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the one case, is to its selling price in the other, so is the selling price of a gallon in the first case, to its selling price in the other.

Ex. 24. If 15 per cent. be gained by selling sugar at $8.75 per cwt., how much is gained per cent. by selling it at $9-25? Ans. 214 per cent. or $21.58.2 per 100 dolls.

25. If 6 per cent. be gained by selling Canton crape at $13.25 per piece, how much per cent. would be gained by selling the same at $15.37 per piece? Ans. 23 per cent. 26. If 20 per cent. be gained by selling flour at $5.871 per barrel, how much per cent. is gained or lost by selling it Ans. 13.56 per cent. at $4.25 per barrel?

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27. Bought 21300 yards of linen, at 2s. 94d. per yard, and paid for various charges £88 15s. 1d.; and sold one third at 3s., one third at 3s. 4d., and one third at 3s. 2d. per yard required the whole gain, and the gain per cent.

Ans. £310 12s. bd. and £10 2s. 10 d.

CHAPTER X.

Division into Proportional Parts; Fellowship; and Alligation.

DIVISION INTO PROPORTIONAL PARTS.

PROBLEM. To divide a given quantity into parts which shall have to each other given ratios.

164. RULE.--As the sum of the numbers expressing the ratios, is to any one of these numbers, so is the entire quantity to be divided, to the part corresponding to the number used as the second term of the analogy.*

* When all the parts, except one, have been determined, that one may be found by adding the rest together, and taking the sum from the number to be divided. In this case, the operation will be proved by finding that part by the general rule: the agreement of the two results will prove the accuracy of the work.

The reason of this rule, which is of frequent use in Arithmetic, and other branches of Mathematics, is obvious from this principle, that the sum of all the antecedents of any number of equal ratios, is to the sum of alt the consequents, as any one of the antecedents is to its consequent,

The operation is proved by adding the several results together if the sum be equal to the quantity to be divided, the work is right.

Example. A farm of 756 acres, 2 roods, 14 poles, is to be divided into two parts, such that the one may be threefourths of the other. What are the parts ?

Here, the parts are evidently in the ratio of 4 to 3, the sum of which is 7; therefore,

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Exercises.-1. Suppose a traveller to proceed from NewYork to Washington City, at the rate of 6 miles per hour, and another at the same time from Washington City for NewYork, at the rate of 5 miles per hour: where will they meet, the distance between the two places being 228 miles? Ans. 124 miles from New-York. 2. Divide 398 into three parts, which will be to one another as the numbers 5, 7, and 11.

Ans. 861, 121, and 190. 3. Divide $800 into four parts, in the ratio of 10, 19, 9, and 2. Ans. 200, 380, 180, and 40. 4. Divide $38000 among three persons, in such a manner that the share of the second may be one half greater than that of the first, and the share of the third one half greater than that of the second.* Ans. $8000, 12000 and 18000.

5. Pure water is composed of two gases, or kinds of air, called oxygen and hydrogen, in such proportions, that the weight of the former is to that of the latter as 15 to 2. Required the weight of each contained in a cubic foot, or 1000 ounces Avoirdupois weight of water.

Ans. 882 oz., and 1171 oz. 6. How much copper and how much tin will be required to make a cannon, weighing 16cwt. 1qr. 20lbs., gun metal being composed of 100 parts of copper, and 11 of tin?

Ans. 14cwt. 3qrs. 573 lbs.; and 1cwt. 2qrs. 14,38 lbs. 7. The British standard gold for coinage consists of 11 parts of pure gold, and 1 part of alloy, usually a mixture of

* In this exercise, it is easy to see, that the parts will be as the num bers 1, 11, and 21; or as 4, 6, and 9.

silver and copper: how much pure gold and how much alloy are contained in aguinea, which weighs 5 dwts. 9 grains ? Ans. 4dwts. 221grs., and 10grs.

8. The British silver coin consists of 37 parts of silver and 3 of copper; how much of each does the half crown, worth 2s. 6d. contain, each pound, Troy weight, being coined into 66 shillings? Ans. 8dwts, 8.93+grs., and 164grs. 9. How much tin and copper are contained in a bell, weighing 150 lbs. ; bell metal being composed of 3 parts of copper and 1 part of tin? Ans. 112 lbs., and 371lbs. 10. Pewter is composed of 112 parts of tin, 15 of lead, and 6 of brass; how much of each ingredient is requisite to make a ton of pewter ?

Ans. 16cwt. 3qrs. 10 lbs., 2cwt. 1gr. 01 lb., and 3qrs. 17lbs. S 11. Proof spirits are composed of 48 parts of alcohol, or pure spirit, and 52 parts of water. How much of each of these contained in 84 gallons of proof spirits?

Ans. 40galls., and 4317galls. 12. 76 parts of nitre, 14 of charcoal, and 10 of sulphur, compose gunpowder: how much of these ingredients will be requisite to form one hundred weight of gunpowder? Ans. 3qrs. 1lbs., 1517lbs., and 11lbs.

Fellowship.

