Aptitude · Boats and Streams · Model 2
Find the distance once, then use it twice
Once a real distance or a real time appears in the question, the ratios are no longer enough. Convert the boat and the stream into two travelling speeds, use the leg you were given to pin the distance, and the other leg follows — or skip the distance entirely with the time ratio.
Turn a time into a distance and back →01 The idea
The distance is the bridge between the two legs
Model 1 lived entirely in ratios and never needed a kilometre. Model 2 starts the moment the question hands you a genuine distance or a genuine time, and it wants a genuine distance or time back. The formulas do not change — you still have D = b + s and U = b − s — but now they feed straight into distance = speed × time.
The one habit worth building is to convert first. Write D and U down before anything else, and then forget the river exists. A boat at 20 km/h on a 12 km/h current is not a river problem at all after the first line; it is a vehicle that does 32 km/h one way and 8 km/h the other. Every mistake in this model comes from carrying b into a division where D belonged.
The second habit is to notice what the two legs share. They do not share a speed and they do not share a time, but they share the distance — the boat goes to a point and comes back to where it started. So a speed and a time in one direction fix the distance, and the distance then fixes everything in the other direction. One number does the whole crossing.
And because the distance is shared, you can often avoid computing it. Time is inversely proportional to speed over a fixed distance, so if D : U is 4 : 1 then the times are 1 : 4, and a 16-hour trip down means a 64-hour trip back — without ever mentioning the 512 kilometres in between. Use the long route when the distance is the answer and the short route when it is not.
02 Worked example
Twenty on a twelve, sixteen hours down
One boat runs the whole lesson. A boat moves at 20 km/h in still water while the current flows at 12 km/h. It takes 16 hours to travel a certain distance downstream. How much time will it take to cover that same distance upstream?
The two routes cost very different amounts of writing for the same answer. If the question had asked how far the point was, you would need the 512 km; because it asked for a time, the flip does it in one multiplication. The general rule is worth stating plainly: compute the distance only when the distance is what you were asked for, or when the two legs are different lengths so the flip does not apply.
03 The method
Two routes, and how the given arrives
The method is short. Most of the difficulty in exam versions is in the first line, where the speeds are handed to you as a percentage, a ratio, or a distance over an awkward time.
| How the speeds arrive | First line | Then |
|---|---|---|
| b and s outright | add and subtract | D = 32, U = 8 |
| A distance and a time | divide for the speed | 90 km in 4 h → D = 22.5 |
| b : s given as 17 : 3 | D = 20 parts, U = 14 parts | D = 40 → 1 part = 2, U = 28 |
| “b is 250% higher than s” | b : s = 350 : 100 = 7 : 2 | D = 9 parts, U = 5 parts |
| 2 hours 45 minutes | write it as 11/4 hours | never 2.45 |
| Equal legs, time wanted | flip the speed ratio | 16 h × 4 = 64 h |
| Unequal legs | the flip is invalid | find both speeds instead |
05 Cheat sheet
Model 2 on one page
Both routes and the conversions that feed them. The example column is the lesson boat: b = 20, s = 12, 16 hours downstream.
| Step | Rule | On b 20, s 12, 16 h down |
|---|---|---|
| Travelling speeds | D = b + s, U = b − s | 32 and 8 km/h |
| Shared distance | D × t(down) | 32 × 16 = 512 km |
| Return leg | distance / U | 512 / 8 = 64 h |
| Same thing, no distance | t(down) × D/U | 16 × 4 = 64 h |
| Round trip | t(down) + t(up) | 80 hours |
| Round-trip average | (b² − s²)/b | (400 − 144)/20 = 12.8 |
| Using b in a division | never — use D or U | not 512/20 |
06 Where & why
Where this shows up
Model 2 is the workhorse of the chapter in bank and staff-selection papers, because it lets an examiner bolt a percentage or a fractional time onto arithmetic that is otherwise easy.
b, s and a downstream time, asking for the upstream time. Two lines with the flip. The mark is for not using b as a travelling speed.
“110 km in 2 hours 45 minutes, with b : s = 17 : 3.” Convert the time to 11/4 first, then price the parts. Decimalising 2 hours 45 minutes as 2.45 is a standard trap answer.
“250% higher”, “25% of”, “80% more”. Decode into parts on line one and the question collapses into this model.
Any round trip through a helping-then-hindering medium: a drone against the wind, a courier route with a gradient, a ferry across a tide. The out leg times the route and the back leg follows from the speed ratio.
07 Interview questions
What gets asked
Ten, from why you convert first through to the case where the shortcut everyone reaches for is not available.
Why convert to D and U before doing any distance arithmetic?
A boat does 20 km/h in still water on a 12 km/h current and takes 16 hours downstream. How long back?
What is the shortcut that avoids the distance?
When should you compute the distance anyway?
How do you handle “110 km downstream in 2 hours 45 minutes”?
“A boat’s speed in still water is 250% higher than the stream’s.” What ratio?
Given b : s = 17 : 3 and 110 km downstream in 11/4 hours, how long for 98 km upstream?
The two distances are different. Does the time flip still work?
What tells you whether a question is Model 1 or Model 2?
When would you actually use this?
08 Practice problems
Six on times and distances
Convert to D and U on the first line every time, and convert minutes into fractions of an hour before you divide anything. One of these deliberately breaks the time-flip shortcut.