Multiplying both sides by (x+y):

Expanding the left-hand side:

Rearranging the terms:

mx-ax=by-my

Taking common factors:

x(m-a)=y(b-m) 

Therefore,

This is the fundamental formula used in the Alligation Method.

4. Shortcut Trick: Cross-Alligation Method

The cross-alligation method helps us find the ratio in which two varieties should be mixed to obtain a desired average price or concentration.

Remember: Subtract the mean from the higher value to get the quantity of the cheaper variety, and subtract the lower value from the mean to get the quantity of the costlier variety.

5. Important Exam Tips

• The average price of a mixture must lie between the prices of the two varieties when both quantities are positive.

• If equal quantities are mixed, the average price is the arithmetic mean of the two prices.

• If the desired mean is closer to the cheaper variety’s price, a larger quantity of the cheaper variety is required.

• If the desired mean is closer to the costlier variety’s price, a larger quantity of the costlier variety is required.

• The alligation method can also be applied to milk and water, profit percentages, marks, and solutions of different concentrations.

Exam practice: Always identify the lower value, higher value and desired mean before applying cross-alligation. This reduces calculation time in Banking, SSC and Railway examinations.