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Case III.

When one commodity is rated above the ready money price, to find the bartering price of the other.

RULE. As the ready money price of the one is to its bartering price, so is that of the other to its bartering price.

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6. Loss and Eain.

Loss AND GAIN is a rule which enables merchants to ascertain the profit, or loss, in buying and selling their goods, and also teaches them how much to raise or fall in the price, in order to gain or loseso much per cent.

Case I.

To know what is gained or lost, per cent.

RULE. Find the whole gain or loss by subtraction; then as the price it cost is to the whole gain or loss, so is $100 to the gain or loss per cent.

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To know how a commodity must be sold to gain or lose so much per cent,

RULE. AS $100 is to the price, so is $100 with the profit added or loss subtracted to the gaining or losing price.

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Alligation teaches to mix commodities of different qualities, so that the composition may be of a middle quality. Alligation is of two kinds, Medial and Alternate.

4. ALLIGATION MEDIAL.

Alligation Medial is the method of finding the rate of the compound, from having the prices and quantities of the several commodities given.

RULE.-Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities, the quotient will be the rate of the compound required.

*The truth of this rule is too evident to need a demonstration; for multiplying each quantity by its price, and adding the products gets the price of the whole quantity; then as the sum of the quantities, or number of bushels, is (by the Rule of Three) to the whole price of the mixture, so is one bushel of the mixture to its price. But as 1 is the third term in this and all similar cases, we have only to divide the whole price by the whole quantity.

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2. ALLIGATION ALTERNATE.

Alligation Alternate is the method of finding what quantity of each of the articles, whose rates are given, will compose a mixture of a given rate.

Alligation Alternate is the reverse of Alligation Medial, and may be proved by it.

Rule 1.*

1. Write the rates of the several articles in a column under each other, beginning with the least, and place the rate of the compound at the left hand.

2. Connect each rate which is less than that of the compound with one that is greater, and each that is greater with one that is less. 3. Write the difference between each rate and that of the compound against the number with which each is connected.

4. Then if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be more than one, their sum will be the quantity.

* As a less rate is in all cases connected with a greater, and the differences between them and the mean rate placed alternately there is just as much gained by one quantity as there is lost by the other. Thus in getting answer first,

Examples.

1. A farmer wishes to mix rye at 4s corn at 3s. barley at 2s. 6d. and oats at 23 per bushel, so that the mixture may be worth 2s. 10d. per bushel; how much of each sort must he take?

bush.

24

bush.
14 oats.

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example first, 24 and 30, which are below the mean, 34, are linked to 48 and 36, which are above the mean, and the difference, 10. between 24 and 34, is placed against 48, and 14, the difference between 34 and 48, is placed against

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