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Alligation Method (GRE and GMAT)

Stumped by those pesky solution problems? Let's introduce an age-old method that solves these problems quickly: alligation!

We can use alligation to quickly find the amount of a mixture X that is x% of an ingredient and a mixture Y that is y% of the same ingredient needed to make a mixture Z that is z% of the same ingredient.

We do so as follows: we find the absolute value of the percent of the target mixture minus the percent of each mixture; in other words, find the absolute values of both z - x and of z -y. The ratio of X to Y in mixture Z is then:

$\frac{|z-y|}{|z -x|}$

Or, in other words, |z-y| parts of X for every |z -x| parts of Y.***

Let's use this technique on an example problem. You could do it with this question, for one:

GRE users -- GRE Solution Problem

GMAT users -- GMAT Solution Problem

Here, the question is asking us how much of a 30% mixture of alcohol, solution Y, do we need to add to a 10% mixture of alcohol, solution X, to make a 25% mixture.

Let x = 10, y = 30, and z = 25

Then the ratio of X to Y is:

$\frac{|z-y|}{|z -x|}$

$\frac{|25 - 30|}{|25-10|}$

$\frac{5}{15}$

$\frac{1}{3}$

So if we have 200 ml of solution X, we want 3 times as much of solution Y, so we need to add 3*200 = 600 ml of solution Y to the mixture.

Then we're done!

***Note that |z-y| is the parts of X, not of Y, and vice-versa.