Question

There are $ 180 $ multiple-choice questions in a test. If a candidate gets $ 4 $ mark for each correct answer and for each unattempted or wrong answer one mark is deducted from the total score of the correct answer. If a candidate scores $ 450 $ in the test how many questions did he answer correctly?

Answer 1

Hint: The exam rule for giving marks is based upon two things working simultaneously that is awarding $ 4 $ marks for each correct answer and deducting $ 1 $ mark for the unanswered or wrongly answered question. The total number of questions that can be answered is given to us ( $ 180 $ ) . Also the mark a candidate obtained is given to us ( $ 450 $ ). We are now unknown about the numbers of answers correctly given and the questions that are not attempted or left unanswered by the candidate. So, we will use variables and express the given conditions in terms of mathematical equations. We will solve the constructed equations to get appropriate values of the variables.

Complete step-by-step answer:
We are given that there are $ 180 $ multiple choice questions in a test and a candidate scores $ 450 $ in the test, where the rule of awarding and reducing marks is $ 4 $ marks for each correct answer and $ 1 $ mark for unanswered or wrongly answered questions respectively.
Let us assume that the candidate appeared $ x $ questions correctly and $ y $ questions were unanswered or answered wrongly by the candidate to secure $ 450 $ marks in the test.
So, this forms a mathematical equation for us as:
 $ 4x - 1y = 450 $ --(1)
Again we have a total of $ 180 $ multiple choice questions in a test so we get another equation:
 $ x + y = 180 $ --(2)
From the above equation we have
 $ x = 180 - y $
Putting this value in (1) we get:
 $ 4(180 - y) - 1y = 450 $
 $ \Rightarrow 4 \times 180 - 4y - 1y = 450 $
 $ \Rightarrow 720 - 5y = 450 $
 $ \Rightarrow 720 - 450 = 5y $
 $ \Rightarrow 270 = 5y $
 $ \Rightarrow y = 54 $ --(3)
Putting the above value in (2) we get:
 $ x = 180 - 54 $
 $ \Rightarrow x = 126 $ --(4)
From (3) and (4) we have the values of $ x $ and $ y $ as $ 126 $ and $ 54 $ respectively.
Which means the candidate has answered $ 126 $ questions correctly and $ 54 $ questions were unanswered or answered wrongly
Therefore, the candidate has answered $ 126 $ questions correctly

Note: Here the most important thing is to choose the coefficient of the variable for the equations. Since, variable $ y $ deals with the negative marking in the first case (for generating (1)) so its coefficient is a negative term, don’t assume $ y $ value to be negative.