Boats and Streams Model 3: The Double Time Trick

Boats and Streams · 25 min

Aptitude · Boats and Streams · Model 3

A single multiplier prices the whole river

“It took him three times as long to come back.” That one clause, with no distance and no speed attached, already fixes the ratio of the boat to the current. One formula converts it, and one real number turns the ratio into kilometres per hour.

Turn a multiplier into two speeds →
If the upstream leg takes n times as long, then b : s = (n + 1) : (n − 1) — and D : U = n : 1, which is a different ratio.

01 The idea

One number does all the work

This model is Model 1 with the time ratio handed to you in words. Instead of “4 hours down and 12 hours up” the question says “three times as long coming back”, and that is the same information with the distances and the hours stripped out. Call the multiplier n and the whole question reduces to it.

Because the distance is the same in both directions, speed is inversely proportional to time. So if the return leg takes n times as long, the outward speed must be n times the return speed: D : U = n : 1. Everything so far is the flip you already know.

The step that has its own formula is the next one. b is the sum of the two speed parts and s is their difference, so b : s = (n + 1) : (n − 1). With n = 3 that is 4 : 2, which reduces to 2 : 1. Notice how different that is from the 3 : 1 you started with — the 3 : 1 belongs to D and U, and mixing the two ratios up is the standard wrong answer in this model.

The formula is also indifferent to how n arrives. Two exact times give n = tup / tdown, which turns the same rule into b : s = (tup + tdown) : (tup − tdown). A percentage works too: “40% more time upstream” is simply n = 1.4, and 2.4 : 0.4 reduces to 6 : 1. Decimals in n are not a problem, only an invitation to multiply both sides by ten.

n is the upstream time divided by the downstream time. D : U = n : 1, and b : s = (n + 1) : (n − 1). Everything else is anchoring.
n, the time multiplierHow many times longer the upstream leg takes than the downstream one over the same distance. It must be strictly greater than 1 — at n = 1 there is no current, and below 1 the river would have to help in both directions.
The master ratiob : s = (n + 1) : (n − 1). The sum goes on top because the boat is always the bigger of the two, and the difference underneath because the stream is the smaller.
What n cannot tell youAny distance, and any actual time. n fixes only ratios, so a question that wants hours or kilometres must supply a second real number somewhere.

02 Worked example

Three times as long, on a four km/h current

One question runs the whole lesson. A rower takes three times as long to row a stretch of river upstream as he takes to row the same stretch downstream. The current flows at 4 km/h. Find his rowing speed in still water.

1
Pull the multiplier out of the sentence“Three times as long” is the upstream time over the downstream time, and nothing else in the sentence matters yet.n = 3
2
Flip it into speedsThe stretch is the same both ways, so three times the time means one third of the speed.D : U = 3 : 1
3
Add and subtract for the master ratioThe boat is the sum of the parts and the stream is their difference. Reduce afterwards.b : s = (3 + 1) : (3 − 1) = 4 : 2 = 2 : 1
4
Anchor with the currentThe stream is the one real number and it is worth exactly one part of the reduced ratio.1 part = 4 km/h  ⇒  b = 2 × 4 = 8 km/h
5
Check the multiplier comes back outRebuild D and U and take their ratio. It has to return the n you started with.D = 8 + 4 = 12  ·  U = 8 − 4 = 4  ·  12/4 = 3 ✓

The two ratios in this question are 3 : 1 and 2 : 1, and they mean completely different things: 3 : 1 is downstream to upstream, 2 : 1 is boat to stream. Answering 12 km/h — treating the boat as three parts — is the mistake this model is built to catch, and it is always in the option list. Also notice what was never needed: no distance, no time, and no mention of how long either leg actually took.

03 The method

One formula, four wordings

The formula is a single line. Its value is that it absorbs every way an examiner can phrase a time comparison, including the ones with decimals in them.

b : s = (n + 1) : (n − 1), where n = tup / tdown. Equivalently, straight from two times, b : s = (tup + tdown) : (tup − tdown) — divide both by tdown and the two forms are identical.
Do not confuse the two ratios. D : U = n : 1 and b : s = (n + 1) : (n − 1), and only the second one answers a question about the boat and the current. Also check n > 1 before you divide: at n = 1 the formula divides by zero, which is the algebra telling you that equal times mean no river at all.
The wordingnb : s
Twice as long upstream23 : 1
Three times as long upstream34 : 2 = 2 : 1
Four times as long upstream45 : 3
1.5 times as long1.52.5 : 0.5 = 5 : 1
40% more time upstream1.42.4 : 0.4 = 6 : 1
6 hours up, 4 hours down6/4 = 1.510 : 2 = 5 : 1
Same time both ways1 — impossibledivides by zero

05 Cheat sheet

Model 3 on one page

One formula, the two ratios it produces, and the boundary it breaks at. The example column is the lesson question: n = 3 with a 4 km/h current.

