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Trains and streams

A 150 m train at 54 km/h clears a 300 m platform in how long? Relative speed, boats against the stream, round-trip averages.

  • Quantitative reasoning
  • Untimed
  • Endless levels

How it plays

Each question is something moving: a train past a post or through a station, two trains passing, a boat up and down a river, two drivers meeting, one catching another, a round trip at two speeds.

New kinds unlock as you climb and the numbers grow without end. Speeds in km/h are chosen so they turn into whole metres per second: × 5/18.

The wrong options are the real slips: km/h used as if it were m/s, the platform or the other train's length forgotten, speeds added where they should be subtracted, and the plain average of two speeds where the time-weighted one is asked.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes speed, relative-speed and stream problems of these kinds quicker and surer. It claims nothing wider.

Learn

How to get better at trains and streams: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Put every number in one set of units before anything else. Train lengths are in metres and times in seconds, so turn km/h into m/s first; road and river questions stay in km, km/h and hours.
  2. Ask what distance has to be covered. Past a post, it is the train’s own length; through a platform, the platform plus the train; for two trains, both lengths; to meet or catch up, the gap.
  3. Ask what speed covers it. One thing moving: its speed. Two moving towards each other: add. Same direction: subtract. A boat: still-water speed plus the stream going down, minus it going up.
  4. Divide distance by speed, and check the answer is in the unit asked for — seconds, hours, km/h or km.

Techniques and tricks

  • km/h to m/s: × 5/18

    1 km/h is 1,000 m in 3,600 s, which is 5/18 m/s. Every speed here is a multiple of 18 km/h, so divide by 18 and multiply by 5. Going back, m/s × 3.6 gives km/h.

    e.g. 18 → 5, 36 → 10, 54 → 15, 72 → 20, 90 → 25 m/s

  • The train travels its own length too

    A train is past a post when its back has gone by, so it travels its full length. Through a platform, the front must go the platform’s length and then the train’s.

    e.g. 150 m train at 54 km/h through a 300 m platform: (300 + 150) ÷ 15 = 30 s

  • Relative speed: add if towards, subtract if same way

    Two trains passing close the distance between them at the sum of their speeds if they face each other, and at the difference if one overtakes the other. The distance to close is both lengths.

    e.g. 200 m and 300 m trains at 54 and 36 km/h, towards each other: 90 km/h = 25 m/s, 500 ÷ 25 = 20 s

  • Downstream and upstream: half the sum, half the difference

    Downstream is b + s and upstream is b − s. The still-water speed b is half their sum; the stream s is half their difference.

    e.g. down 15 km/h, up 9 km/h: b = 24 ÷ 2 = 12, s = 6 ÷ 2 = 3

  • Round-trip average: 2ab ÷ (a + b)

    Same distance out and back at speeds a and b is not their plain average, because you spend longer at the slower speed. The true average is 2ab ÷ (a + b), always below the plain one.

    e.g. out at 40 km/h, back at 60: 2 × 40 × 60 ÷ 100 = 48 km/h, not 50

  • Meeting and catching: close the gap

    Two setting off towards each other close the gap at the sum of their speeds. To catch up, first find the head start in km, then close it at the difference of speeds.

    e.g. A leaves at 40 km/h, B follows an hour later at 60: head start 40 km, closed at 20 km/h, so 2 hours

As the levels climb

  • The kinds unlock in this order: a train past a pole and there-and-back averages; then a train through a platform and boats in a stream; then two trains passing and two drivers meeting; then catching up. Four in ten questions come from the newest kind.
  • The numbers grow as you climb — faster trains, longer platforms, longer trips — but wherever metres and seconds are involved, the speed that matters — the train’s, or two trains’ combined or relative speed — is a multiple of 18 km/h, so × 5/18 comes out whole. Know the table past 180 km/h = 50 m/s.
  • Two trains come in two forms: towards each other, and one overtaking the other. Read the direction before anything else; it decides whether you add or subtract.
  • Each step of size adds 8% to the time, up to twelve steps. Typing the answer gives a quarter more again, and counts within 0.05.

Traps to avoid

  • km/h used as if it were m/s: 150 m at 54 km/h is not 150 ÷ 54 seconds. Convert first — 54 km/h is 15 m/s — and the answer is 10 s.
  • The platform or the other train forgotten: dividing only the platform’s length, or only one train’s. A train is clear only when its back has passed, so its own length counts every time.
  • Speeds added where they should be subtracted: two trains going the same way overtake at the difference of speeds. Adding them gives a time far too short.
  • The plain average of two speeds for a round trip: 50 km/h for out at 40 and back at 60. It is 48, because you drive the slower leg for longer.
  • The catch-up time counted from the wrong start: the question asks how long after the second one sets off. Do not add the head-start hour back on.

How to practise

  • Use “Kinds of question” to drill “A train passes a pole” alone until × 5/18 is instant; every train question starts there.
  • Pair “Two trains” with “Boats and streams” in “Kinds of question”: both are about adding or subtracting speeds, and practising them together sharpens which is which.
  • Switch “Answer by” to “Typing the answer” to produce the number yourself rather than spot it among the slips.
  • Raise “Options shown” towards 8 to see the forgotten-length and wrong-direction answers side by side with the right one.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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