Question

The mean marks (out of 100) of boys and girls in an examination are 70 and 73 respectively. If the mean marks of all the students in the examination is 71, find the ratio of number of boys to the number of girls.
(A) 1:1
(B) 2:1
(C) 1:2
(D) 2:3

Answer 1

Hint: We solve this question by first assuming the number of boys and girls as $x$ and $y$. Then we consider the formula for mean, $Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$ and use it to find the total marks of boys by substituting the mean marks of boys and number of boys. Then we follow the same procedure and find the total marks of girls. Then we add both the values to find the total marks of students. Then we use the formula for mean and substitute the value of mean marks of students and solve it to find the relation between $x$ and $y$, thereby we find the ratio of number of boys to number of girls.

Complete step by step answer:
We are given that mean marks of boys and girls in an examination are 70 and 73 respectively.
We are also given that mean marks of all the students in that exam is 71.
We need to find the ratio of number of boys to number of girls.
Now, let us assume that number of boys are $x$ and number of girls are $y$.
Then the total number of students is $x+y$.
Now let us consider the mean marks of boys, that is 70.
Now, let us consider the formula for mean
$Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$
Using this we can write the mean marks of boys as,
$\Rightarrow Mean\ marks\ of\ boys=\dfrac{Sum\ of\ marks\ of\ boys}{Total\ number\ of\ boys}$
Now let us substitute the number of boys and mean marks of boys in it. Then we get,
$\begin{align}
  & \Rightarrow 70=\dfrac{Sum\ of\ marks\ of\ boys}{x} \\
 & \Rightarrow Sum\ of\ marks\ of\ boys=70x............\left( 1 \right) \\
\end{align}$
Now let us consider the mean marks of girls, that is 73.
Now, let us consider the formula for mean
$Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$
Using this we can write the mean marks of girls as,
$\Rightarrow Mean\ marks\ of\ girls=\dfrac{Sum\ of\ marks\ of\ girls}{Total\ number\ of\ girls}$
Now let us substitute the number of girls and mean marks of girls in it. Then we get,
$\begin{align}
  & \Rightarrow 73=\dfrac{Sum\ of\ marks\ of\ girls}{y} \\
 & \Rightarrow Sum\ of\ marks\ of\ girls=73y............\left( 2 \right) \\
\end{align}$
So, from equations (1) and (2) we get,
$\begin{align}
  & \Rightarrow Total\ Marks=Sum\ of\ marks\ of\ boys+Sum\ of\ marks\ of\ girls \\
 & \Rightarrow Total\ Marks=70x+73y \\
\end{align}$
Now, let us consider the formula for mean
$Mean=\dfrac{Sum\ of\ observations}{Total\ number\ of\ observations}$
Using this we can write the total mean marks of students as,
\[\Rightarrow Mean\ marks\ of\ students=\dfrac{Total\ sum\ of\ marks\ of\ students}{Total\ number\ of\ students}\]
Now let us substitute the number of students and mean marks of students and total marks of students in it. Then we get,
\[\begin{align}
  & \Rightarrow 71=\dfrac{70x+73y}{x+y} \\
 & \Rightarrow 71x+71y=70x+73y \\
 & \Rightarrow 71x-70x=73y-71y \\
 & \Rightarrow x=2y \\
 & \Rightarrow \dfrac{x}{y}=\dfrac{2}{1} \\
\end{align}\]
So, we get the ratio of the number of boys to the number of girls as 2:1.

So, the correct answer is “Option B”.

Note: We can solve this question in an alternate process by using a simpler formula.
When two groups of sizes $m$ and $n$ with means $\overline{x}$ and $\overline{y}$ are mixed, then the mean of the mixed group is,
$\dfrac{m\overline{x}+n\overline{y}}{m+n}$.
After assuming the number of boys and girls as x and y we can use the above formula and equate the mean obtained to the given total mean, 71 and solve the equation as above to find the ratio required.