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T 99. In assessing taxes, it is necessary to have ventory of the property, both real and personal, of the wnole 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 $3241935'90, to be assessed on property. $64530: $1935'90 :: $1: '03; or, 1830 1935'90 = '03, tax on $1.

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Now, to find A's tax, his real estate being $1340, I find,

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$40'20.

26'22

1'20

Amount, $67'62

In like manner I find the tax on his personal

property to be

2 polls at '60 each, are

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? 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?

at $4657?

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?

100. The pasturage of 8 cows for 3 months is the same as of 24 cows for 1 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. $51-288, B's share. $43 711, A's share.

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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?

.

ALLIGATION.

T 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 Medial.

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.

4 bushels, at 150 cents, cost 600 cents.

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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. 214. 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. degrees.

Ans. 69

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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1. A man has oats worth 40 cents per bushel, which he wishes to mix with corn worth 50 cents per bushel, so that the mixture may be worth 42 cents per bushel; what proportions, or quantities of each, must he take?

Had the price of the mixture required exceeded the price of the oats, by just as much as it fell short of the price of the corn, it is plain, he must have taken equal quantities of oats and corn; had the price of the mixture exceeded the price of the oats by only as much as it fell short of the price of the corn, the compound would have required 2 times as much oats as corn; and in all cases, the less the difference between the price of the mixture and that of one of the simples, the greater must be the quantity of that simple, in proportion to the other; that is, the quantities of the simples must be inversely as the differences of their prices from the price of the mixture; therefore, if these differences be mutually exchanged, they will, directly, express the relative quantities of each simple necessary to form the compound required. In the above example, the price of the mixture is 42 cents, and the price of the oats is 40 cents; consequently, the difference of their prices is 2 cents: the price of the corn is 50 cents, which differs from the price of the mixture by 8 cents. Therefore, by exchanging these differences, we have 8 bushels of oats to 2 bushels of corn, for the proportion required.

Ans. 8 bushels of oats to 2 bushels of corn, or in that proportion.

The correctness of this result may now be ascertained by the last rule; thus, the cost of 8 bushels of oats, at 40 cents, is 320 cents; and 2 bushels of corn, at 50 cents, is 100 cents; then, 320 +100420, and 420, divided by the number of bushels, (8+2,) = 10, gives 42 cents for the price of the mixture.

2. A merchant has several kinds of tea; some at 8 shillings, some at 9 shillings, some at 11 shillings, and some at 12 shillings per pound; what proportions of each must he mix, that he may sell the compound at 10 shillings per pound?

Here we have 4 simples; but it is plain, that what has just been proved of two will apply to any number of pairs, if in each pair the price of one simple is greater, and that of the other less, than the price of the mixture required Hence we have this

RULE.

The mean rate and the several prices being reduced to the same denomination,-connect with a continued line each price that is LESS than the mean rate with one or more that is GREATER, and each price GREATER than the mean rate with one or more that is LESS.

Write the difference between the MEAN rate, or price, and the price of EACH SIMPLE opposite the price with which it is connected; (thus the difference of the two prices in each pair will be mutually exchanged;) then the sum of the differences, standing against any price, will express the RELATIVE QUANTITY to be taken of that price.

By attentively considering the rule, the pupil will perceive, that there may be as many different ways of mixing the simples, and consequently as many different answers, as there are different ways of linking the several prices. We will now apply the rule to solve the last question :

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Here we set down the prices of the simples, one directly under another, in order, from least to greatest, as this is most convenient, and write the mean rate, (10 s.) at the left hand. In the first way of linking, we find, that we may take in the proportion of 2 pounds of the teas at 8 and 12 s. to 1 pound at 9 and 11 s. In the second way, we find for the answer, 3 pounds at 8 and 11 s. to 1 pound at 9 and 12 s.

3. What proportions of sugar, at 8 cents, 10 cents, and 14 cents per pound, will compose a mixture worth 12 cents per pound?

Ans. In the proportion of 2 lbs. at 8 and 10 cents to 6 lbs. at 14 cents.

Note. As these quantities only express the proportions of each kind, it is plain, that a compound of the same mean price will be formed by taking 3 times, 4 times, one half, or any proportion, of each quantity. Hence,

When the quantity of one simple is given, after finding

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