# Making Maths Count

## Maths Week Scotland 2019 – The Test

October 1, 2019 by John Swinney MSP 3 Comments | Category Education, Games, Maths

**Problem 2: The Test**

a) In 2018, a 26-question test was given to some survey volunteers. Questions that were answered correctly received 8 points, while those answered incorrectly were given a 5-point penalty. If Dipak tackled all the questions and scored zero overall, how many questions did he get correct?

b) For the 2019 test, the same questions were used again but with four new questions added. The scoring system was the same as for 2018. All the volunteers tackled all of the questions but none of them scored zero. Explain why this was bound to happen.

**Solutions to Problem 2:**

Answers a) 10 b) n/a

Tags: John Swinney, Maths Week

I agree with Angus re question A

Not 100% what an integer is so I will show my working re part B.

Proposition 1) Only every fifth multiple of 8 can end with 0 – 0, 40, 80, 120, 160, 200, 240.

Proposition 2) Only every 8th multiple of 5 will equal a multiple of 8

If 8 wrong answers cancel 5 correct ones then equilibrium can only be achieved where the total number of questions is a multiple of 13.

My working for why it can’t be done:

1) to score zero requires a multiple of 8 to equal a multiple of 5 AND for the sum of the multipliers to be 30.

2) A multiple of 8 will never end in 5 so equilibrium can only be achieved where both totals end in 0.

3) With a multiplier of up to 30, the possible positive scores ending in 0 are 0, 40, 80, 120, 160, 200, 240

4) The number of wrong answers must be greater than the number of right ones to achieve equilibrium so the positive score must be under 120 (15 correct) for a zero score to be achieved

5) 80 implies 10 correct answers and twenty wrong = 100 so the sum would be minus 20.

Thus equilibrium is impossible given 30 questions and a rule that you must answer.

The closest you can get to equilibrium is 96 (12 right) minus 90 (18 wrong) giving a score of 6.

A) 10

B) Because to score zero, the number of correctly answered questions is not an integer and that is impossible.