Olympic Training

Problem

Rose and Ron are training for the Olympic Games. They started off swimming from different ends of the pool and when they reached the opposite end they immediately turned round.

When they met the second time, Ron was swimming to the right end. At that moment the ratio of their distance from the left end to the right end was 3:2. What was the ratio of Ron’s speed to Rose’s speed?

Teaching sequence

1. Interest the students in the problem with a brief discussion of well-known swimmers. You could perhaps rank the best 4 swimmers that they know of.
2. Pose the problem to the class.
3. After giving the students time to think about the problem ask for ideas about how the problem might be solved. Ask the students to briefly explain why they made their suggestion, for example, they might recognise the problem as similar to one they have previously solved.
4. Let the students work on the problem in pairs. Check that they are recording their work so that they can communicate their solution to others.
5. As the students work ask questions that focus on their understanding of ratios and the strategy they have used.
   What does a 3:2 ratio look like in terms of their position in the pool?
   How did you figure that out?
6. Depending on their experience the students may need some support with the forming of a linear equation.
   How far has Rose swum when they met?
   How can you record that?
   How far has Ron swum when they met?
   How can you record that?
7. Share solutions.

Extension to the problem

This time, the ratio of the distance from the left side of the pool when they met the second time, to the distance from the right side of the pool when they met the first time was 6:5. What is the ratio of Ron’s speed to Rose’s speed this time?

Solution

Let the length of the pool be L. When they meet Ron has travelled L + 3/5L and Rose has travelled L + 2/5L. So the ratios of their speeds when

Extension:

When they first meet Ron has swum 5 units and together Rose and Ron have swum the length of the pool, L units. When Rose and Ron next meet, together they have swum 3L units. But for every length that they swim together, Ron swims 5 units. So at their second meeting, Ron has swum 3 x 5 = 15 units.

But when they meet the second time, Ron has also swum L + 6 units. So 15 = L + 6, so L = 9.

Now we know that when Ron and Rose first meet them, Ron has swum 5 units and Rose L – 5, which is 9 – 5 = 4. So the ratio of Ron’s speed to Rose's speed is 5:4.