# Mentally Solve Problem of Averages in Seconds

In all competitive examinations and otherwise also in our day to day calculations we encounter issues related to averages. Especially when the composition of the group changes it becomes difficult to do computations. Let us discuss quick and easy solutions to such problems.

### When a person leaves a group and another person replaces him, then there can be 2 scenarios:

**Scenario I:** When the average age increases then,

**Age of the new comer** = age of person who left + (no. of persons in the group * increase in average age)

**Scenario II:** When the average decreases then,

**Age of the new comer** = age of the person who left – (no. of persons in the group * decrease in average age)

Let us try to understand these with a simple example –

**Question:** The average age of 45 persons is decreased by 1/9 year when one of them whose age is 60 years is replaced by a new comer. What is the age of the new comer.

**Solution: **

**Age of the new comer** = age of the person who left – (no. of persons in the group * decrease in average age)

Age of the new comer = 60 – (45 * 1/9) = **55 years**

Isn’t that easy to calculate using the formula I just gave you. However, to intuitively answer such questions you need to ** keep following things in mind** –

Firstly, the average age is reducing, which means an older man is replaced by relatively younger person. Hence the answer has to be lesser than 60. Hence we subtract from 60 and not add to it.

Secondly, average reduction of 1/9 kg for a group of 45 people means, in total there is a wait reduction of 1/9 per person x 45 persons = 5 kg for the entire group. This is result of the replacement of an old person by a relatively younger person in the group. That’s the reason we’re subtracting 5 from 60, to get the age of the new comer.

### B. When a person joins a group without any replacement, then there can be 2 scenarios:

**Scenario I:** When the average age increases then,

**Age of the incoming person** = previous average age of the group + no. of persons including the person who joined * increase in the average age value

**Scenario II:** When the average age decreases then,

**Age of the incoming person** = previous average age of the group – no. of present persons including the person who joined * decrease in the average age value

I will explain this further with an example,

**Question: **The average age of 20 teachers is 45 years which is decreased by 6/7 years when a student joins the group. Then what is the age of the student?

**Solution: **

**Age of the outgoing person** = previous average age of the group – new no. of persons including the person who joined * decrease in the average age value

Therefore, age of the student = 45 – 21 * 6/7 = 27 years

Here also you need to observe that since the average age is going down when the new person is joining, the age of new comer will be lesser than the average, hence subtraction.

Moreover, the decrease is equal to 6/7 per person for each person in the new group. That is 21 times 6/7 = 18 kgs, which needs to be subtracted from the overall average to get the age of the new comer.

### C. When a person leaves a group without any replacement, then there can be 2 scenarios:

**Scenario I:** When the average age increases then,

**Age of the outgoing person** = previous average age of the group – no. of persons excluding the person who left * increase in the average age value

**Scenario II:** When the average age decreases then,

**Age of the outgoing person** = previous average age of the group + no. of present persons excluding the person who left * decrease in the average age value

Hope we’ve covered all scenarios and with the help of the above logical formulas we will be able to do our calculations.

### Practice questions for you to solve (try to solve mentally)

Question 1: In a boat there are 8 men whose average weight is increased by 1 kg when a man of 60 kg is replaced by a new man. What is the weight of the new comer?

Question 2: In a class there are 30 boys whose average weight is decreased by 200 grams when one boy whose weight was 25 kgs leaves the class and a new comer is admitted. Find the weight of the new comer?

Question 3: A cricketer has a certain average for 9 innings. In the 10^{th} innings, he scores 100 runs, thereby increasing his average by 8 runs. His new average is?