Q) A test consists of ‘True’ or ‘False’ questions. One mark is awarded for every correct answer while 1/4 mark is deducted for every wrong answer. A student knew answers to some of the questions. Rest of the questions he attempted by guessing. He answered 120 questions and got 90 marks.

1. If answer to all questions he attempted by guessing were wrong, then how many questions did he answer correctly?
2. How many questions did he guess?
3. If answer to all questions he attempted by guessing were wrong and answered 80 correctly, then how many marks he got?
4. If answer to all questions he attempted by guessing were wrong, then how many questions answered correctly to score 95 marks?

Ans:

1. Number of correctly answered questions:

Step 1: Let X be the number of questions which are answered correctly

and Y be the number of questions which are answered by guessing.

Therefore, total no. of questions in the test = Question answers correctly + Questions answered by guessing

∴ 120 = X + Y ……….. (i)

Step 2:  The student gets 90 marks with 1 marks for correct answer and for incorrect answer.

∵ Marks for 1 correct answer = 1

∴ Marks for X correct answers = 1 (X) = X

Similarly, marks for 1 incorrect answer = –

∴ Marks for Y incorrect answer = – (Y)

Therefore, total marks = Marks for correct answers + Marks for incorrect answers

∴ 90 = X – Y

∴ 360 = 4 X – Y …….. (ii)

Step 3:  Let’s solve equations (i) and (ii) and find the values of X and Y.

To solve these, we add (i) with equation (ii), we get:

120 + 360 = (X + Y) + (4 X – Y)

∴ 480 = X + Y + 4 X – Y

∴  480 = 5 X

∴ X =

∴ X = 96

Therefore, the student answered 96 questions correctly.

2. Number of guessed questions:

By substituting value of X = 96 in equation (i), we get:

120 = X + Y

∴ 120 = 96 + Y

∴ Y = 120 – 96 = 24

Therefore, the student answered 24 questions by guessing.

3. Marks with 80 correct answers:

∵ Correctly answers questions = 80

∴ No. of guessed answers or incorrect answers  = Total no. of questions – Correctly answered questions

= 120 – 80 = 40

∵ Marks for 1 correct answer = 1

∴ Marks for 80 correct answers = 1 (80) = 80

Similarly, marks for 1 incorrect answer = –

∴ Marks for 40 incorrect answer = – (40) = – 10

Therefore, total marks = Marks for correct answers + Marks for incorrect answers

= 80 – 10 = 70

Therefore, student will get 70 marks.

4. Number of correctly answered questions when marks scored 95:

Let’s consider, correctly answers questions by student = X

∴ No. of guessed answers or incorrect answers  = Total no. of questions – Correctly answered questions

= 120 – X

∵ Marks for 1 correct answer = 1

∴ Marks for X correct answers = 1 (X) = X

Similarly, marks for 1 incorrect answer = –

∴ Marks for (120 – X) incorrect answer = – (120 – X) =

Since, total marks = Marks for correct answers + Marks for incorrect answers

95 = X –

∴ 95 =

∴ 380 = 480 – X

∴ X = 480 -380 = 100

Therefore, the student answered 100 questions correctly.

Check: let’s calculated total marks with X = 100. Since X = 100, Y = 120 – 100 = 20.

Marks from correctly answered questions = 100 x 1 = 100; Marks from incorrect answers = 20 x = – 5

Total marks = 100 – 5 = 95. This matches with the given condition. hence our answer of X = 100 is correct. 

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