Average is one of the most fundamental and frequently tested topics in quantitative aptitude for competitive exams. It is asked in almost every major exam including CAT, SSC CGL, SSC CHSL, Bank PO, Bank Clerk, Railway RRB and CSAT. A strong understanding of average concept, formulas and tricks is essential for scoring well in these exams. In this post we cover everything from the basic definition of average and mean value, average of combined groups using both direct method and alligation method, effect on average when items are added or removed, replacement of items and change in average, and special average age problems involving families — all explained with multiple methods and solved examples.
Average
Average:- The average is sum of all observations divided by the number of observations. This is also known as mean value.
Ex:- Find the average of following numbers
40, 50, 60, 70, 80
Sol:- Average = \(\frac{{40 + 50 + 60 + 70 + 80}}{5} = \frac{{300}}{5}\) = 60 Answer
Method(2):
● Average of given terms always lies in the range of given data.
i.e. Lowest value ≤ Average ≤ Highest value
● Average of different groups
Sometimes, the average of two different groups arae known and the average of a third group (made by combining these groups) is to be found out out.
Ex:- The average weight of 15 girls is 45 kg and that of 20 boys is 60 kg. Find the average weight of the whole class.
Sol:-
Method(1): Average = \(\frac{{15 \times 45 + 20 \times 60}}{{15 + 20}}\) = 53.57 kg Answer
Method(2):
Addition and Removal of items and change in Average:-
Ex:- The average of 50 students in a class is 20 years. When 10 new students are admitted, the average is increased by 0.5 years. Find the average of new students.
Sol:-
Method(1):
Average of newly added students = 20 + \(\left( {1 + \frac{{50}}{{10}}} \right)\×0.5
= 23 years Answer
Using Alligation–Method(2):
Ex:- The average salary of 12 teachers is Rs. 6500 per month. 4 teachers left the school and the average salary of remaining teachers dropped by Rs. 150. Find the total salary of teachers who left the school.
Sol:-
Method(1):
Average salary of the teachers who left = A – \(\left( {1 – \frac{N}{n}} \right).x\)
= 6500 – \(\left( {1 – \frac{{12}}{4}} \right).150\)
= 6800
∴ total salary of 4 teachers who left = 6800 × 4 = 27200 Rs. Answer
Method(2):
Using Alligation:
● Replacement of Some of the items:-
Sometimes, when a number of items are removed from a group and these are replaced with equal number of different items then the average of the group ↑ or ↓ by x.
Ex:- When a man weighing 70 kg is replaced by another man in a group of 5 persons, the average weight decreases by 4 kg. What is the weight of new man ?
Sol:-
Weight of new man – 70 = -5 × 4
∴ wdight of new man = 70 – 20 = 50 Answer
Method(2): The other way of thinking
One person leaves while the other person joins the group
⟹ mean number orf persons in the group does not change.
Now the average is to be reduced by 4 that mean 4-4 kg. wight should be reduced from each of 5 persons i.e. total 20 kg weight should be reduced from the group to get the desired result but we have reduced it by 70 kg. So we shoul make a person weighing 50 kg join the group so that the overall weight loss fromo the group be 20 kg.
Hence 50 kg Answer
Ex:- Average age of 8 men is increased by 3 years when two of them whose age are 30 and 34 years are replaced by 2 persons. What is the average age of the 2 persons ?
Sol:-
Sum of age of 2 new person – (30 + 34) = +8 × 3
x – 64 = 24
x = 88
∴ average = \(\frac{{88}}{2}\) = 44 Answer
Method(2): The other way of thinking
persons leaves & 2 joins i.e. number of person does not change.
So if average is to be ↑ by 3 then 3 should be added to the weight of every person means 3 × 8 = 24 kg weight should be increased of the group but in place of ↑ we are removing (30 + 34) = 64 kg weight. So the new 2 person which comes should manage these weights i.e. weight of 2 new persons = 64 + 24 = 88
∴ average = \(\frac{{88}}{2}\) = 44 Answer
● Before ‘t’ years, the average age of ‘N’ members of a family was ‘T’ years. If the average remains same even after one more member joins the family, then the present age of new member = T – Nt
Ex:- 4 years ago, the average age of 6 members of a family was 26 years. On the birth of a child in the family, the average remains the same. find the present age of the child.
Sol:
t = 4
T = 26
N = 6
∴ Present age of child = T – Nt = 26 – 6 × 4 = 2 years Answer
● when all the numbers are divided or multiplied by an arbitrary number, then the average also get divided or multiplied accordingly.
● when an arbitrary number is added or subtracted to all the numbers, then the average also ↑ or ↓ by the value of arbitrary number respectively.