Q) Tara scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each wrong answer, then Tara would have scored 50 marks. Assuming that Tara attempted all questions, find the total number of questions in the test.

Ans:

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

and Y be the number of questions which are answered incorrectly.

Therefore, total no. of questions in the test = X + Y

Step 2:  Let’s take 1st case: Tara gets 40 marks with 3 marks for correct answer and – 1 for incorrect answer.

∵ Marks for 1 correct answer = 3

∴ Marks for X correct answers = 3 (X) = 3 X

Similarly, marks for 1 incorrect answer = – 1

∴ Marks for Y incorrect answer = – 1 (Y) = – Y

Therefore, Tara’s total marks = Marks for correct answers + Marks for incorrect answers

∴ 40 = 3 X – Y …….. (i)

Step 3:  Let’s take 1st case: Tara gets 50 marks with 4 marks for correct answer and – 2 for incorrect answer.

∵ Marks for 1 correct answer = 4

∴ Marks for X correct answers = 4 (X) = 4 X

Similarly, marks for 1 incorrect answer = – 2

∴ Marks for Y incorrect answer = – 2 (Y) = – 2 Y

Therefore, Tara’s total marks = Marks for correct answers + Marks for incorrect answers

∴ 50 = 4 X – 2 Y …….. (ii)

Step 4: Let’s solve equations (i) and (ii) and find the values of X and Y.

To solve these we multiply equation (i) by 2 and subtract equation (ii) from it, we get:

(2 x 40) – (50) = 2 (3 X – Y) – (4 X – 2 Y)

80 – 50 = 6 X – 2 Y – 4 X + 2 Y

30 = 2 X

X = = 15

By substituting value of X = 15 in equation (i), we get:

40 = 3 X – Y

∴ 40 = 3 (15) – Y

∴ 40 = 45 – Y

∴ Y = 45 – 40 = 5

Step 5: Now we have values of X and Y,

∴ Total number of questions = X + Y

= 15 + 5 = 4=20

Therefore, total no. of questions in the test are 20. 

Check: let’s put x = 15 and y = 5 in, equation (i), we get: 3 (15) – (5) = 45 – 5 = 40 …This also matches with the given condition. hence values of X and Y are correct. 

Similarly, from equation (ii), we get: 4(15) – 2(5) = 60 – 10 = 50… This also matches with the given condition. hence values of X and Y are correct. 

Since both conditions are satisfied, hence our answer is correct.

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