Question

A question paper has descriptive questions carrying 3 marks each and multiple-choice questions carrying 1 mark each. There are 24 questions in all and the total marks of the paper is 50. Find the number of questions that carry 3 marks.

Answer 1

Hint: We first assume the variables for the 3 marks and 1 marks questions. From the given conditions we find two mathematical expressions. We solve them to find the solutions for the variables.

Complete step by step solution:
We assume that there are x number of questions carrying 3 marks each and y numbers of multiple-choice questions carrying 1 mark each.
We have two conditions and we form them in mathematical expressions.
There are 24 questions in all and the total marks of the paper is 50.
Therefore, $x+y=24$ and $3\times x+1\times y=50\Rightarrow 3x+y=50$.
Now we subtract the first equation from second equation and get
$\left( 3x+y \right)-\left( x+y \right)=50-24$.
We take the variables together and the constants on the other side.
Simplifying the equation, we get
$\begin{align}
  & \left( 3x+y \right)-\left( x+y \right)=50-24 \\
 & \Rightarrow 2x=26 \\
 & \Rightarrow x=\dfrac{26}{2}=13 \\
\end{align}$
The value of $x$ is 13. Now putting the value in the equation $x+y=24$, we get
$\begin{align}
  & x+y=24 \\
 & \Rightarrow y=24-13=11 \\
\end{align}$.
Therefore, the number of questions that carry 3 marks is 13.

Note: We can also find the value of one variable $y$ with respect to $x$ based on the equation
$x+y=24$ where $y=24-x$. We replace the value of $y$ in the second equation of
$3x+y=50$ and get
\[\begin{align}
  & 3x+y=50 \\
 & \Rightarrow 3x+\left( 24-x \right)=50 \\
 & \Rightarrow 3x+24-x=50 \\
\end{align}\]
We get the equation of $x$ and solve
\[\begin{align}
  & 3x+24-x=50 \\
 & \Rightarrow 2x=50-24=26 \\
 & \Rightarrow x=\dfrac{26}{2}=13 \\
\end{align}\]
Putting the value of $x$ we get $x+y=24\Rightarrow y=24-13=11$.