Posted in Mathematics, Study Material

# Averages: A Complete Guide

Averages is an extremely important section for CLAT. Most of you have already completed this topic as it is one of the easiest topics in Maths. However, there still might be some people who have not done it. I expect this article to be useful for both the groups.

Almost every one of us is quite familiar with this primary class formula for determining Average, i.e. Sum of quantities / Number of quantities. But to use this formula you need a pen and a paper. Jotting down the numbers given in the question and then trying to solve them will unnecessarily take up your time. Therefore, in this article, I’ll try to explain how to solve most of the questions related to averages asked in CLAT (there still will be some questions which will require a pen and a paper) without using pen and paper or the above-given formula and save your ‘precious’ time. So, let us start:

While solving questions on averages, keep one thing in mind, i.e. assume the average of quantities as quantities in possession of equal points. For example, if it is given that average of 10 quantities is 15, assume that there are 10 people/things each having 15 points in its possession. Now, let us move forward by solving some examples –

Q.1. A batsman scores of 87 runs in the 17th match and thus increases his average by 3. Find his average after 17th match.

Sol. As mentioned earlier, assume 17 matches as seventeen people with equal runs, i.e. average. Now, by scoring 87 runs in 17th match, he increased his average by 3. So, 3 runs flow from the 17th person to all other 16 persons, i.e. a total of 16*3=48 runs from 17th person to other persons. Now, 17th person has 87-48=39 runs which is the average.

Q.2. Average weight of 10 people increased by 1.5 kg when one person of 45 kg is replaced by a new man. What is the weight of this new man?

Sol. One thing is clear from the question that the weight of the new person is more than 45 kg as his entry increases the average weight. Also, if the average, i.e. equal points of every person, increases it must flow from the new person. Now,the total extra weight that this man has brought with him is 1.5*10=15 kg. So,the weight of this new person is 15+45 = 60 kg.

Q.3. Average of five numbers is 27. If one number is excluded the average becomes 25. Find the excluded number.

Sol. Since exclusion of one number leads to a reduction in average, the number must be greater than the average, i.e. 27. Extra quantity this number takes away with it is 2 from each of the remaining four numbers as the average after exclusion is 25. So, total extra quantity taken away is 2*4=8. Hence, the number which was excluded is 8+27 = 35.

Q.4. Average of 10 matches is 32. How many runs should one score to increase his average by 4 runs?

Sol. To increase the average by 4, i.e. to make it 36, one should score (36 + 4*10) = 76 runs. Here, by scoring 76 runs in 11th match, one can give away 4 each to each of 10 other matches thereby increasing the average of 10 other matches to 36 and keeping 76-40= 36 for the 11th match.

Q.5. The average age of the mother and her six children is 12 years which is reduced by 5 years if the age of the mother is excluded. How old is the mother?

Sol. Before mother’s exclusion, the average age is 12 years. Mother’s exclusion takes away 5*6=30 years from the children. So, the age of the mother is 12+30= 42 years.

CLAT questions are generally similar to the above-given examples which can be quite easily solved without using any formula or pen-paper. Other types of questions, where you will be given numbers and asked to compute the average, it can be done by using the formula –

Sum of Quantities / Number of Quantities

For natural numbers, the average of n natural numbers is given by: (n+1)/2. (As sum of n natural numbers is n (n+1)/2). 