How many questions could he have omitted?

Question

A quiz has 25 questions with four points awarded for each correct answer and one point deducted for each incorrect answer, with zero for each question omitted. John scored 77 point. How many questions could he have omitted?

Answer 1

Let R be the number of right answera and W be the number of wrong answers submitted. Then he scored 4R-W points.

Therefore we need to first solve the Linear Diophantine equation:

 

4R - W = 77

 

under the conditions R ≥ 0, W ≥ 0 and R + W ≤ 25

 

Solution to equation:

 

Clearly R ≤ 25. Also, 4R ≥ 77 so R ≥ ⌈77/4⌉ = 20

 

We can now list the solution pairs (R,W) by trying R = 20,21,...,25 and checking the constraints.

 

The only pair satisfying all constraints is (20,3).

 

So he must have gotten 20 questions right, 3 questions wrong, and has omitted 2 questions.