Boats and Streams Formula for Bank Exams
The boats and streams formula is: downstream speed = boat speed + stream speed and upstream speed = boat speed − stream speed. So the boat's speed in still water is half the sum of the downstream and upstream speeds, and the stream's speed is half their difference.
🔥 Key Exam Takeaway: If downstream = D and upstream = U, then boat = (D + U) ÷ 2 and stream = (D − U) ÷ 2. If going up takes k times as long as coming down the same distance, then boat : stream = (k + 1) : (k − 1). Example: up takes twice as long → boat : stream = 3 : 1.
Boats and streams gives 1 to 2 questions in IBPS/SBI Clerk and PO prelims (often inside a word-problem set or quantity comparison) and appears in SSC CGL and CHSL arithmetic. Every answer in this note has been checked in Python.
Upstream and Downstream Meaning (Basic Terms)
| Term | Meaning | Symbol used here |
|---|---|---|
| Still water | Water that is not flowing | — |
| Speed of boat in still water | Speed of the boat (or swimmer) with no current | b |
| Speed of stream (current) | Speed at which the river water flows | s |
| Downstream | Moving with the current, so the current helps | D = b + s |
| Upstream | Moving against the current, so the current slows you | U = b − s |
A boat can move upstream only if b > s. So in every valid question, the boat's speed is more than the stream's, and downstream speed is always more than upstream speed.
Boats and Streams Formulas List (All Important Formulas)
| # | Situation | Formula |
|---|---|---|
| 1 | Downstream speed | D = b + s |
| 2 | Upstream speed | U = b − s |
| 3 | Speed of boat in still water | b = (D + U) ÷ 2 |
| 4 | Speed of stream | s = (D − U) ÷ 2 |
| 5 | Upstream takes k times the downstream time (same distance) | b : s = (k + 1) : (k − 1) |
| 6 | Same distance: upstream time t₁, downstream time t₂, boat speed b known | s = b(t₁ − t₂) ÷ (t₁ + t₂) |
| 7 | Same distance: upstream time t₁, downstream time t₂, stream speed s known | b = s(t₁ + t₂) ÷ (t₁ − t₂) |
| 8 | Goes d km and comes back in total time T | d = T(b² − s²) ÷ (2b) |
| 9 | Average speed for a full round trip | (b² − s²) ÷ b = D × U ÷ b |
| 10 | Time for a round trip of d km each way | T = d ÷ (b + s) + d ÷ (b − s) = 2bd ÷ (b² − s²) |
Why formula 5 works: for the same distance, time is inversely proportional to speed. If upstream time is k times downstream time, then D = k × U, so (b + s) = k(b − s), which gives b(k − 1) = s(k + 1), that is b : s = (k + 1) : (k − 1).
Boats and Streams Tricks (Shortcut Methods)
Trick 1: Average and Half-Difference
Read the downstream and upstream speeds, then average them for the boat and take half the difference for the stream. No equations needed.
Trick 2: Time Ratio to Speed Ratio
If times are in ratio t₁ : t₂ (up : down) for the same distance, speeds are in ratio t₂ : t₁, that is U : D = t₂ : t₁. Then use Trick 1 on the ratio parts.
Trick 3: Round Trip in Total Time
When a boat goes and returns in a total time T, use d = T(b² − s²) ÷ 2b directly instead of setting up two fractions.
Trick 4: Option Check (Fastest in Prelims)
For "find the boat speed" questions with options, test each option: the distance divided by (b + s) plus distance divided by (b − s) must equal the given time. Usually the right option gives whole numbers.
Boats and Streams Questions for Bank Exams with Solutions
Example 1 (Basic, IBPS Clerk level). A boat goes downstream at 18 km/h and upstream at 12 km/h. Find the speed of the boat in still water and the speed of the stream.
Solution: b = (18 + 12) ÷ 2 = 15 km/h; s = (18 − 12) ÷ 2 = 3 km/h.
Example 2 (Round trip time). A boat's speed in still water is 10 km/h and the stream flows at 2 km/h. How long will it take to go 48 km downstream and come back?
Solution: Downstream speed = 12, upstream = 8. Time = 48 ÷ 12 + 48 ÷ 8 = 4 + 6 = 10 hours.
Example 3 (Time ratio, SBI Clerk level). A man rows at 9 km/h in still water. It takes him twice as long to row upstream as to row downstream the same distance. Find the stream speed.
Solution: k = 2, so b : s = (2 + 1) : (2 − 1) = 3 : 1. With b = 9, s = 3 km/h.
Example 4 (Distances given). A man rows 30 km downstream in 3 hours and 18 km upstream in 3 hours. Find his speed in still water and the stream speed.
Solution: D = 30 ÷ 3 = 10 km/h, U = 18 ÷ 3 = 6 km/h. b = (10 + 6) ÷ 2 = 8 km/h, s = (10 − 6) ÷ 2 = 2 km/h.
Example 5 (Go and return in total time). A boat's still-water speed is 7 km/h and the stream speed is 3 km/h. It goes to a place and comes back in 7 hours. How far is the place?
Solution: d = T(b² − s²) ÷ 2b = 7 × (49 − 9) ÷ 14 = 7 × 40 ÷ 14 = 20 km. Check: 20 ÷ 10 + 20 ÷ 4 = 2 + 5 = 7 hours. ✔
Example 6 (Average speed of round trip). For the boat in Example 5, what is the average speed for the whole trip?
Solution: (b² − s²) ÷ b = 40 ÷ 7 = 40/7 km/h (about 5.71 km/h). Check: total distance 40 km ÷ 7 hours = 40/7. ✔ Trap: the average speed is not (10 + 4) ÷ 2 = 7. Average speed is total distance divided by total time.
