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Tut. You are right. When the prices of two commodities are given; to know how much of one commodity must be given for a given quantity of the other com. modity

Find the whole price of the given commodity, and divide it by the price of the other commodity, and the quotient will be the

answer.

How much sugar, at $5, per cwt. must be given for 350 yards of linen, at 25 cents per yard ?

Ans. 17 cwt. 2 qrs. When the bartering price of one commodity is rated above its ready money price, to find the bartering price of the other;

Say, as the ready money price of one commodity is to the ready money price of the other, so is the bartering price of one, to the bartering price of the other.

A. and B. barter; A. has 150 gallons of brandy at $1,375 per gallon, ready money, but in barter he will have $1.50; B. has linen at 44 cents per yard, ready money; how must B. sell his linen per yard in proportion to A's bartering price, and how many yards are equal to A's brandy ?

Ans. barter price is 48 cents, and he must give A. 468 yds. 3 qrs.

There are other methods of bartering, but they are very simple, and you will easily understand them, from what I have taught you of the above methods.

ALLIGATION.

What is alligation ?

Pup. Alligation is the mixing of simples of different qualities.

Tut. Alligation is a method of mixing two or more simples of different qualities or prices, so as to make a middle quality or price. Alligation is of two kinds, medial and alternate.

N

ALLIGATION MEDIAL

Is a method of finding the rate of the compound, when the rates and quantities are given ; for which we have the following rule.

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

If I mix 20 bushels of wheat at $1, per bushel, 16 bushels of rye at 75 cents per bushel, 12 bushels of barley at 50 cents per bushel, and 8 bushels of oats at 40 cents per bushel; what is the value of one bushel of this mixture ?

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A goldsmith melted together 8oz. of gold, 22 carats* fine, llb. 8oz. of 21 carats fine, and 10oz. of 18 carats fine; what is the fineness of the composition ?

Ans. 221, carats fine.

It frequently happens, that we have a number of simples, with their rates given, and wish to know what quantity of each it will take to make a mixture of a given rate; so that this is the reverse of alligation medial, and is called

* A carat is a twenty fourth part; 22 carats fine, means of pure metal. A carat is also divided into four parts, called grains of a carat.

ALLIGATION ALTERNATE.

If a man have corn worth $1, per bushel, and oats worth 50 cents per bushel, what quantity of each must he take, to make a mixture worth 75 cents per bushel ?

Now if you take one bushel of corn, and mix it with one bushel of oats, there will be two bushels, the whole price of which will be $1,50, and one bushel will be 75 cents; and if even quantities of each be mixed, the price of the m'xture will be 75 cents per bushel, whatever the quantity may be.

Again, if you have corn at 84 cents per bushel, rye at 72 cents, barley at 48 cents, and oats at 36 cents per bushel, and you wish to make a mixture which shall be worth 60 cents per bushel, you must go accommodate the mixture, that what is lost by diminishing the quantities, whose prices exceed the price of the mixture, may be gained by increasing those quantities whose prices are less than the price of the mixture; which may be done by linking the prices together in the following manner, and taking the difference between the price of each simple, and the price of the mixture, and writing it against the price with which this price is linked.

36 24 bushels at 36 cts.

48 12 bushels at 48 cts. 60

72 12 bushels at 72 cts.

84 24 bushels at 84 cts. Proof by alligation medial.

36 X 24 = $8,64
48 x 12 = 5,76
72 X 12 = 8,64
84 X 24 = 20,16

72 ) 43,20(60

432

Hence the following rule ;

Write the prices of the simples in a column under each other, the least uppermost, and place the price of the compound at the left of the column. Connect or link with a continued line the price of each simple, which is less than that of the compound, with one

or any pumber of those that are greater than the compound; and each greater price with one or any number of the less. Write the difference between the price of the mixture, and that of each of the simples, opposite to the prices with which they are respectively linked. Then if only one difference stand against any price, it will be the quantity belonging to that price ; but if there be severa al, their sum will be the quantity.

A goldsmith would mix gold of 19 carats fine with some of 15, 18, 23 and 24 carats fine, so that the compound may be 21 carats fine; what quantity of each must he take ?

Ans. 5 oz. of 16 carats fine, 5 oz. of 18, 5 oz. of 19, 10 oz. of 23, and 10 oz. of 24 carats fine.

How much barley at 40 cents, rye at 60 cents, and wheat at 80 cents per bushel, must be mixed together, that the compound may be worth 624 cents per bushel ?

(174 barley, Ans. 173 rye,

25 wheat.

When the whole composition is limited to a certain quantity, find an answer by linking as before; then say, as the sum of the quantities, thus found, is to the given quantity, so is each quantity, found by linking, to the required quantity of each.

Suppose you have corn at 75 cents, and oats at 40 cents per bushel; how many bushels of each must be taken to make 120 bushels worth 60 cents per bushel ?

S50 15 25 : 120 :: 10 : 48 bushels of corn. 60

2 7510 25 : 120 :: 10 : 72 bushels of oats.

25 A goldsmith has gold of 14, 16, 18, and 22 carats fine, and wishes to melt together, of all these sorts, so much as will make a mass of 50 oz. 20 carats fine; how much must he take of each sort? Ans. 5.5 oz 14 carats fine; 5. 16; 5 oz. 18; and 33} oz. 22 carats fine.

When one of the ingredients is limited to a certain quantity;

Find the difference between each price and the price of the mixture as before ; then say, as the difference of that simple,

whose quantity is given, is to the rest of the differences severally, so is the quantity given to the several quantities required.

How much gold of 16, 20, and 24 carats fine, and how much alloy, must be mixed with 10 oz. of 18 carats fine, that the composition may be 22 carats fine ?

Alloy, 00

16
22 18-

20
24

2
2
2
2 +4 +6+ 22 = 34.

2 : 2 :: 10 : 10 oz. alloy.
2 : 2 :: 10 : 10 oz. 16 carats fine.
2 : 2 :: 10 : 10 oz. 20 carats fine.
2 : 34 :: 10 : 170 oz. 24 carats fine.

Questions.

How is barter useful ?

When the prices of two commodities are given, bow do you find how much of one must be given for the other ?

When the bartering price of one commodity is rated above its ready money price, how do you find the bartering price of the other? What is alligation ? How many kinds of alligation are there? What is alligation medial? What is the rule for it? What is alligation alternate?

When you have several commodities, with their rates or prices given, how do you find what quantity of each must be taken to make a mixture of a given rate ?

When the whole composition is limited to a certain quantity, how do you find the required quantity of each commodity ?

When one of the ingredients is limited to a certain quantity, how do you find the required quantity of each commodity ?

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