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.