GK 2018 Q24
24. Viola has been practising the long jump. At one point, the average distance she had jumped was 3.80 m. Her next jump was 3.99 m and that increased her average to 3.81 m. After the following jump, her average had become 3.82 m. How long was her final jump?
- A 3.97 m
- B 4.00 m
- C 4.01 m
- D 4.03 m
- E 4.04 m
Grey Kangaroo 2018, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 4.01 m
Let the number of jumps Viola has already made be \(n\) and let the total distance she has jumped in these \(n\) jumps be \(T\) m. Since the total distance jumped is equal to the average distance jumped multiplied by the number of jumps, the information in the question tells us that \(T = 3.8n\) and that \(T + 3.99 = 3.81(n + 1)\). Therefore \(3.8n + 3.99 = 3.81n + 3.81\) and hence \(0.18 = 0.01n\), which has solution \(n = 18\). In order to increase her average distance to 3.82 m with her final jump, she must jump \(x\) m, where \(T + 3.99 + x = 3.82(n + 2)\). Therefore \(3.8n + 3.99 + x = 3.82n + 7.64\) and hence \(x = 0.02n + 3.65\). Since we have already shown \(n = 18\), we have \(x = 0.36 + 3.65 = 4.01\). Hence the distance she needs to jump with her final jump is 4.01 m.
Solution © UK Mathematics Trust, from the GK 2018 solutions.