Question

In a quarterly examination, a student secured 30% of the total marks and failed by 12 marks. In the same test another student secured 40% of total marks and got 28 more marks than required to pass the examination. Find the percentage a student has to secure in order to pass that examination?
(a) 24%
(b) 28%
(c) 25%
(d) 33%

Answer 1

Hint: Let the total marks be M. Let the passing percentage be P %. Then write the equations with the help of given information.
(i) 30 % of M is 12 less than P % of M.
$\begin{align}
  & 30\%\,\text{of}\,M+12=P\%\,\text{of}\,M \\
 & \Rightarrow \dfrac{30}{100}M+12=\dfrac{P}{100}M \\
\end{align}$
(ii) 40 % of M is 28 more than P % of M.
$\begin{align}
  & 40\%\,\text{of}\,M-28=P\%\,\text{of}\,M \\
 & \Rightarrow \dfrac{40}{100}M-28=\dfrac{P}{100}M \\
\end{align}$
Solve the above two equations for P and M and then answer P.

Complete step-by-step answer:
Let M be the total marks of the examination and P be the percentage of total marks needed to be obtained by a student in order to pass the examination.
The question says that one student secured 30 % of total marks which means
Marks scored by 1st student is $30\%\,\text{of}\,M=\dfrac{30}{100}M$
Marks need to be scored to pass the examination is $p\%\,\text{of}\,M=\dfrac{p}{100}M$
First student fails by 12 marks. That means he scored 12 marks less than passing marks. So,
$\dfrac{30}{100}M=\dfrac{P}{100}M-12\,\,\,\,\,\,\,\,\,\,\,\,\cdot \cdot \cdot \text{(i)}$
Similarly 2nd student secured 40 % of total marks which means
Marks scored by 2nd student is $40\%\,\text{of}\,M=\dfrac{40}{100}M$
2nd student got 28 marks more than the passing marks. So,
$\dfrac{40}{100}M=\dfrac{P}{100}M+28\,\,\,\,\,\,\,\,\cdot \cdot \cdot \text{(ii)}$
Now solving the equation (i) and equation (ii) for P and M,
Subtracting equation (i) by equation (ii) we get
$\begin{align}
  & \dfrac{40}{100}M-\dfrac{30}{100}M=\dfrac{P}{100}M-\dfrac{P}{100}M+28-(-12) \\
 & \Rightarrow \dfrac{10}{100}M=28+12=40 \\
 & \Rightarrow M=\dfrac{100}{10}\cdot 40 \\
 & \Rightarrow M=400 \\
\end{align}$
Putting this value of M in equation (i), we get
$\begin{align}
  & \dfrac{30}{100}400=\dfrac{P}{100}400-12 \\
 & \Rightarrow 120=4P-12 \\
 & \Rightarrow 4P=120+12=132 \\
 & \Rightarrow P=\dfrac{132}{4}=33 \\
\end{align}$
Hence the passing percentage of the examination is 33 %.
So, option (d) is correct.

Note:One smartest way to answer this question is that according to question 30 % is less than passing percentage and 40 % is more than passing percentage. So the passing percentage must lie between 30 % and 40 %. There is only one option between 30 % and 40 % that is 33 %. Hence option (d) is correct.