Time, speed, and distance shortcuts for competitive exams

Time-speed-distance (TSD) questions are really ratio problems in disguise. Once you stop treating speed, time, and distance as three separate things to solve for and start treating two of them as inversely or directly proportional, most questions collapse into a one-line calculation.

The core relationship: distance stays fixed → speed and time are inversely proportional

If distance is constant, speed and time move in opposite directions — double the speed, halve the time. This single idea replaces most of the algebra in TSD word problems.

Worked example: a train covers a distance in 5 hours at 60 km/h. What speed is needed to cover it in 4 hours?

• Distance = 60 × 5 = 300 km (constant)
• New speed = 300 / 4 = 75 km/h

Unit conversion shortcut: km/h to m/s

To convert km/h to m/s, multiply by 5/18. To go the other way, multiply by 18/5. Memorize the fraction, not a formula — it comes directly from 1000 m / 3600 s reduced to lowest terms.

Relative speed: same direction vs opposite direction

When two objects move toward each other, their speeds add. When they move in the same direction, you use the difference. This one distinction covers almost every "two trains" or "two people" question.

Worked example: two trains, 150 m and 100 m long, moving in opposite directions at 40 km/h and 32 km/h, cross each other. Find the time taken.

• Relative speed = 40 + 32 = 72 km/h = 72 × 5/18 = 20 m/s
• Total distance to cover = sum of lengths = 150 + 100 = 250 m
• Time = 250 / 20 = 12.5 seconds

The two lengths are added because each train must clear the other's entire length — a detail exam questions test specifically to see if you're just plugging into a formula without understanding it.

Same-direction (catching up) shortcut

For a faster object catching up to a slower one with a head start, use the difference in speeds against the gap distance.

Worked example: A starts 5 km ahead of B. A walks at 4 km/h, B walks at 6 km/h in the same direction. How long until B catches A?

• Relative speed = 6 − 4 = 2 km/h
• Time to close a 5 km gap = 5 / 2 = 2.5 hours

Average speed: don't just average the two speeds

The most common trap in TSD questions — when equal distances are covered at two different speeds, the average speed is not the simple average. It's the harmonic mean, which has a clean shortcut form.

Worked example: a car covers a distance at 40 km/h and returns over the same distance at 60 km/h. Find the average speed for the whole trip.

• Average speed = 2 × 40 × 60 / (40 + 60) = 4800 / 100 = 48 km/h

This mirrors the weighted-average idea used for combining two data groups: the slower speed gets more "weight" because the same distance takes longer to cover at it, so it pulls the average toward itself.

Boats and streams: same relative-speed logic, different labels

Boats-and-streams questions are relative speed problems wearing a costume. Downstream speed adds the current; upstream speed subtracts it.

Worked example: a boat's speed in still water is 15 km/h, and the stream flows at 3 km/h. Find the time to cover 54 km downstream.

• Downstream speed = 15 + 3 = 18 km/h
• Time = 54 / 18 = 3 hours

From two known values — say downstream and upstream speed — you can back out the boat's own speed and the stream's speed just as fast: boat speed is the average of the two, stream speed is half the difference.

Time saved / time lost: reduce to a percentage-style setup

"If speed increases by x%, how much time is saved on the same distance?" questions are easiest solved by converting the percentage change in speed directly into a ratio, since distance is fixed.

Worked example: a man usually takes 40 minutes to reach work. One day he increases his speed by 25%. How long does the trip take now?

• New speed is 125% of usual, i.e. ratio 5:4 (usual:new = 4:5)
• Time is inversely proportional to speed, so new time ratio = 5:4 flipped from speed → time ratio = old:new = 5:4
• New time = 40 × (4/5) = 32 minutes

Where this actually helps

TSD questions in bank and competitive exams almost always test whether you understand the underlying proportion, not whether you can plug numbers into a memorized formula. Trains, boats, races, and "increased speed" questions are all the same handful of relationships reused with different labels.