Leg it
How far from the finishing line should these runners start to make the race 'fair'?
Age
11 to 14
Challenge level
Problem
Amina and Bhavi are both training for a 100 metre race. Amina's average time to run 100 metres is 12 seconds, and Bhavi's is 15 seconds. To make their training sessions more competitive, they want to run 'fair' races against each other - so that if they ran at their average speeds, they would finish at the same time.
a) If Amina starts 100 metres from the finish line, how many metres in front of her should Bhavi start to make the race 'fair'?
b) If Bhavi starts 100 metres from the finish line, how many metres behind her should Amina start to make the race 'fair'?
Student Solutions
Answer: a) 20 metres, b) 25 metres
Using ratio
a) If Amina runs 100 metres, then the race will be about 12 seconds long. Bhavi needs to run for about 12 seconds.
100 metres in 15 seconds
20 metres in 3 seconds
80 metres in 12 seconds
Head start of 20 metres.
b) If Bhavi runs 100 metres, then the race will be about 15 seconds long. Amina needs to run for about 15 seconds.
100 metres in 12 seconds
25 metres in 3 seconds
125 metres in 12 seconds
Start 25 metres behind Bhavi.
Using distance-time graphs
This distance-time graph shows the race if Amina and Bhavi both start at the same time.
Image
a) The graph below shows the race if Bhavi starts in front of Amina
Image
The time Bhavi takes is split into 12 and 3. Split the distance axis in the same way.
Share 100 metres in the ratio 12 : 3 = 4 : 1 = 80 : 20
Head start of 20 metres
b) The graph below shows the race if Amina starts behind Bhavi
Image
Amina runs for 3 seconds before the start line and 12 seconds after the start line.
3 : 12 = ? : 100
1 : 4 = ? : 100
? = 25, start 25 metres behind