Question

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

Answer 1

Let the marks obtained by Shefali in English and Mathematics be x and y respectively. For two equations with the data given in the question, solve the equation to obtain the value of x and y.

Complete step-by-step answer:
Let the marks obtained in English $= x$
Let the marks in Maths $= y$
According to the question, the sum of marks in both subjects is $30$.
Therefore,
$x + y = 30 \\\ y = 30 - x \\\$ (1)
If she had got two more marks in maths, her marks would have increased to $y + 2$ and if she had got three marks less in English, her marks would have been reduced to $x - 3$.
According to question, product of the marks would have been $210$, i.e.,
$\left( {x - 3} \right)\left( {y + 2} \right) = 210 \\\ xy + 2x - 3y - 6 = 210 \\\$ (2)
Substituting the value of y from equation (1) in equation (2), we get,

$$x\left( {30 - x} \right) + 2x - 3\left( {30 - x} \right) - 6 = 210 \\\ 30x - {x^2} + 2x - 90 + 3x - 6 = 210 \\\ \- {x^2} + 35x - 96 - 210 = 0 \\\ {x^2} - 35x + 306 = 0 \\\$$

By splitting the middle term,
${x^2} - 17x - 18x + 306 = 0 \\\ x\left( {x - 17} \right) - 18\left( {x - 17} \right) = 0 \\\ \left( {x - 17} \right)\left( {x - 18} \right) = 0 \\\ x = 17,18 \\\$
That is,
When
$x = 17 \\\ y = 30 - x = 30 - 17 = 13 \\\$
And when,
$x = 18 \\\ y = 30 - x = 30 - 18 = 12 \\\$
Hence, if her marks in English is $17$, then in Maths she got $13$. If her mark in English is $18$, then in Maths she got $12$.

Note: This problem could also have been solved using one variable by letting the marks in English be x and in maths $\left( {30 - x} \right)$. Similar to this question, you would have got a quadratic equation in x. Here too, x would have had two roots.