Level 50: Three students ride bicycles from school to the District Gymnasium to learn to swim. Your first friend’s speed is 3 km/h less than your second’s speed. The time it takes for the first, second, and third friends to get from school to the District Gymnasium is 30 minutes, \({2 \over 5}\) hours and \({4 \over 9}\) hours, respectively. How many kilometers is the distance to the District Gymnasium?

Let the speeds of the three students be a, b, c (km/h) respectively (Condition: a, b, c > 0)

We have: b – a = 3. Convert 30 minutes = \({1 \over 2}\) hours.

Because the distance traveled by the three of you is the same, according to the problem, we have: \(a. {1 \over 2} = b. {2 \over 5} = c. {4 \over 9}\)

So we have: \({a \over 2} = {{2b} \over 5} = {{4c} \over 9}\) and 2b – 2a = 6. So \({{2a} \over 4 } = {{2b} \over 5} = {{4c} \over 9}\) and 2b – 2a = 6

According to the property of the series of equal ratios, we have:

\(\eqalign{ & {{2a} \over 4} = {{2b} \over 5} = {{4c} \over 9} = {{2b – 2a} \over {5 – 4}} = {6 \over 1} = 6 \cr & {{2a} \over 4} = 6 \Rightarrow a = {{4.6} \over 2} = 12;{{2b} \over 5} = 6 \Rightarrow b = {{ 5.6} \over 2} = 15;{{4c} \over 9} = 6 \Rightarrow c = {{9.6} \over 4} = 13.5 \cr} \)

So the speed of the first student is 12 km/h

The distance from the school to the County gymnasium is: \(12. {1 \over 2} = 6(km).\)