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EXAMPLES 1. A farmer mixed together 4 bushels, of rye, worth 90 cents per bushel, 6 bushels of corn, worth 50 cents per bushel, and 8 bushels of oats, worth 30 cents per bushel; what is a bushel of this mixture worth?

4 bushels at 90 cost 360


18 bushels cost

900 As 18 : 900 :: 1: 50 cents, price of 1 bushel. 2. A grocer mixed 5cwt. of sugar, worth $10 per cwt., 8cwt. worth $12 per cwt., with 3cwt, worth $9 per cwt. ; what will be the cost of 4cwt. of this mixture ?

Ans. $43,25. 3. A vintner mixed together 16 gallons of wine at $1,12c. per gallon, 12 gallons at $ ,90 per gallon, and 20 gallons at $ ,95 per gallon ; what is the price of a gallon of the mixture ?

Ans. $ ,99, 4m. + 4. A goldsmith melted together 5 ounces of gold 21 cara ats fine, and 3 ounces 19 carats fine ; what is the fineness of the mixture; that is, of one ounce of this mixture ?

Ans. 201 carats finè: 5. A Grocer mixed 20 gallons of water with 85 gallons of rum, worth 76 cents per gallon, what is a gallon of the mixture worth?

Ans. 61ctš. 5m. + 6. A Refiner melted together 12lb. of silver bullion of 6oz. fine, 8lb. of 7oz. fine, and 10lb of 8oz. fine, I demand the fineness of the mixture. Ans. 6oz. 18pwt. 16gr.

7. Suppose that 3lbs. of gold of 22 carats fine, 5lbs. of 20 carats fine, and ilb. alloy be melted together, what will be the fineness of the compound? Ans. 18 carats fine.

ALLIGATION ALTERNATE, Is when the prices of the several simples and the mean price or rate are given, to find what proportion of each must be taken to compose a mixture of the given rate. It is therefore the reverse of Alligation medial, and may be proved by it.


When the mean rate and the rates of the several ingredients

are given, without any limited quantity.


1. Reduce the several prices to the same denomination.

2. Set the several prices under each other in a column, and place the mean rate on the left.

3. Connect or link each price which is less than the mean rate, with one or any number of those which are greater than the mean rate, and each price greater than the mean rate with one, or any number of the less.

4. Place the difference between the mean rate and that of each of the simples, opposite the price with which they are connected.

5. Then if only one difference stand against any rate, it will be the quantity belonging to that price, but if there be several, their sum will be the quantity


1 Octs. per





1 2

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i. A Grocer would mix the following qualities of sugar, viz: at 8cts., 9cts., 11cts., and 12cts. per lb., what quantity must he take of each sort, that the mixture may be worth lb.


8 cts.







1 Here we set down the prices of the simples in order; from the least to the greatest; placing the mean rate at the left hand. And in the first way of linkng the prices, and taking the difference between the several prices and mean rate, and placing each difference at the opposite end of the link, we find that we must take in proportion of 2lbs. at 8cts., ilb. at Octs., 1lb. at 11cts., and 2lbs. at 12cts. ; and in the 2d operation, we have the proportion of llb. at 8cts., 21b. at 9cts., 2lb. at 11cts., and ilb. at 12cts. It will be seen that by linking any two of the prices together, and placing the

difference between those prices and the mean rate alternately, that is, placing the difference between the greater price and the mean rate, against the lesser price, and placing the difference between the lesser price and the mean rate, against the greater price, the difference of the prices become mutually changed ; and these differences express the relative quantities of each simple, necessary to form the compound, and what is lost on one quantity, is gained on another, Hence the balance of loss and gain between any two, will be equal, consequently the same on the whole.

2. A Merchant has three sorts of tea, viz. : one sort at 5 shillings per lb., another at 7 shillings per lb., and another at 8 shillings per lb., what proportion of each kind must he take to make a mixture worth 6s.


lb. ?

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3. How much vinegar at 42cts., 60cts. and 67cts. per gallon, must be mixed together, so that it may be worth 64 cents per gallon ?

