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?

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

Proof 144 \times 4320 \div 30

\text{As } 36 : 30 :: 144 : 120
\text{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.