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Q.
Mean marks scored by the students of a class is $53$. The mean marks of the girls is $55$ and the mean marks of the boys is $50$. What is the percentage of girls in the class?
Statistics
Solution:
Let $n(G)$ and $n(B)$ be the number of girls and boys in a class respectively and $m(G)$ and $m(B)$ be the total marks scored by girls and boys respectively.
$\therefore \frac{m\left(G\right)+m\left(B\right)}{n\left(G\right)+n\left(B\right)}=53\quad\ldots\left(i\right)$
Also, $m\left(G\right) = 55 \,n\left(G\right)$, $m\left(B\right) = 50 \,n\left(B\right)\quad \ldots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$, we get
$\frac{55 \,n\left(G\right)+55 \,n\left(B\right)}{n\left(G\right)+n\left(B\right)}= 53$
$\Rightarrow n\left(B\right) = \frac{2}{3} n\left(G\right)$
$\therefore $ Percentage of girls
$= \frac{n\left(G\right)}{n\left(B\right)+n\left(G\right)}\times100$
$= \frac{n\left(G\right)}{\left(\frac{2}{3}+1\right)n\left(G\right)}\times100=60\%$