Example 7 (Up and down times known). A boat with still-water speed 12 km/h takes 5 hours to go upstream and 3 hours to come back downstream over the same distance. Find the stream speed and the distance.
Solution: s = b(t₁ − t₂) ÷ (t₁ + t₂) = 12 × (5 − 3) ÷ (5 + 3) = 24 ÷ 8 = 3 km/h. Distance = (12 − 3) × 5 = 45 km = (12 + 3) × 3 = 45 km. ✔
Example 8 (Two journeys, IBPS PO level). A boat covers 24 km upstream and 28 km downstream in 6 hours. It covers 30 km upstream and 21 km downstream in 6.5 hours. Find the speed of the boat in still water and of the stream.
Solution: Let 1/U = x and 1/D = y. 24x + 28y = 6 … (i) 30x + 21y = 6.5 … (ii) Multiply (i) by 3 and (ii) by 4: 72x + 84y = 18 and 120x + 84y = 26. Subtract: 48x = 8, so x = 1/6, U = 6 km/h. Put in (i): 24/6 + 28y = 6, so 28y = 2, y = 1/14, D = 14 km/h. b = (14 + 6) ÷ 2 = 10 km/h, s = (14 − 6) ÷ 2 = 4 km/h.
Example 9 (Unknown boat speed, stream given). A boat covers a distance downstream in 1 hour and returns in 1.5 hours. The stream speed is 2 km/h. Find the boat's speed and the distance.
Solution: (b + 2) × 1 = (b − 2) × 1.5 → b + 2 = 1.5b − 3 → 0.5b = 5 → b = 10 km/h. Distance = 12 × 1 = 12 km. Shortcut: b = s(t₁ + t₂) ÷ (t₁ − t₂) = 2 × 2.5 ÷ 0.5 = 10. ✔
Example 10 (SSC CGL style). A man can row 6 km/h in still water. It takes him twice as long to row up as to row down. Find the stream speed.
Solution: b : s = 3 : 1, so s = 6 ÷ 3 = 2 km/h.
Common Mistakes in Boats and Streams Questions
| Mistake | Correct approach |
|---|---|
| Taking the average of upstream and downstream speeds as the average speed of the round trip | Average speed = total distance ÷ total time = (b² − s²) ÷ b |
| Forgetting units (m/s vs km/h) | Convert first: km/h × 5/18 = m/s |
| Using b = D − U | b = (D + U) ÷ 2; s = (D − U) ÷ 2 |
| Ignoring that "rows against the current" means upstream | Against = upstream (b − s); with = downstream (b + s) |
| Accepting s ≥ b | A boat can go upstream only if b > s; reject such options |
Boats and Streams Quick Revision Table
| Given | Find | Use |
|---|---|---|
| D and U | b and s | b = (D + U)/2, s = (D − U)/2 |
| b and s | Time for distance d each way | d/(b + s) + d/(b − s) |
| Up time = k × down time | b : s | (k + 1) : (k − 1) |
| b, t₁ (up), t₂ (down) | s | b(t₁ − t₂)/(t₁ + t₂) |
| s, t₁ (up), t₂ (down) | b | s(t₁ + t₂)/(t₁ − t₂) |
| b, s, total time T | One-way distance | T(b² − s²)/(2b) |
Boats and Streams Practice Questions (with Answers)
1. A boat's downstream speed is 22 km/h and upstream speed is 14 km/h. Find the boat speed and the stream speed. Answer: b = 18 km/h, s = 4 km/h.
2. Boat speed 15 km/h, stream speed 3 km/h. Time to go 36 km and return? Answer: 36/18 + 36/12 = 2 + 3 = 5 hours.
3. A swimmer swims at 5 km/h in still water; the river flows at 1 km/h. How long to swim 24 km downstream and back? Answer: 24/6 + 24/4 = 4 + 6 = 10 hours.
4. Rowing upstream takes 3 times as long as rowing downstream. The stream flows at 4 km/h. Find the boat speed. Answer: b : s = 4 : 2 = 2 : 1, so b = 8 km/h.
5. Boat speed 8 km/h, stream 4 km/h. The boat goes to a point and returns in 9 hours. Find the one-way distance. Answer: 9 × (64 − 16) ÷ 16 = 27 km. (Check: 27/12 + 27/4 = 2.25 + 6.75 = 9.)
6. A boat goes 40 km downstream in 2 hours and the same distance upstream in 4 hours. Find b and s. Answer: D = 20, U = 10, so b = 15 km/h, s = 5 km/h.
7. A man rows at 9 km/h in still water; the river flows at 3 km/h. How long to row 6 km upstream and come back? Answer: 6/6 + 6/12 = 1.5 hours.
8. A boat with still-water speed 18 km/h covers 48 km downstream in 2 hours. How long will it take to cover 48 km upstream? Answer: D = 24, so s = 6; U = 12; time = 48/12 = 4 hours.
Boats and Streams FAQs
What is the formula of upstream and downstream?
Downstream speed = b + s and upstream speed = b − s, where b is the boat's speed in still water and s is the stream's speed.
How do you find the speed of the stream?
Speed of stream = (downstream speed − upstream speed) ÷ 2.
What is the trick if upstream time is double the downstream time?
Boat : stream = (2 + 1) : (2 − 1) = 3 : 1. In general, for k times, it is (k + 1) : (k − 1).
How many questions come from boats and streams in bank exams?
Usually 0 to 2 in IBPS/SBI prelims, often as part of arithmetic word problems, quantity comparison or data sufficiency.
Is boats and streams part of speed, time and distance?
Yes. It is speed, time and distance with one extra speed (the current) that adds downstream and subtracts upstream.
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