Question

In a class test, the sum of Moulika’s marks in Mathematics and English is 30. If she got 2 marks more in Mathematics and 3 marks less in English, the product of her marks would have been 210. Find her marks in the two subjects.

Answer 1

Hint: After reading the question, it is clear that we will get two variables since the question talks about marks in two different subjects and their sum and product. Our approach would be to write one of the variables in the form of another, substitute it in the question and get a quadratic equation, and then solve that equation with the best suitable method.

Complete step by step answer:
Let us assume that the marks scored by Moulika in Mathematics is x and the marks scored by Moulika in English is y.
The question states that the sum of Moulika’s marks in Mathematics and English is 30.
Then, in terms of variable, the statement can be written as
$\begin{align}
  & x+y=30 \\
 & \Rightarrow x=30-y\text{ }\ldots \left( i \right) \\
\end{align}$
Further the question states that, if she got 2 marks more in Mathematics and 3 marks less in English, the product of her marks would have been 210.
If she got 2 marks more in Mathematics, then her score in mathematics can be written as x+2.
If she got and 3 marks less in English, then her score in English can be written as y-3.
In this case, the product of her marks is given by 210. It can be written as
$\left( x+2 \right)\left( y-3 \right)=210$
Putting the value of x from equation (i), we get
\[\begin{align}
  & \left( 30-y+2 \right)\left( y-3 \right)=210 \\
 & \Rightarrow \left( 32-y \right)\left( y-3 \right)=210 \\
 & \Rightarrow 32y-96-{{y}^{2}}+3y=210 \\
 & \Rightarrow -{{y}^{2}}+35y-306=0 \\
 & \Rightarrow {{y}^{2}}-35y+306=0 \\
\end{align}\]
Splitting the middle term, we get
\[\begin{align}
  & {{y}^{2}}-35y+306=0 \\
 & \Rightarrow {{y}^{2}}-17y-18y+306=0 \\
 & \Rightarrow y\left( y-17 \right)-18\left( y-17 \right)=0 \\
 & \Rightarrow \left( y-18 \right)\left( y-17 \right)=0 \\
 & \Rightarrow y=17,18 \\
\end{align}\]

Thus, the mark scored by her in English is either 17 or 18.
Then from equation (i), the mark scored by her in Mathematics is either 30-17 or 30-18, that is, 13 or 12.

Hence, marks scored by Moulika is either 13 in Mathematics, 17 in English or 12 in Mathematics, 18 in English.

Note: The best approach to solve this question is to break down it into simpler pieces and then define and substitute variables accordingly. Once, we get a quadratic equation or two linear equations, we can solve it accordingly.