165. Fellowship is the method of determining of the partners in a mercantile company.

166. Fellowship is usually distinguished into two kinds, simple and compound, or single and double.

167. In simple or single fellowship, the stocks or sums contributed by the several partners, all continue in trade for the same time.

168. In compound or double fellowship, the stocks continue in trade for different periods.

SIMPLE FELLOWSHIP AND BANKRUPTCY.

169. RULE. As the whole stock is to the whole gain or loss, so is the stock of any partner to his gain or loss.

In the same way, the estate of a bankrupt may be divided among his creditors by this analogy-As the

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sum of all the claims on the estate is to its value, so is the claim of any creditor to his dividend, or share of the estate.*

Example 1. Three merchants, A, B, and C, form a joint capital, of which A contributes $700, B $1000, and C $1600. What is the share of each in a gain of $880?

Here, the sum of the stocks is $3300, the whole capital. Then, as $3300: $880, or, by contraction, as 15: 4 :: $700 $186 66, A's share; and as 15: 4::$1000: $266 662, B's share; and, lastly, as 15: 4 :: $1600 $426 663, C's share. The sum of these shares is exactly $880, which proves the operation to be correct.

2. A bankrupt owes to A $900, to B $850, to C $640, to D $150, to E $750, and to F $310; but his whole estate amounts to $1500. Required the share of each creditor.

Here, the sum of the debts is $3600. Then, as $3600: $1500, or, by contraction, as 36: 15: $900 $375, A's dividend; and 36: 15: $850 $354,163, B's dividend. In the same manner, we find C's share to be $266.66%; D's $62.50; E's $312.50; and F's $129.163. The sum of all these is $1500, which proves the work to be correct.†

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Exercises.-1. A's and B's stocks are $375 and $425, respectively required the share of each in a gain of $240. Ans. A's $112 50, and B's $127 50. 2. Three merchants, A, B, and C, enter into partnership: A puts into the stock $1555.50, B $12591.75, and C $1876.75. Required the share of each in a gain of $2000.

Ans. A's $194.14.6, B's $1571.16.1, and C's $234.24.2. 3. If a bankrupt, whose property amounts to £2100, owe to A £826 12s., to B £1263 9s. 6d., to C £724 15s. 10d., to D £1000, and to E £242 16s. 4d.; how much can he pay in the pound, and what is the dividend of each creditor?

* This rule is merely a particular application of that contained in the fast article, and therefore requires no separate illustration. The method of proof is also the same.

In the division of a bankrupt's estate, it is usual first to find how much in the dollar, or pound, he can pay; that is, how much the creditors will receive for each dollar, or pound, of their respective claims: thus, resuming the same example, we have this analogy-as 3600: 1500, or as 36: 15: $1: 15, the sum that each creditor is to receive in the dollar. Then, as $1 $15:: $900 $375, as before. In the same way the rest of the dividends may be found.

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Ans. 10s. 41d., nearly, in the pound; and the dividend of A £427 15s. 11d.; B £653 178. 10gd.; C £375 28.

1d.; D £517 108. Sad.; and of E £125 138. 4d.+ 4. A debtor, the value of whose effects is only $1075, owes to A $350.50, to B $750.75, to C $1270, and to D $1875.50. What is the dividend of each and how much is paid in the dollar? Ans. A $88.72.312324, B$190.04.015520 C$321.48.3,8278, D $474.75.1787.

5. If two persons purchase a house jointly for $4000, and afterwards let it for the yearly rent of $650; what share of the yearly profit is each to receive, the one having contributed $1850, and the other $2150; and the taxes and other expenses being $35.50 per year?

Ans. $330.29.38, and $284.20.64. 6. If the capital stock of a banker, amounting to $500000, gain in 6 months $22500, what is the gain per cent.,* and what does a stockholder receive, who holds 6 shares of $50 each? Ans. $4.50 per cent., and $13.50. 7. Three merchants, A, B, and C, freight a ship from Madeira for Liverpool, 216 tuns of wine; of which A had 96, B 72, and C 48; the mariners, meeting with a storm at sea, were constrained, for the safety of their lives, to cast 45 tuns thereof overboard: how many of the 45 tuns has each merchant lost according to the rate of his adventure? Ans. A 20, B 15, and C 10 tuns.

COMPOUND FELLOWSHIP.

170. RULE.-Let all the times be of the same denomination, and multiply each stock by the time of its continuance in trade: then, using the products as stocks, proceed according to the rule for simple Fellowship.

Example. A and B enter into partnership: A contributes $600 for 13 months, and B $800 for 10 months. Required the share of each in a gain of $650.

Here, the products are $7800 and $8000, the sum of which is $15800. Then, as $15800: $650, or, by contraction, as $316 13: $7800: $320.89, A's share; and as $316: $13 :: $8000:

$600 × 13=$7800 800 × 10 8000

$15800

*It is proper to observe, that in banking concerns, the gain or loss per cent. is first calculated, and the dividend is declared at so much per cent; so that each person's share of the profit is found by multiplying his part of the capital by the gain per cent,

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