simples are given, to find the proportions or quantities of each simple, the process is called Alligation Alternate: alligation alternate is, therefore, the reverse of alligation medial, and may be proved by it.

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