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Question: In a class test, the sum of the marks obtained by \[P\] in Mathematics and Science is 28. Had he got...

In a class test, the sum of the marks obtained by PP 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.

Explanation

Solution

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 PP.

Complete step-by-step answer:
Let the marks obtained in mathematics be xx.
Since the sum of the marks obtained by PP in Mathematics and Science is 28, then the marks obtained in Science is 28x28 - x.
PP had got 3 marks more in mathematics, the marks would be x+3x + 3.
PP had got 4 marks less in science, the marks would be 28x428 - x - 4.
Given the product of his marks would have been 180. So,

(x+3)(28x4)=180 (x+3)28(x+3)x(x+3)4=180 28x+84x23x4x12=180 21x+72x2=180 x221x+18072=0 x221x+108=0 \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 xx, we have

x221x+108=0 x212x9x+108=0 x(x12)9(x12)=0 (x12)(x9)=0 x=9,12 \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 PP had scored 9 marks in mathematics then, marks scored in science is 28x=289=1928 - x = 28 - 9 = 19
If PP had scored 12 marks in mathematics then, marks scored in science is 28x=2812=1628 - x = 28 - 12 = 16
Thus, marks obtained byPP in mathematics and science are 9 and 19 respectively.
And marks obtained by PP 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., xx). Similarly, the marks obtained in science.