Averages and Mixtures and Alligations

Averages and Mixtures and Alligations

Averages and alligation are interconnected topics that appear frequently in placement aptitude tests. The alligation rule is particularly powerful — it solves mixture problems in seconds using a cross method that many candidates are unaware of, giving you a decisive speed advantage in every placement test you take.

Averages

Average equals the sum of all observations divided by the number of observations. When a new item is added, the new average equals old total plus new value, divided by the new count. When one item is replaced, adjust the total by the difference between new and old values, then divide by the same count. Weighted average equals the sum of weight times value for each item, divided by the sum of all weights.

The Alligation Cross Method

Draw a cross with the cheaper value on the left, the expensive value on the right, and the desired mean in the centre. Subtract diagonally: the left diagonal gives expensive minus mean, and the right diagonal gives mean minus cheaper. These two results form the ratio of cheaper to expensive in the required mixture. This visual method works for any mixing problem involving price, concentration, or any averaged quantity.

Solved Example

In what ratio should tea at Rs. 80 per kg be mixed with tea at Rs. 120 per kg to get a mixture worth Rs. 96 per kg? 120 minus 96 equals 24, and 96 minus 80 equals 16. Ratio of cheaper to expensive equals 24 to 16, which simplifies to 3 to 2.

Interview Tips

The alligation cross method reduces 3-step mixture problems to a 10-second calculation once mastered. For repeated dilution problems, use: remaining pure substance equals original amount times 1 minus replacement divided by total, raised to the power of repetitions. Memorizing this formula eliminates the need to work through each step individually for every dilution cycle.