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.

Example. A vintner has wine at 3s. per gallon, and would mix it with water, so as to make a composition of 144

gallons, worth 2s. 6d. per gallon: How much wine, and how much water, must he take?

\begin{array}{r|l|l|l} \left\{ \begin{array}{c} 36 \\ 0 \\ 6 \end{array} \right. & 30 & \left. \begin{array}{l} 120 \text{ of wine,} \\ 24 \text{ of water.} \end{array} \right\} \text{Ans.} \\ \hline 36 & & 144 \text{ total.} \\ 120 \times 36 = 4320 & & \\ 24 \times 0 = 0 & & \end{array}

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.