Question

In a class test, the sum of Arun’s marks in hindi and English is 30. Had he got 2 marks more in Hindi and 3 marks less in English , the product of the marks would have been 210. Find his marks in the two subjects.

Answer 1

Hint: Assume the marks scored by Arun in Hindi and English. Then make the linear equations of marks according to the given conditions. At last solve the linear equations to get the assumed marks.

Complete step-by-step answer:
Lets say Arun scored marks in Hindi = x
And he scored marks in English = y
 He scored total marks in hindi and English = 30
\[x + y{\text{ }} = {\text{ }}30\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\] .....(1)
If arun scored two marks more in hindi than his score would be \[ = {\text{ }}\left( {x + 2} \right)\]
If he scored 3 marks less in English than his score would be \[ = {\text{ }}\left( {y - 3} \right)\]
Product of the marks would be 210
\[\left( {x + 2} \right)\left( {y - 3} \right){\text{ }} = {\text{ }}210\;\;\;\;\;\;\;\;\] .........(2)
Solving equation 1 and 2
We will find the value of y from equation 1 and put that value in equation 2
\[\;x + y{\text{ }} = {\text{ }}30\]
\[{\text{y }} = {\text{ }}30 - x\]
Put value of y in equation 2
\[\Rightarrow \left( {x + 2} \right)\left( {30 - x - 3} \right){\text{ }} = {\text{ }}210\]
\[\Rightarrow \left( {x + 2} \right)\left( {27 - x} \right)\; = {\text{ }}210\]
$\Rightarrow 27x - {x^2} + 54 - 2x = 210 $
$\Rightarrow 27x - {x^2} + 54 - 2x - 210 = 0 $
$\Rightarrow 25x - {x^2} - 156 = 0 $
$\Rightarrow {x^2} - 25x + 156 = 0 $
$\Rightarrow {x^2} - 12x - 13x + 156 = 0 $
$\Rightarrow x(x - 12) - 13(x - 12) = 0 $
$\Rightarrow (x - 12)(x - 13) = 0 $
\[x - 12{\text{ }} = {\text{ }}0\;\;\;\;\;x - 13 = {\text{ }}0\]
x = 12 and x =13
Since we took 2 variables so definitely we will get two values so Arun scored either 12 or 13 in hindi and similarly if he scored 12 in hindi than his score in English 18 and if he scored 13 in hindi then his score in English is 17.

Note: Revise the concept to represent the given conditions in linear equations. Also revise the methods to solve a system of linear equations with two variables.If the quadratic equation so formed cannot be solved by splitting middle term once can use sridharas method.