ItemRuleOn n = 3, s = 4
The multipliern = t(up) / t(down)n = 3
Travelling speedsD : U = n : 13 : 1
Master ratiob : s = (n+1) : (n−1)4 : 2 = 2 : 1
From two exact times(t up + t down) : (t up − t down)same answer
Anchoringgiven / its part count1 part = 4, b = 8
CheckD/U must equal n12/4 = 3
Using n : 1 as b : sthe standard wrong answer12, not 8
Percentages of extra time are just n“40% more time upstream” means the upstream leg is 140% of the downstream one, so n = 1.4 and b : s = 2.4 : 0.4 = 6 : 1. “Twice as much extra time” would be n = 3, not n = 2 — read whether the percentage is of the downstream time or more than it.
The formula is symmetric in disguiseb : s = (n + 1) : (n − 1) can be inverted: given b : s = p : q, the multiplier is n = (p + q)/(p − q). A 5 : 3 boat-to-stream ratio therefore means n = 8/2 = 4, so the return leg takes four times as long.
n alone never gives an hourEvery quantity n produces is a ratio. A question that wants a real time or distance must supply one absolute number — a speed, a distance or a leg time. If you find yourself trying to get hours out of n on its own, re-read the question for the given you missed.

06 Where & why

Where this shows up

This is the fastest-to-solve model in the chapter once you recognise it, which is precisely why examiners dress the multiplier up so heavily.

TCS NQT · Capgemini
“Twice as long to return”

Five seconds with the formula. The whole question tests whether you know that (n + 1) : (n − 1) is the boat-to-stream ratio and n : 1 is not.

SSC CGL Mains
Exact hours instead of a multiplier

“4 hours down, 6 hours up.” Use the sum-over-difference form directly — 10 : 2 = 5 : 1 — and skip working out n at all.

Bank PO
A decimal or fractional multiplier

“1.5 times”, “40% more time”, “1 and 7/9 times”. The formula is unchanged; multiply both sides of the ratio by whatever clears the fraction.

Any asymmetric round trip
Commutes, uploads, flights

If the trip home takes 1.4 times as long as the trip out, the same formula recovers the ratio of your own pace to whatever is slowing you — traffic, wind, contended bandwidth.

The general lesson is that a ratio of times is a fully quantitative given, even though it contains no units. A great many aptitude questions look under-specified until you notice that a comparison is data.

07 Interview questions

What gets asked

Ten, from the definition of n through to the boundary case where the formula refuses to answer.

What exactly is n in this model?
The upstream time divided by the downstream time, over the same distance. “It takes three times as long to come back” means n = 3. It is a pure number with no units, and it is the only thing you need to extract from the wording.
Why is b : s equal to (n + 1) : (n − 1) rather than n : 1?
Because n : 1 is the ratio of D to U, not of b to s. Since b = (D + U)/2 and s = (D − U)/2, in parts the boat is the sum n + 1 and the stream is the difference n − 1. The halves cancel out of the ratio, which is why no 2 appears.
A rower takes 3 times as long upstream. The current is 4 km/h. Find his still-water speed.
8 km/h. n = 3 gives b : s = 4 : 2 = 2 : 1, so with the stream at 4 km/h one part is 4 and the boat is 8. Check it: D = 12 and U = 4, and 12/4 = 3 as required. Answering 12 means you used the 3 : 1 ratio for the boat and the stream.
What if the question gives two exact times instead of a multiplier?
Use the equivalent form b : s = (t(up) + t(down)) : (t(up) − t(down)) and skip n entirely. For 6 hours up and 4 down that is 10 : 2 = 5 : 1, so a 3 km/h stream means a 15 km/h boat.
How do you handle “40% more time upstream”?
The upstream leg is 140% of the downstream one, so n = 1.4 and b : s = 2.4 : 0.4. Multiply both sides by ten to get 24 : 4, then reduce to 6 : 1. Watch the difference between “40% more time” and “40% of the time” — the second would be n = 0.4, which is impossible.
Why must n be greater than 1?
Because upstream is the slow direction, so it always takes longer. At n = 1 the two times are equal, which forces s = 0, and the formula divides by n − 1 = 0 to tell you so. Below 1 you would need a negative current, which means the two legs have been swapped in the reading.
A boat does 10 km/h in still water and takes four times as long upstream. Find the current.
6 km/h. n = 4 gives b : s = 5 : 3, so five parts are 10 km/h, a part is 2, and the stream is 3 × 2 = 6. Sanity check: D = 16 and U = 4, and 16/4 = 4. Note the stream is more than half the boat speed here, which is unusual but perfectly legal.
Can n on its own give you a distance or a time?
No. Every output of the formula is a ratio, so a question wanting kilometres or hours must give you one absolute number as well — a speed, a distance, or one of the two leg times. If nothing absolute is given, the only answerable question is a ratio.
How is this different from the flip in Model 1?
It is not, mathematically. Model 1 gives you two times and you flip them; Model 3 gives you their ratio in words. The reason it gets its own formula is speed of recognition: going straight from “three times as long” to 4 : 2 skips two written lines, and in a timed paper that is the whole point.
When would you actually use this?
Whenever a return trip is slower by a known factor. If your commute home reliably takes 1.4 times the trip in, the same formula gives the ratio of your free-flowing speed to the delay, without measuring either. The transferable idea is that (sum) : (difference) converts a ratio of totals into a ratio of components.

