Question

In a class test, the sum of the marks obtained by $P$ in Mathematics and Science is 28. Had he got 3 marks more in Mathematics and 4 marks less in Science, the product of his marks would have been 180. Find his marks in two subjects.

Answer 1

Hint: Take the marks obtained in one of the subjects as a variable. Then find the product of his marks according to the given conditions so that you will obtain an equation. Solve the equation for finding the marks obtained by $P$.

Complete step-by-step answer:
Let the marks obtained in mathematics be $x$.
Since the sum of the marks obtained by $P$ in Mathematics and Science is 28, then the marks obtained in Science is $28 - x$.
$P$ had got 3 marks more in mathematics, the marks would be $x + 3$.
$P$ had got 4 marks less in science, the marks would be $28 - x - 4$.
Given the product of his marks would have been 180. So,

$$\Rightarrow \left( {x + 3} \right)\left( {28 - x - 4} \right) = 180 \\\ \Rightarrow \left( {x + 3} \right)28 - \left( {x + 3} \right)x - \left( {x + 3} \right)4 = 180 \\\ \Rightarrow 28x + 84 - {x^2} - 3x - 4x - 12 = 180 \\\ \Rightarrow 21x + 72 - {x^2} = 180 \\\ \Rightarrow {x^2} - 21x + 180 - 72 = 0 \\\ \Rightarrow {x^2} - 21x + 108 = 0 \\\$$

Solving for $x$, we have

$$\Rightarrow {x^2} - 21x + 108 = 0 \\\ \Rightarrow {x^2} - 12x - 9x + 108 = 0 \\\ \Rightarrow x\left( {x - 12} \right) - 9\left( {x - 12} \right) = 0 \\\ \Rightarrow \left( {x - 12} \right)\left( {x - 9} \right) = 0 \\\ \therefore x = 9,12 \\\$$

So, the marks scored in mathematics can be 9 or 12.
If $P$ had scored 9 marks in mathematics then, marks scored in science is $28 - x = 28 - 9 = 19$
If $P$ had scored 12 marks in mathematics then, marks scored in science is $28 - x = 28 - 12 = 16$
Thus, marks obtained by$P$ in mathematics and science are 9 and 19 respectively.
And marks obtained by $P$ in mathematics and science are 12 and 16 respectively.
Note: In the solution the obtained equation is a quadratic equation. So, we have got two solutions for finding the marks obtained in mathematics (i.e., $x$). Similarly, the marks obtained in science.