the name of a method of solving all questions that relate to the mixture of one ingredient with another. Though writers on arithmetic generally make alligation a branch of that science; yet, as it is plainly nothing more than an application of the common properties of numbers, in order to solve a few questions that occur in particular branches of business, we choose rather to keep it distinct from the science of arithmetic.

Alligation is generally divided into medial or alternate.

**Alligation Medial**, from the rates and quantities of the simples given, discovers the rate of the mixture.

**Rule.** As the total quantity of the simples, To their price or value; So any quantity of the mixture, To the rate.

**Exampl.** A grocer mixeth 30 lb. of currants, at 4d. per lb. with 10 lb. of other currants, at 6d. per lb.: What is the value of 1 lb. of the mixture? **Ans.**

| lb. | d. | d. | |-----|----|----| | 30 | at 4 amounts to 120 | | 10 | 6 |

Total 40 lb. 180

If 40 : 180 :: 1 : 4½

Note 1. When the quantity of each simple is the same, the rate of the mixture is readily found by adding the rates of the simples, and dividing their sum by the number of simples. Thus,

Suppose a grocer mixes several sorts of sugar, and of each an equal quantity, viz. at 50 s. at 54 s. and at 60 s. per cwt. the rate of the mixture will be 54 s. 8d. per cwt.; for

\[ \frac{50 + 54 + 60}{3} = 54 \text{ s. 8d.} \]

Note 2. If it be required to increase or diminish the quantity of the mixture, say, As the sum of the given quantities of the simples, to the several quantities given; so the quantity of the mixture proposed, to the quantities of the simples sought.

Note 3. If it be required to know how much of each simple is an assigned portion of the mixture, say, As the quantity of the mixture, to the several quantities of the simples given; so the quantity of the assigned portion, to the quantities of the simples sought.

Thus,

Suppose a grocer mixes 10 lb. of raisins with 30 lb. of almonds and 40 lb. of currants, and it be demanded, how many ounces of each sort are found in every pound or in every 16 ounces of the mixture, say,

| oz. | 80 : 10 : 16 : 2 raisins. | |-----|--------------------------| | | 80 : 30 : 16 : 6 almonds. | | | 80 : 40 : 16 : 8 currants. |

**Proof 16**

Note 4. If the rates of two simples, with the total value and total quantity of the mixture, be given, the quantity of each simple may be found as follows, viz.

Multiply the lesser rate into the total quantity, subtract the product from the total value, and the remainder will be equal to the product of the excess of the higher rate above the lower, multiplied into the quantity of the higher-priced simple; and consequently the said remainder, divided by the difference of the rates, will quote the said quantity. Thus,

Suppose a grocer has a mixture of 400 lb. weight, that cost him 7l. 10s. consisting of raisins at 4d. per lb.

And almonds at 6d. how many pounds of almonds were Alligation in the mixture?

| lb. | Rates. | |-----|-------| | 400 | 6d. | | 4 | 4d. | | 7 | 10 | | 100 | 1600 | | 1600| 1600d.|

Total 400

**Alligation Alternate**, being the converse of alligation medial, from the rates of the simples, and rate of the mixture given, finds the quantities of the simples.

**Rules.** I. Place the rate of the mixture on the left side of a brace, as the root; and on the right side of the brace set the rates of the several simples, under one another, as the branches. II. Link or alligate the branches, so as one greater and another less than the root may be linked or yoked together. III. Set the difference betwixt the root and the several branches right against their respective yoke-fellows. These alternate differences are the quantities required. Note, 1. If any branch happen to have two or more yoke-fellows, the difference betwixt the root and these yoke-fellows must be placed right against the said branch, one after another, and added into one sum. 2. In some questions, the branches may be alligated more ways than one; and a question will always admit of so many answers as there are different ways of linking the branches.

Alligation alternate admits of three varieties, viz.

