# Alligation

The technique of alligation is applicable in all the cases where two extreme values are given and one average value is given. It is a very useful technique which can be applied in chapters like Percentage, Simple interest, Ratio & proportion, Average etc.

This technique enables us to calculate the ratio in which extreme values/ prices/ interests/ ratios and averages should be mixed so that a given average value/price/interest/ratio and average can be obtained.

Allegation is the rule that enables us to find the proportion in which the two or more ingredients at the given price must be mixed to produce a mixture at a given price.

We represent it as under (Cheaper quantity): (Dearer quantity)=(d-m) : (m-c).

where d is the dearer price, m is the mean price and c is the cheaper price.

Note that:
1. Mean price is always less then dearer price and is always more than cheaper price.
2. The price of the first kind should always be on the left hand side.

Illustration 1:

In what proportion must a grocer mix tea at Rs 1.25 per kg with tea at Rs 1.50 per kg, so that the mixture is worth Rs 1.30 a kg?

Solution

Ignore the minus sign
Write the prices right on top as shown. The mean price is written in the centre; in this case it is Rs 1.30. Follow the direction of the arrows and subtract the mean price from the two CPs. Write the subtracted figure at the bottom. The bottom figure represents the ratio of the quantities that the two must be mixed. In this case the ratio of dearer to cheap is 20 : 5. They must be mixed in the ratio 4:1

Illustration 2:

In what proportion should tea worth Rs. 20 be mixed with tea worth Rs. 14, so that the average price per kg of tea is Rs. 16 per kg?

Solution

The two extreme prices are Rs. 20 and Rs. 14 and average price is Rs. 16. Put the extreme values on two ends and average in the centre. Then subtract in the direction of the arrows and arrive at the resultant values (given at the bottom of the cross):

Ratio ⇒ 2 : 4 = 1 : 2
The difference between the average and the extreme from both the sides is calculated and is written down the line.
but take care of the following two points.

First, the differences are written in the direction of the arrow. Second, the ratio arrived at is what is written under what e.g. in the above Illustration, answer is 1 : 2 and not 2 : 1.

It is clear from this diagram that answer of the ratios is 1 : 2.

Keep in mind the simple point that the order of the ratio follows the order of what is written at the top.

Example

How many kg of sugar worth Rs 24 should be mixed with 60 kg of sugar worth Rs 30 so as to have an average price of sugar worth Rs. 26?

Solution
From the above, ratio is 2 : 1, which means for 1 kg of 2nd sugar we require 2 kg of 1st sugar So for 60 kg of 2nd sugar we require 60 × 2/1 kg of 1st sugar i.e. 120 kg is the answer.

Illustration 4:

In an organisation the average salary of employees is Rs 800. The average salary of 10 officers is Rs 3000 and the average salary of clerks is Rs 600. What is the number of clerks in the organisation?

Solution

From the above, the ratio is 11 : 1 means for 1 officer there are 11 clerks in the organization.
So for 10 officers there would be 11 × 10 /1 = 110 clerks in the organisation.

Illustration 5:

In a school a sum of Rs 600 was divided among 300 students. Every boy was given Rs 2.25 and every girl was given Rs 1.25. What is the number of boys in the school?

Solution

Every Boy was given Rs 2.25 and every girl was given Rs 1.25 and the average amount per student was 600/300 = Rs. 2

Number of boys out of total 300 students = 300 × 3⁄4 = 225 Number of girls out of total 300 students = 300 × 1⁄4 = 75

In sums where there is a mixture and a certain quantity is removed and water mixed in it every time:

Quantity of water left in the vessel = m (1 – n/m)t

Where m = total quantity, n = quantity drawn every time and t = no of times This formula is given above also in th section on dealing with liquids.

Illustration 5:

In a vessel, there are 80 litres of water. 8 litres of water are withdrawn and replaced with milk, then 8 litres of mixture are withdrawn and replaced with milk. What is the quantity of water left in the mixture?

Solution

Quantity of water left in the vessel = 80(1 – 8/80)2.= 80 × 9/10 × 9/10 = 64.8 litres
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