How to Solve Boats Round-Trip Time Problems
Solve boats and streams round-trip time problems, derive the average speed formula, and practice with a worked example and MCQs.
Expected Interview Answer
A round trip’s total time is the sum of the upstream leg time and the downstream leg time computed separately, total time = d/(b−c) + d/(b+c), which simplifies to a single formula involving the boat’s still-water speed and the current’s speed rather than a simple average.
For a round trip of one-way distance d, upstream time is d/(b−c) and downstream time is d/(b+c); summing gives total time = d × [2b / (b²−c²)], and average speed for the whole round trip is total distance (2d) divided by that total time, giving the compact formula (b²−c²)/b. This average speed is always less than the still-water speed b, because the boat spends more time on the slower upstream leg than the faster downstream leg for the same distance — a subtlety that trips up anyone who tries to simply average the upstream and downstream speeds. Interviewers often ask candidates to derive this from first principles rather than recall the formula, so understanding the d/(b−c) + d/(b+c) derivation matters more than memorizing (b²−c²)/b.
- Derives the round-trip average speed formula from first principles
- Avoids the common mistake of simply averaging upstream and downstream speeds
- Formula (b²−c²)/b generalizes to any round-trip speed pair problem
- Clarifies why round-trip average speed is always less than still-water speed
AI Mentor Explanation
A player running between the wickets covers 20 meters there and 20 meters back, but if fatigue slows the return leg, the average speed for the full there-and-back run is not the simple average of the two leg speeds — it is total distance over total time, which weights the slower leg more heavily since more time is spent on it. Boats round-trip problems compute average speed the identical way: total distance divided by the sum of upstream and downstream times, never a naive average of the two speeds.
Worked example
Upstream leg
- 24 km ÷ 8 km/h = 3 h
Downstream leg
- 24 km ÷ 12 km/h = 2 h
Average speed
- 48 km ÷ 5 h = 9.6 km/h
- (b²−c²)/b = 96/10 = 9.6
Step-by-Step Explanation
Step 1
Compute each leg time separately
Upstream time = d/(b−c); downstream time = d/(b+c).
Step 2
Sum for total time
Total time = d/(b−c) + d/(b+c) = 2bd/(b²−c²).
Step 3
Compute total distance
Round-trip distance = 2d.
Step 4
Derive average speed
Average speed = 2d ÷ total time = (b²−c²)/b, always less than b.
What Interviewer Expects
- Correct separate computation of upstream and downstream leg times
- Recognizing average speed is total distance over total time, never a simple mean
- Ability to derive (b²−c²)/b from first principles, not just recall it
- Understanding why round-trip average speed is always below still-water speed
Common Mistakes
- Averaging upstream and downstream speeds directly instead of using total distance over total time
- Forgetting the round-trip distance is 2d, not d
- Sign or algebra errors when combining the two fractions d/(b−c) and d/(b+c)
- Assuming average speed equals still-water speed when current is nonzero
Best Answer (HR Friendly)
“I never average the upstream and downstream speeds directly for a round trip — instead I compute the time for each leg separately using distance over speed, add those two times together, and then divide the total round-trip distance by that total time. That naturally leads to the formula (b squared minus c squared) over b, and it’s always a bit less than the boat’s still-water speed because more time gets spent on the slower upstream leg.”
Follow-up Questions
- How would you find the current speed if only the round-trip average speed and boat speed are given?
- Why is round-trip average speed always less than the still-water speed?
- How does this formula change if the outbound and return distances are different?
- How would you extend this to a three-leg trip with varying current speeds?
MCQ Practice
1. A boat's still-water speed is 12 km/h and the current is 3 km/h. What is the average speed for a round trip over the same route?
Average speed = (b²−c²)/b = (144−9)/12 = 135/12 = 11.25 km/h.
2. A boat covers 15 km upstream in 3 hours and 15 km downstream in 1.5 hours. The round-trip average speed is?
Total distance = 30 km, total time = 3 + 1.5 = 4.5 h, average speed = 30/4.5 = 20/3 km/h.
3. Why is round-trip average boat speed always less than the still-water speed when current is nonzero?
For equal distances, the slower upstream leg takes more time, weighting the total-distance-over-total-time average toward the lower speed.
Flash Cards
Round-trip average speed formula? — Average speed = (b² − c²) / b, where b = still-water speed, c = current speed.
How is average speed correctly computed? — Total round-trip distance divided by total round-trip time — never a simple mean of the two leg speeds.
Why is average speed always less than b? — More time is spent on the slower upstream leg for the same distance, pulling the average down.
Total time formula for a round trip? — Total time = d/(b−c) + d/(b+c) = 2bd/(b²−c²).