A boat's speed in still water is 15 km/h. It goes 30 km upstream and returns in 4 hours 30 minutes. What is the speed of the stream?
5 km/h. Solve 30/(15 - s) + 30/(15 + s) = 4.5. This gives 900/(225 - s^2) = 4.5, so 225 - s^2 = 200 and s = 5. Check: 30/10 + 30/20 = 3 + 1.5 = 4.5 hours.
Explain it simply
When the unknown is the stream speed, you cannot just add and halve. Each leg has its own time, and the two times must add to the total. That gives an equation with s in two denominators. In an exam it is often faster to test the options than to solve it algebraically.
Example
Question: b = 15 km/h; 30 km up and back in 4.5 h. Find s.
Algebra:
- 30/(15 - s) + 30/(15 + s) = 4.5.
- Combine: 30[(15 + s) + (15 - s)] / (225 - s^2) = 900 / (225 - s^2).
- 900 / (225 - s^2) = 4.5, so 225 - s^2 = 200, s^2 = 25, s = 5 km/h.
Option testing: try s = 5. Upstream 10 km/h takes 3 h; downstream 20 km/h takes 1.5 h. Total 4.5 h. It works.
const t = s => 30 / (15 - s) + 30 / (15 + s);
console.log(t(5)); // Output: 4.5
console.log(t(3).toFixed(2)); // Output: 4.17s = 3 gives about 4.17 h, too short, which confirms the stream must be faster.
What to say in the interview
"Each leg's time is distance over its own speed, and they add to 4.5 hours. Combining the fractions gives 900 over 225 minus s squared, so s squared is 25 and the stream is 5 km/h. Plugging in 5 gives 3 plus 1.5 hours, which checks."
Common mistakes
- Assuming the round trip takes the same time as in still water (4 hours).
- Taking the negative root; a speed must be positive.