Time, Speed and Distance

Overview

Time, Speed and Distance appears about 1.8 times per paper across the last four AFCAT solved papers, placing it in the high yield band of Numerical Ability.

Time, Speed and Distance is one of the most predictable scorers in the AFCAT Numerical Ability section. The topic carries about 1.75 marks per paper, with items spread across train crossings, relative motion, average speed, upstream-downstream, circular tracks, escalators and short race problems. Every sub-type sits on the same single equation — Distance equals Speed multiplied by Time — and a tight unit-conversion habit. Candidates who lose marks here usually lose them to two avoidable errors: mixing kilometres-per-hour with seconds in the same expression, and using the arithmetic mean when the question demands the harmonic mean. This page builds the topic from the conversion table upward so that train, boat, track and race problems all look like the same problem in different costumes.

Why Time, Speed and Distance pays AFCAT marks

The AFCAT Numerical Ability section runs to roughly 18 to 22 questions out of 100, and Time-Speed-Distance reliably contributes one to two of them. Each correct answer is worth +3, each wrong answer costs −1, so a clean two-question pickup here is worth six marks against a paper where the cut-off typically lives between 130 and 160.

Three features make TSD a high-yield topic for the AFCAT candidate.

• The formulas are short, finite and reusable. Once you internalise the conversion factor, the relative-speed rule and the harmonic mean, you have covered roughly ninety per cent of what AFCAT will ask.
• The numbers chosen are usually clean — speeds of 36, 54, 72, 90 km/h, distances of 100, 150, 200, 250 metres, times in whole seconds. The arithmetic is rarely the bottleneck.
• The traps are predictable. AFCAT recycles the same four or five trap patterns year after year, so a candidate who studies the trap list will avoid most negative marks.

The objective for the rest of this page is simple — finish every standard TSD item in sixty to ninety seconds, recognise the trap items inside ten seconds, and skip nothing on this topic in the actual paper.

Core formula and unit conversions

Every Time-Speed-Distance problem reduces to one statement.

Distance = Speed × Time

Rearranged, this gives Speed = Distance ÷ Time and Time = Distance ÷ Speed. The skill is not in remembering the formula but in keeping the units consistent. If speed is in kilometres per hour, time must be in hours and distance in kilometres. If speed is in metres per second, time must be in seconds and distance in metres. Mixing the two systems in one equation is the single most common arithmetic error in AFCAT TSD items.

The conversion factor between the two systems is fixed.

• 1 km/h = 5/18 m/s — multiply km/h by 5/18 to get m/s.
• 1 m/s = 18/5 km/h — multiply m/s by 18/5 to get km/h.

The 5/18 factor comes from dividing 1000 metres by 3600 seconds. Memorise it as a fraction, not a decimal, because cancellation is faster.

A third unit, miles per hour (mph), shows up only rarely in AFCAT, usually inside a comprehension-style item. The conversion is 1 mph ≈ 1.609 km/h, but the paper almost always restates the speed in metric units before asking the question.

Standard speed conversion table

AFCAT loves clean speed values that round to whole-number m/s. Memorising the table below saves five to eight seconds per item.

If the speed in the question is not in this table, multiply by 5/18 directly. The fraction usually cancels neatly because AFCAT chooses speeds that are multiples of 18 or factors that simplify against the 5.

Relative motion — same and opposite direction

Relative speed is the rate at which the distance between two moving objects changes. The rule is short.

The intuition: when two trains move toward each other, both are closing the gap, so their speeds add. When one train chases another in the same direction, only the speed difference closes the gap.

This single rule powers train-crossing problems, two-runner circular-track problems, race problems with a head start, and even pursuit problems where one vehicle leaves later than another.

Train problems

Train problems are the most common TSD format in AFCAT. The trick is to identify what counts as the distance to be covered before applying Distance = Speed × Time.

Train crossing a stationary point or person

A point — a pole, a signal, a stationary person — has no length. The train has to move forward by exactly its own length for its rear to pass the point.

• Distance = length of the train.
• Speed = the train's own speed (converted to m/s).
• Time = length ÷ speed.

Train crossing a platform or bridge

A platform has length. The train has to travel its own length plus the platform's length before its rear clears the far end of the platform.

• Distance = length of train + length of platform (or bridge or tunnel).
• Speed = the train's own speed in m/s.

Two trains crossing each other

When two trains pass, the relative motion rule applies and the distance is the sum of both lengths because each train must clear the other end-to-end.

• Opposite directions: relative speed = sum of speeds; distance = sum of lengths.
• Same direction (overtaking): relative speed = larger speed − smaller speed; distance = sum of lengths.

Train overtaking a slower train

This is the same-direction case above. The faster train approaches at the speed difference and must cover both lengths combined to overtake.

Average speed

Average speed is total distance divided by total time. It is almost never the simple arithmetic mean of the individual speeds.

The most asked sub-case is the round-trip: a vehicle travels from A to B at one speed and back at another. Equal distances both ways, so use the harmonic mean.

Time-and-distance with reduced or increased speed

A class of problems gives a fixed distance and tells you that changing the speed changes the time. The fastest setup is to write two equations for the same distance.

Pattern: If a person increases their speed by x km/h, they reach y minutes earlier; if they reduce by x km/h, they reach y minutes later.

Let the original speed be S km/h and the distance be D km. The original time is D/S hours.

• At speed S + x, time = D / (S + x). Difference = D/S − D/(S + x) = y/60 hours.
• At speed S − x, time = D / (S − x). Difference = D/(S − x) − D/S = y/60 hours.

Cross-multiply and solve. Most AFCAT items of this type give two of the three unknowns (speed, distance, time difference) and ask for the third.

