GK 2020 Q24
24. Anne, Bronwyn and Carl ran a race. They started at the same time, and their speeds were constant. When Anne finished, Bronwyn had 15 m to run and Carl had 35 m to run. When Bronwyn finished, Carl had 22 m to run. What was the length of the race?
- A 135 m
- B 140 m
- C 150 m
- D 165 m
- E 175 m
Grey Kangaroo 2020, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 165 m
First note that Carl ran (35 − 22) m = 13 m while Bronwen ran 15 m. Let the length of the race be \(x\) m. Since their speeds were constant, the ratio of the distances they ran in any time is also constant. Therefore, since Carl ran \((x - 35)\) m while Bronwen ran \((x - 15)\) m, we have \(\dfrac{x - 35}{x - 15} = \dfrac{13}{15}\). Therefore \(15x - 15 \times 35 = 13x - 13 \times 15\) and hence \(2x = 15 \times 35 - 13 \times 15\). Therefore \(2x = 22 \times 15\) and hence \(x = 11 \times 15\). Therefore the distance they ran is 165 m.
Solution © UK Mathematics Trust, from the GK 2020 solutions.