Ans. 3gals. at 42cts., 3do. at 60c. and 26do. at 67c. 4. How much sugar at 9cts. and 15cts. per lb. must be mixed together, so that the compound may be worth 12 cents per lb. ? Ans. An equal quantity of each sort.

5. À Goldsmith mixed together gold of 17, 19, 21, and 24 carats fine, so that the composition was 22 carats fine what proportion did he take of each?

Ans. 2 of each of the first 3 sorts, and 9 of the last. 6. A Merchant has spices at 7d. 8d. 10d. and 11d. per Ib. which he would mix together so that the whole composition may be sold at 9d. per lb., what proportion must he take of each kind ? 2lb. at 7 cts.

ilb. at 7cts. 3lb. at 7 cts.
1 8
2 8

2 8
1 10
2 10

2 10


3 11 4th Ans. 3lb. of each sort. These four answers arise from the various ways of link

ing the prices of the ingredients together. Hence we may see that questions in this rule admit of a great variety of answers; for having found one answer, we may take any other numbers which have the same proportion between themselves, as the numbers have which compose the



When one of the ingredients is limited to a certain quantity to find what quantity of each of the others must be taken in proportion to the given quantity.

RULE. 1. Link the prices and take their differences as in Case 1. which will give the unlimited proportions.

2. Then as the proportion whose quantity is limited, is to its limited quantity, so is each of the other proportions to its required quantity.



1. A man wishes to mix 5 bushels of wheat worth 90 cents per bushel, with rye at 56cts. per bushel, barley at 36cts., and oats at 30cts., so that the composition may be sold for 45cts. per bushel:

90 15 the proportion whose quantity cts. 56 9

is limited. mean rate 45 36


then as 15 : 5 :: 9

bashels of rye. 15 : 5:;11:3 barley. 15 : 5 :: 45 : 15

oats. 2. How much water at 0 per gallon, must be mixed with 100 gallons of brandy worth 180cts. per gallon, to reduce it to 150cts. per gallon ?

Ans. 20gal. 3. How much gold of 15, of 17, and of 22 carats fine, must be mixed with 5oz. of 18 carats fine, so that the composition may be 20 carats fine ?

Ans. 5oz. of 15, 5oz. of 17, and 25oz. of 22 carats fine.

4. With 95 gallons of wine worth 96cts. per gallon, I mixed wine at 80cts. per gallon, and some water, so that it

stood me in 76cts. per gallon, how much wine and how much water did I take?

Ans. 95gals. wine at 80cts., and 30gals. water


When the whole composition is limited to a given quantity.


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İ. Link the prices and find the proportions as in Case 1. Then, as the sum of the proportion is to the given quantity or whole composition, so is each proportion to its required quantity.

EXAMPLES. i. A Grocer has sugar at 5cts., 7cts., 10cts. and 13cts. per lb., and would mix them together so as to fill a cask of 200lbs. worth Scts. per pound.


cts. 7 2
mean rate, 8 10 1

13 3

11=sum of the proportion.
As 11 : 200 :: 5 9011lbs. at 5cts.
11 : 200 :: 2 364

11 : 200 ::1 184 10cts.

11 : 200 ::3 5411 13cts. 2. How much water of no value must be mixed withi spirits at 90cts. per gallon, so as to fill a cask of 120gals. that may be so for 60cts. per gallon ?

Ans. 40gals. of water, and 80gals, of spirits. 3. A Grocer has teas at 25., 38., 5s., and 6s. per lb. and would mix them together so that the composition may be worth 4s. per lb. What quantity must he take of each, to fill a chest that will hold 90lbs. ?

Ans. 30lbs., 15lbs., 15lbs., and 30lbs. 4. How much gold of 15, of 17, of 18, and of 22 carats fine, must be mixed together, to make a composition of 400z. that will be 20 carats fine ?

Ans. 5oz. of 15 5 of 17, 5 of 18, and 25oz. of 22.

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