# One and three

Two motorboats are travelling up and down a lake at constant speeds and turning at each end without slowing down. They leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A and, for the second time, 400 metres from B.

How long is the lake (the distance from A to B) and what is the ratio of the speeds of the boats?

Two motorboats are travelling up and down a lake at constant speeds and turning at each end without slowing down. They leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A and, for the second time, 400 metres from B. How long is the lake (the distance from A to B) and what is the ratio of the speeds of the boats?

We give the solution by **Nisha Doshi** , Year 9,
The Mount School, York. Well done Nisha this is a really good
solution.

Call the boats $A\prime$ and $B\prime$. The first time they meet, the distances gone are: 600 metres by $A\prime$ and $(x + 400)$ metres by $B\prime$.

The second time the distances are: $(x + 400 + 400)$ metres by $A\prime$ and $(600 + 600 + x)$ metres by $B\prime$.

Since the two speeds are constant, the ratios of the distances travelled each time must be constant, so $$ \frac{600}{x+400} = \frac{x+800}{x+1200} $$ which leads to

The ratio of the speeds must be 600 : 800 or 3 : 4.

FOOTNOTE: You can also use the fact that at the first meeting the combined distance travelled is one length of the lake and at the second it is 3 lengths. It follows that if $A\prime$ travels 600 metres by the first meeting it will have travelled three times that distance by the second meeting, that is 1800 metres. This gives $x = 1800 - 600 - 400 - 400 = 400$ and hence the length of the lake is 1400 metres.

This problem stimulates students into creating some kind of diagram as an aid to visualisation. Hopefully the phrase "ratio of the speeds" will encourage students to think through how the simulation is not significantly altered if the speeds of both boats were, for example, doubled or halved. This is a useful problem for connecting speed, distance and time, and may also offer students a useful context in which to use algebra to express relationships containing known and unknown quantities.