A useful shortcut: if the ratio of new speed to old speed is a : b, then the ratio of new time to old time is b : a, because time is inversely proportional to speed for fixed distance.

Boats and streams

A boat in a flowing river has two speeds — its own speed in still water, and the speed the river adds or subtracts. AFCAT items use only three combinations.

Let b be the speed of the boat in still water and s be the speed of the stream.

• Downstream speed = b + s — the stream pushes the boat forward.
• Upstream speed = b − s — the boat fights the stream.
• Still water: the stream is absent, so the boat's speed is just b.

Recovering b and s from downstream and upstream speeds

If downstream speed is D and upstream speed is U, then:

• b = (D + U) / 2 — boat speed is the average of the two.
• s = (D − U) / 2 — stream speed is half the difference.

Ratio shortcut for equal-distance up-and-down trips

If the boat covers the same distance downstream in time t_d and upstream in time t_u, then:

b / s = (t_u + t_d) / (t_u − t_d)

This avoids solving simultaneous equations. AFCAT frequently asks for the ratio of boat speed to stream speed directly, in which case this shortcut wins.

Circular tracks

Two runners on a circular track of length L metres meet repeatedly. The frequency depends on direction.

Two runners, same direction

The faster runner gains L metres on the slower runner each time they meet. Time between meetings = L / (faster speed − slower speed).

Two runners, opposite direction

Each meeting closes a gap of L metres at the combined speed. Time between meetings = L / (faster speed + slower speed).

Meeting at the starting point

Both runners return to the start when each has completed a whole number of laps. Time for each runner to complete one lap is L/speed. They meet at the start at the LCM of those two lap times.

Three runners on a circular track

Compute the time between meetings for each pair using the same direction or opposite direction rule. All three meet together at the LCM of the three pairwise meeting times. They meet at the starting point at the LCM of the three individual lap times.

Escalator problems

An escalator carries a person at its own speed while the person may also walk in the same or opposite direction. Treat the escalator and the walker as two motions to be combined.

• Let the escalator have N visible steps when stationary.
• Let the escalator's speed be e steps per second, and the walker's speed be w steps per second.
• When the walker climbs up a moving up-escalator, effective speed = e + w; steps taken by the walker = w × time; total steps = N.
• When the walker walks down a moving up-escalator, effective speed = e − w (if e > w); steps taken by walker = w × time.

The standard AFCAT version: a person counts the number of steps they take on a moving escalator at two different walking speeds, and the question asks the total number of visible steps. Set up two equations using N = (e + w) × t and the relation "steps taken by walker = w × t" for both scenarios, then solve simultaneously.

Race problems

Race problems use a short vocabulary that must be read precisely.

The standard solution uses the ratio of speeds equal to the ratio of distances covered in the same time. If A beats B by x metres in a race of L metres, then in the time A covers L, B covers L − x. So the ratio of A's speed to B's speed is L : (L − x).

For head-start problems, set up the equation so that the head start exactly cancels the speed advantage at the finish line.

Trap patterns AFCAT uses

• Unit confusion: the speed is given in km/h and the length in metres; candidates who do not convert lose the mark. Always check units before writing the equation.
• Arithmetic mean instead of harmonic mean: equal-distance round-trip problems where the wrong-answer choice is the simple average of the two speeds. The right formula is 2ab / (a + b).
• Forgetting to add the train's own length: in train-crossing-a-platform problems, the distance includes both the train and the platform. In train-crossing-a-pole problems, the distance is only the train.
• Treating the relative speed wrongly: two trains in the same direction use the difference of speeds, not the sum. Same with two runners on a track.
• Boat-stream sign error: downstream is b + s, upstream is b − s. Some candidates flip the signs under time pressure.
• Head-start misreading: head start in metres is a distance advantage, head start in seconds is a time advantage. The two require different equations.
• Average-speed for unequal distances: if the two legs are not equal, the harmonic mean formula does not apply — go back to total distance over total time.

Time budget and pacing

Allocate sixty to ninety seconds per TSD item in the actual paper.

• 0–10 seconds: read the stem, identify the sub-type (train, average, boat, race, circular, escalator), check units.
• 10–20 seconds: convert km/h to m/s if needed; write the formula relevant to the sub-type.
• 20–60 seconds: plug numbers, simplify, pick the answer.
• 60–90 seconds: sanity check — does the answer have the right order of magnitude? A train at 72 km/h covering a 500 m crossing should take roughly 25 seconds, not 4 minutes.

If you cannot identify the sub-type inside ten seconds, mark and move on. Coming back to a single TSD item with a fresh eye is faster than wrestling with it under accumulated time pressure.

Worked AFCAT-style examples

Exam-day strategy

1. Convert km/h to m/s by multiplying by 5/18 the instant a train, platform length or seconds value appears. Mixing units is the single most common error.
2. Memorise the conversion table for 36, 54, 72, 90 and 108 km/h. These five speeds cover roughly four out of five AFCAT TSD items.
3. For train-crossing problems, ask first: is the obstacle stationary and lengthless (a pole), stationary with length (a platform), or moving (another train)? The distance formula follows automatically.
4. Use the harmonic mean 2ab/(a+b) for equal-distance round trips; use the arithmetic mean only when equal time is spent at each speed.
5. For boats and streams, write downstream as b + s and upstream as b − s on the rough sheet before plugging numbers. The ratio shortcut b/s = (t_u + t_d)/(t_u − t_d) wins when only the ratio is asked.
6. For circular tracks, distinguish 'meet anywhere' (pairwise meeting formula) from 'meet at the starting point' (LCM of lap times). Read the question twice.
7. Aim for sixty to ninety seconds per item. If the sub-type is not clear in ten seconds, mark and move on.

Frequently asked questions

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