08 Practice problems

Six on the multiplier

Write n down as its own line before you use the formula, and keep the two ratios — D : U and b : s — visibly separate. Two of these give you exact times rather than a multiplier.

Straight from the wording

Easy
A man rows to a certain place and comes back. It takes him twice as long to row upstream as it takes to row downstream. What is the ratio of his speed in still water to the speed of the river current?
Follow-up
There is no absolute number anywhere in the question, so the only possible answer is a ratio — which is a useful thing to notice before you start hunting for one.
Show the hint
n = 2, and the master ratio is (n + 1) : (n − 1).

A bigger multiplier

Easy
A man takes four times as long to row a distance upstream as to row the same distance downstream. What is the ratio of the speed of the boat in still water to the speed of the stream?
Follow-up
With n = 4 the two parts of the answer are both greater than one, so the ratio does not reduce to something with a 1 in it. That makes it harder to check by eye and easier to misremember.
Show the hint
5 : 3, if you apply the formula. Confirm it by rebuilding D and U and checking D/U = 4.

Two exact times

Medium
A boat covers a certain distance downstream in 4 hours, but takes 6 hours to return upstream to the starting point. If the speed of the stream is 3 km/h, find the speed of the boat in still water.
Follow-up
You are given hours, not a multiplier, so there are two routes — compute n = 6/4 first, or use the sum-over-difference form directly. Only one of them avoids a fraction.
Show the hint
b : s = (6 + 4) : (6 − 4). Reduce before anchoring.

The boat is the given

Medium
A rower takes 1.5 times as long to row a distance upstream as he does downstream. If his boat speed in still water is 25 km/h, find the speed of the current.
Follow-up
n is a decimal, so the raw ratio has a decimal in it too and has to be cleared before it is useful. The anchor is the boat rather than the stream, so the part count you divide by is the larger one.
Show the hint
2.5 : 0.5 — multiply both sides by ten and reduce.

Multiplier and boat, find the current

Medium
A boat travels at 10 km/h in still water. The time it takes to cover a certain distance upstream is four times the time taken to cover that same distance downstream. What is the speed of the current?
Follow-up
The multiplier is large, so the current turns out to be a big fraction of the boat speed — large enough that a careless reading gives a stream faster than the boat, which is impossible. Check your answer against that.
Show the hint
n = 4 gives b : s = 5 : 3, and the boat is the five.

A percentage, a distance, and the average

Hard
A rower covers a distance upstream taking 40% more time than he takes to cover the same distance downstream. The current flows at 5 km/h. Find the total time for a round trip of 105 km each way, and then find the average speed for the whole journey and compare it with his still-water speed.
Follow-up
Three ideas stacked: a percentage that has to become n, an anchor that turns the ratio into speeds, and then a round trip whose average speed is not the still-water speed. The last part is the one worth the effort — work out both numbers and then see whether the gap between them matches s squared divided by b.
Show the hint
40% more time makes n = 1.4, so b : s = 2.4 : 0.4 = 6 : 1. Anchor on the 5 km/h stream, build D and U, then time each 105 km leg separately.