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.

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

\begin{array}{r|l|l|l} \left\{ \begin{array}{c} 12 \\ 7 \\ 4 \end{array} \right. & 1, 4 & 5 & \left. \begin{array}{l} 40 \text{ of brandy,} \\ 32 \text{ of rum,} \\ 32 \text{ of gin.} \end{array} \right\} \text{Ans.} \end{array}

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,

\text{If } 5 : 4 :: 40 : 32.

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