Question

An examination consists of 160 questions. One mark is given for every correct option. If one-fourth mark is deducted for every wrong option and half a mark is deducted for every question that is left, then one person scores 79. And if half a mark is deducted for every wrong option and one-fourth marks are deducted for every question left, then the person scores 76. Find the number of questions he attempted correctly.
A. 80
B. 100
C. 120
D. 140

Answer 1

Hint: This is a question of linear equations in two variables. We will assume two variables and form the equations according to what is given in the question. The number of equations to be formed should be equal to the number of variables declared, to get a unique solution.

Complete step-by-step solution -
We have been given that there are a total of 160 questions. Let the number of correct answers be $x$, and the number of wrong answers is $y$. Hence, the rest of the questions are those which have been left and are equal to $(160 - x - y)$.

Now, we have been given that a student scores 79 when the marks obtained for correct, wrong and unattempted answers are one, one-fourth and half respectively. So, we can write that-
Marks by correct answer - marks by wrong answer - marks by unattempted answer = 79
$\begin{align}
  &1 \times {\text{x}} - \dfrac{1}{4} \times {\text{y}} - \dfrac{1}{2}\left( {160 - {\text{x}} - {\text{y}}} \right) = 79 \\
  &{\text{x}} - \dfrac{{\text{y}}}{4} - 80 + \dfrac{{\text{x}}}{2} + \dfrac{{\text{y}}}{2} = 79 \\
  &\dfrac{{3{\text{x}}}}{2} + \dfrac{{\text{y}}}{4} = 159 \\
  &\text{Multiplying both sides by 4} \\
  &6{\text{x}} + {\text{y}} = 636………...\left( 1 \right) \\
\end{align} $
Also, we have been given that the same student scores 76 when the marks obtained for correct, wrong and unattempted answers are one, half and one-fourth respectively. So, we can write that-
Marks by correct answer - marks by wrong answer - marks by unattempted answer = 76
$\begin{align}
  &1 \times {\text{x}} - \dfrac{1}{2} \times {\text{y}} - \dfrac{1}{4} \times \left( {160 - {\text{x}} - {\text{y}}} \right) = 76 \\
  &{\text{x}} - \dfrac{{\text{y}}}{2} - 40 + \dfrac{{\text{x}}}{4} + \dfrac{{\text{y}}}{4} = 76 \\
  &\dfrac{{5{\text{x}}}}{4} - \dfrac{{\text{y}}}{4} = 116 \\
  &\text{Multiplying both sides by 4} \\
  &5{\text{x}} - {\text{y}} = 464………...\left( 2 \right) \\
\end{align} $
Now, we will solve equations (1) and (2) as-
$6x + y = 636$
$5x - y = 464$
Adding equations (1) and (2) we get-
$6x + 5x + y - y = 636 + 464$
$11x = 1100$
$x = 100$
We assumed that x is the number of correct answers, so this is the required answer. The correct option is B.

Note: There is an alternative lengthy method to solve this problem. Instead of two variables, we can assume three variables $x$, $y$ and $z$ for each category of questions. Two out of the three equations will be the same, and a third equation will be added. The equations are-
$\begin{align}
  &{\text{x}} + {\text{y}} + {\text{z}} = 160 \\
  &{\text{x}} - \dfrac{{\text{y}}}{4} - \dfrac{{\text{z}}}{2} = 79 \\
  &{\text{x}} - \dfrac{{\text{y}}}{2} - \dfrac{{\text{z}}}{4} = 76 \\
\end{align} $
When we look closely, we can see that these equations are the same. In the first equation we can write that $z = 160 - x - y$, this is the number of left questions which we assumed. When we substitute the value of $z$ in the other two, we get the same equations and hence get the same answers.