Question

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class are 72, then what is the average of the girls?
(a)73
(b)65
(c)68
(d)74

Answer 1

Hint: To solve this question, firstly we will find the number of girls in the class. After that we will let the average of boys and girls be ${{a}_{1}}$ and ${{a}_{2}}$ respectively. As, an average of students will be equal to total marks secured by class divided by the number of students, and total marks secured by class will be equal to marks secured by boys and marks secured by girls together. Hence, we will reframe the formula for the Average of class and by substituting values, we will get average marks of girls.

Complete step-by-step solution:
Now, in a class, there are a total of 100 students. Also, out of 100 students, it is given that there are a total of 70 boys, so the total number of girls out of 100 students, will be equals to,
Number of girls = Total number of students – Number of Boys
So, Number of Girls = 100 - 70
On simplifying, we get
Number of Girls = 30
Now, it is given that there are 70 boys whose average marks in a subject are 75 and the average marks of the complete class are 72.
So, let the average of marks of boys be equal to ${{a}_{1}}$ and the average of marks of girls be ${{a}_{2}}$.
So, we can say that,
$\text{Average of Class = }\dfrac{70{{a}_{1}}+30{{a}_{2}}}{100}$ .
We already know that ${{a}_{1}}=75$ and Average of class = 72.
So, on substituting values, we get
$\text{72 = }\dfrac{70(75)+30{{a}_{2}}}{100}$
On simplifying, we get
$\text{7200 = 5250}+30{{a}_{2}}$
Again, on simplification, we get
$\text{7200 - 5250 = }30{{a}_{2}}$
$30{{a}_{2}}=1950$
On solving, we get
${{a}_{2}}=65$
So, the average of the girls is 65. Hence, option (b) is correct.

Note: Always remember that, if we have collection of data of n – terms, ${{x}_{1}},{{x}_{2}},{{x}_{3}},......,{{x}_{n}}$ , then average of these n items is denoted by $\bar{x}$ and is given by $\bar{x}=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+......+{{x}_{n}}}{n}$. Also, on average when multiplied by the number of students, we get the total marks of the students. Try not to make any calculation mistakes while solving the question.