Time, Speed and Distance for CAT: 4 Relative Setups

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It is a Tuesday night sectional. Question 14 gives you two trains, opposite directions, lengths in metres, speeds in kilometres per hour, and a crossing time you are asked to find. You know the formula. You write it down, convert the units, get a number that is not in the options, and start again. Four minutes gone on a question a strong solver closes in fifty seconds.

The gap is almost never the formula. It is the setup. Boats, trains, escalators and two people walking towards each other are the same question wearing different clothes, and aspirants who solve arithmetic questions for CAT quickly are the ones who recognise the costume before they start writing. This post gives you that recognition step, then the four setups it feeds.

Why Time, Speed and Distance Questions Feel Slow

Most aspirants meet this topic as a list. Relative speed formula, boats formula, trains formula, escalator formula, circular track formula. Five notebook entries, each with its own worked example, each memorised separately. It looks like coverage. In a mock it behaves like a lookup problem, and lookup is slow.

Under time pressure you read the question, scan that mental list for a matching label, and only then start solving. If the phrasing sits outside the template you memorised, the scan fails and you restart or skip. That is a recognition failure, and more practice on the same list will not fix it.

The alternative is to collapse the list. Every question here rests on one relationship, distance equals speed multiplied by time, and one decision about what to treat as stationary. Two consequences of that relationship turn most calculation problems into ratio problems.

The Two Ratio Consequences Worth Memorising

• Fixed distance means an inverse ratio. If two bodies cover the same distance, a speed ratio of 4:5 becomes a time ratio of 5:4.
• Fixed time means a direct ratio. If two bodies travel for the same duration, the distances they cover sit in the ratio of their speeds.
• Meeting questions are always the second case. Two people who start together and meet have travelled for equal time, whatever their speeds.
• Percentage speed changes are always the first case. A 25 percent faster speed is 5/4 of the old, so time becomes 4/5, a 20 percent saving. A stated saving of twelve minutes therefore fixes the original journey at sixty.

The Reference Frame Method: 4 Setups That Cover the Chapter

A reference frame is whatever you decide to treat as stationary. Change what stands still and a two body problem becomes a one body problem. Every question here reduces to one of four choices, and choosing takes five seconds once you know the list.

Setups 1 and 2: The Two Directions

Two bodies moving the same way close the gap at the difference of their speeds. A car at 60 behind a car at 45 closes at 15 kilometres per hour, so overtaking and lapping are that one number applied to a starting gap. Moving towards each other, they close at the sum instead, which is the same structure with one operator changed.

• Same direction on a circular track: time between meetings is track length divided by the speed difference.
• Opposite directions on a circular track: time between meetings is track length divided by the speed sum.
• Straight road with a head start: divide the head start distance by the same relative speed.
• Both starting from the same point at different times: convert the time gap into a head start distance first.
• One body stationary: the frame collapses and relative speed becomes the moving body's own speed.

Setups 3 and 4: Mediums and Lengths

Water, wind and moving walkways look like new topics and are not. The medium carries you, so it behaves as a second body whose speed adds when you move with it and subtracts against it. Written that way, the boats chapter disappears into the two setups above.

Length is the only genuinely new idea here. A point crossing a point covers zero distance. A train crossing a pole covers its own length. Crossing a platform means its length plus the platform. Crossing another train means the sum of both lengths.

Both setups sit on top of the first two rather than beside them. That is the whole reason the chapter collapses: you never choose between four unrelated methods, you choose a direction and then ask whether a medium or a length has been added.

• Medium present, no length: boats, aircraft with wind, walkways.
• Length present, no medium: trains, processions, a convoy passing a marker.
• Both present: rare in CAT, and solved by applying the medium first and the length second.
• Neither present: a plain relative speed question, which is setup 1 or 2 alone.

Boats and Streams: The Same Frame With Different Names

Let the boat speed in still water be b and the stream speed be s. Downstream is b plus s, upstream is b minus s. Two lines, and every standard question here is a rearrangement of them. Three pairings cover almost everything CAT asks.

