Algebra

Linear Equation

Medium

In an examination, there were $75$ questions. $3$ marks were awarded for each correct answer, $1$ mark was deducted for each wrong answer and $1$ mark was awarded for each unattempted question. Rayan scored a total of $97$ marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is

Entered answer:

✅ Correct Answer: 24

Rayan took an exam with 75 questions where:

Correct answer = +3 marks

Wrong answer = -1 mark

Unattempted = +1 mark

Total score = 97 marks

Key constraint: Unattempted questions > Attempted questions

We need to find the maximum number of correct answers possible.

Let's define:

x = number of correct answers

y = number of wrong answers

z = number of unattempted questions

From the given information, we can write:

Total questions:

$x + y + z = 75$

Total marks:

$3x - y + z = 97$

Constraint condition:

$z > x + y$

From equation 1: $z = 75 - x - y$

Substituting into equation 2:

$3x - y + (75 - x - y) = 97$

$3x - y + 75 - x - y = 97$

$2x - 2y + 75 = 97$

$2x - 2y = 22$

$x - y = 11$

This tells us that $y = x - 11$

From equation 1, we can express z in terms of x:

$z = 75 - x - y$

$z = 75 - x - (x - 11)$

$z = 86 - 2x$

Using the constraint $z > x + y$ :

$86 - 2x > x + (x - 11)$

$86 - 2x > 2x - 11$

$86 + 11 > 4x$

$97 > 4x$

$x < 24.25$

Since x must be a whole number, $x \leq 24$ .

We also need $y \geq 0$ , which gives us $x \geq 11$ . Since we found $x \leq 24$ , and $24 > 11$ , this condition can be satisfied.

The maximum number of correct answers Rayan could have given is 24.

We used the constraint that unattempted questions exceed attempted questions to find the upper limit on correct answers. This constraint is what makes the problem interesting - without it, Rayan could potentially answer more questions correctly.

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