1. The question may be unlimited, with respect both to the quantity of the simples and that of the mixture. 2. The question may be limited to a certain quantity of one or more of the simples. 3. The question may be limited to a certain quantity of the mixture.

**Variety I.** When the question is unlimited, with respect both to the quantity of the simples and that of the mixture, this is called **Alligation Simple**.

**Exampl.** A grocer would mix sugars, at 5 d., 7 d. and 10 d. per lb. so as to sell the mixture or compound at 8d. per lb.: What quantity of each must he take?

Here the rate of the mixture 8 is placed on the left side of the brace, as the root; and on the right side of the same brace are set the rates of the several simples, viz. 5, 7, 10, under one another, as the branches; according to Rule I.

The branch 10 being greater than the root, is alligated or linked with 7 and 5, both these being less than the root 5, as directed in Rule II.

The difference between the root 8 and the branch 5, viz. 3, is set right against this branch's yoke-fellow 10.

The difference between 8 and 7 is likewise set right against the yoke-fellow 10. And the difference betwixt 8 and 10, viz. 2, is set right against the two yoke-fellows 7 and 5; as prescribed by Rule III.

As the branch 10 has two differences on the right, viz. Alligation. viz. 3 and 1, they are added; and the answer to the question is, that 2 lb. at 5d., 2 lb. at 7d., and 4 lb. at 10d. will make the mixture required.

The truth and reason of the rules will appear by considering, that whatever is lost upon any one branch is gained upon its yoke-fellow. Thus, in the above example, by selling 4 lb. of 10 d. sugar at 8 d. per lb. there is 8 d. lost; but the like sum is gained upon its two yoke-fellows; for by selling 2 lb. of 5 d. sugar at 8 d. per lb. there is 6 d. gained; and by selling 2 lb. of 7 d. sugar at 8 d. there is 2 d. gained; and 6 d. and 2 d. make 8 d.

Hence it follows, that the rate of the mixture must always be mean or middle with respect to the rates of the simples; that is, it must be less than the greatest, and greater than the least; otherwise a solution would be impossible. And the price of the total quantity mixed, computed at the rate of the mixture, will always be equal to the sum of the prices of the several quantities cast up at the respective rates of the simples.

Variety II. When the question is limited to a certain quantity of one or more of the simples, this is called Alligation Partial.

If the quantity of one of the simples only be limited, alligate the branches, and take their differences, as if there had been no such limitation; and then work by the following proportion:

As the difference right against the rate of the simple whose quantity is given, To the other differences respectively; So the quantity given, To the several quantities sought.

Examp. A distiller would, with 40 gallons of brandy at 12 s. per gallon, mix rum at 7 s. per gallon, and gin at 4 s. per gallon: How much of the rum and gin must he take, to sell the mixture at 8 s. per gallon?

| Gal. | 40 of brandy. | |-----|--------------| | 8 | 5 | | 7 | 4 | | 4 | 4 |

The operation gives for answer, 5 gallons of brandy, 4 of rum, and 4 of gin. But the question limits the quantity of brandy to 40 gallons; therefore say,

If 5 : 4 : 4 :: 40 : 32

The quantity of gin, by the operation, being also 4, the proportion needs not be repeated.

Variety III. When the question is limited to a certain quantity of the mixture, this is called Alligation Total.

After linking the branches, and taking the differences, work by the proportion following:

As the sum of the differences, To each particular difference; So the given total of the mixture, To the respective quantities required.

Examp. A vintner hath wine at 3 s. per gallon, and would mix it with water, so as to make a composition of 144 gallons, worth 2 s. 6 d. per gallon: How much wine, and how much water, must he take?

Gal. | 120 of wine. | |-----|-------------| | 30 | 36 | | 0 | 6 |

Answ. 144 total.

120 × 36 = 4320 24 × 0 = 0

Proof 144/4320(30) As 36 : 30 : : 144 : 120 As 36 : 6 : : 144 : 24.

There being here only two simples, and the total of the mixture limited, the question admits but of one answer.