What you are given	What you can recover immediately
Downstream speed and upstream speed	Boat speed is half their sum. Stream speed is half their difference.
Equal distance each way, plus total time	A single equation in b and s, since time is distance over speed on both legs.
Time downstream and time upstream over the same distance	The ratio of the two speeds, and therefore the ratio of b to s, with no distance needed.

The third row is the one CAT likes. If a boat takes three hours downstream and five hours upstream over the same stretch, the speed ratio is 5:3, so b plus s over b minus s equals 5 over 3. Cross multiply and b comes out as four times s, without the distance ever appearing.

Trains: Why Length Is the Only New Variable

Train questions are relative speed questions with a distance that is not zero. Decide the frame first, then what distance it must cover. Reversing that order is what produces answers missing from the options, because the length attaches to the wrong quantity.

The Checklist for Any Train Question

• Are the bodies moving the same way or opposite ways? That fixes relative speed as a difference or a sum.
• What has to pass what completely? That fixes the distance as one length, two lengths, or a length plus a platform.
• Is one of the bodies stationary? A pole, a signal or a standing man contributes speed zero and length zero.
• Are the units mixed? Convert to metres per second before the equation, never after it.
• Is a man on a platform involved? He is a point, so only the train's length counts.

Work an example. A 180 metre train and a 120 metre train run towards each other at 54 and 36 kilometres per hour. Relative speed is 90 kilometres per hour, which is 25 metres per second. Distance is 300 metres. Time is 12 seconds. Two decisions, one division, and the conversion is the only place left to slip.

A CAT Level Question, Solved Frame by Frame

Setup 3 With Equal Distances

A man rows 30 kilometres downstream and back upstream in a total of eight hours. If the stream flows at 2 kilometres per hour, find his speed in still water. Read it once and note that this is setup 3 with equal distances and a fixed total time, which is the second row of the table above.

1. Name the unknown. Let still water speed be b, so downstream is b plus 2 and upstream is b minus 2.
2. Write each leg as distance over speed: 30 over (b plus 2) and 30 over (b minus 2).
3. Set the sum equal to 8 and clear denominators: 30(b minus 2) plus 30(b plus 2) equals 8(b squared minus 4).
4. Simplify to 60b equals 8b squared minus 32, then to 2b squared minus 15b minus 8 equals 0.
5. Factorise to (2b plus 1)(b minus 8) equals 0. Speed cannot be negative, so b is 8 kilometres per hour.

The work sat in steps one and two, and the rest was mechanical, which is why drilling setups pays more than drilling algebra. If algebra is the bottleneck, spend a week on ratio and proportion practice first.

Where These Arithmetic Setups Break Down

The method is reliable, but five situations catch people out, each with a tell in the stem. Underline the timing words on your first read and most stop being traps.

• Different start times. Relative speed is still correct, but the starting gap is not zero. Compute the head start distance first.
• Rest stops. A stated halt means total time is not travel time. Subtract the halt before using distance over speed.
• Average speed over two legs. It is total distance over total time, never the mean of two speeds. For equal distances at x and y it is 2xy over (x plus y).
• Return journeys with a change of mode. Distances match, times do not, so the inverse ratio applies rather than the direct one.
• Escalators and walkways. A person standing still still moves. Treat the standing case as speed zero plus the medium, not as no motion at all.

Turn the Four Setups Into a Three Week Reflex

Recognition is a trained response, so train it apart from solving. The sequence below takes fifteen minutes a day and front loads the decision rather than the arithmetic, because the decision is where your mock time is going.

1. Week one, setup only. Take twenty questions and write only which setup each one is and what distance the frame must cover. Solve none of them.
2. Week two, solve from your notes. Work the same twenty from your own setup lines and compare your time against your baseline.
3. Week three, unfamiliar wording. Mix them with previous year sets so recognition survives phrasing you have not seen.

Week three is the honest test, because textbook questions announce their type in the first line and the CAT exam does not. Topic-wise CAT exam previous year questions tell you whether the reflex holds when the stem buries the setup.

Keep the formula sheet short. The free CAT formula and cheatsheet library holds the conversions and standard forms, and one page you revise beats a summary you never reopen. If your CAT preparation calendar has no slot for arithmetic revision, the AI study planner for CAT 2026 will place one and move it when you miss it.