The time taken by a boat to go 21 km downstream is equal to the time taken by…

Key idea: Use boat speed in still water and stream speed to form equations from the given times, then compute the required downstream time.

1. Let b = speed of the boat in still water (km/h) and s = speed of the stream (km/h). Downstream speed = b + s, upstream speed = b - s.

2. Given: time for 21 km downstream equals time for 9 km upstream, so 21/(b+s) = 9/(b-s).

3. Solve that equation: 21(b - s) = 9(b + s) ⇒ 21b - 21s = 9b + 9s ⇒ 12b = 30s ⇒ b = 5s/2.

4. The total time for 12 km downstream and 12 km upstream is given as 5 5/7 hours = 40/7 hours. Compute that total time in terms of b and s:

5. Total time = 12/(b+s) + 12/(b-s). Substitute b = 5s/2 to get b+s = 7s/2 and b-s = 3s/2, so

6. 12/(7s/2) + 12/(3s/2) = 24/(7s) + 24/(3s) = 24(1/(7s) + 1/(3s)) = 24(10/(21s)) = 80/(7s).

7. Set this equal to 40/7: 80/(7s) = 40/7 ⇒ 80/s = 40 ⇒ s = 2 km/h. Then b = 5s/2 = 5 km/h.

8. Downstream speed = b + s = 5 + 2 = 7 km/h. Time to go 28 km downstream = 28/7 = 4 hours.

Answer: 4 hours.