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Time & Work
Every CAT Time & Work formula on one page — rates, efficiency ratios, the together-work and LCM methods, pipes and cisterns, and wages — each with a worked example.
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Time, Speed & Distance
CAT'26 QUANT CHEATSHEET
Every time, speed & distance formula you need for CAT 2026 — on one page.
Time, Speed & Distance is one of CAT’s highest-frequency topics, and almost every question is a variation on the same SDT triangle wearing a different costume: trains crossing platforms, boats fighting a current, or two travellers closing the gap between them. The habit that saves the most time is spotting which relative-speed case applies before touching the algebra. This sheet lays out every formula you need for the CAT quant section: speed basics and unit conversion, average speed, relative speed in both directions, trains crossing poles, platforms and each other, boats and streams, and the delay and speed-change shortcuts examiners keep reusing, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-time-speed-distance-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Time, Speed & Distance: every formula you need
1Speed = Distance ÷ Time
How much distance is covered per unit time.
Speed = Distance÷Time | Distance = Speed×Time | Time = Distance÷Speed
Example: 120 km in 2 hrs → 120÷2 = 60 km/hr.
CAT Hack: If stuck, write the SDT triangle and cover the variable you need to find.
2Unit Conversions
Convert cleanly between m/s and km/hr.
km/hr → m/s: ×5/18 | m/s → km/hr: ×18/5
Example: 72 km/hr → 72×5/18 = 20 m/s.
3Average Speed
Total distance over total time, not an average of the speeds.
Average Speed = Total Distance ÷ Total Time
Example: 60km at 30km/hr + 60km at 60km/hr → 120÷3 = 40 km/hr.
4Equal Distance Case
A shortcut for two equal-distance legs at different speeds.
Average Speed = 2ab ÷ (a+b)
Example: 30 km/hr and 60 km/hr → 2×30×60÷90 = 40 km/hr.
CAT Favourite: One of the most repeated formulas on CAT — instantly recognisable whenever a distance is split into two equal legs.
5Relative Speed (Opposite Directions)
Used when two objects move apart or toward each other.
Relative Speed = a + b
Example: 40 km/hr and 60 km/hr moving apart → 100 km/hr.
6Relative Speed (Same Direction)
Used in overtaking questions, where objects move the same way.
Relative Speed = |a − b|
Example: 60 km/hr and 40 km/hr, same direction → 20 km/hr.
Common Mistake: Students often add the speeds here — same direction means subtract, not add.
7Trains Crossing a Pole
A train only needs to cover its own length to pass a point.
Time = Train Length ÷ Speed
Example: 200m train at 20 m/s → 200÷20 = 10 sec.
CAT Hack: A pole (or any point object) has zero length, so only the train's length counts.
8Trains Crossing a Platform
The train must cover its own length plus the platform's.
Time = (Train + Platform Length) ÷ Speed
Example: 200m train, 100m platform, 20 m/s → 300÷20 = 15 sec.
9Two Trains Crossing
Both trains' lengths count, and their relative speed applies.
Time = (L₁+L₂) ÷ Relative Speed
Example: 200m and 300m trains, relative speed 25 m/s → 500÷25 = 20 sec.
10Overtaking
The faster object must fully pass the slower one.
Time = Total Length ÷ Relative Speed
Example: 200m and 300m trains, same direction, relative speed 10 m/s → 500÷10 = 50 sec.
Common Mistake: Same direction means the relative speed is the difference, not the sum, of the two speeds.
11Boats & Streams
The current helps you downstream and works against you upstream.
Downstream = B+S | Upstream = B−S
Example: boat 12 km/hr, stream 3 km/hr → downstream=15, upstream=9 km/hr.
CAT Hack: Water always helps you downstream and always opposes you upstream.
12Boats Formulas
Recover the boat and stream speeds from the two effective speeds.
Boat Speed = (D+U)÷2 | Stream Speed = (D−U)÷2
Example: downstream=15, upstream=9 → boat=(15+9)/2=12, stream=(15−9)/2=3 km/hr.
13Time Gain & Loss
Speed and time always move inversely for a fixed distance.
Speed₁ × Time₁ = Speed₂ × Time₂
Example: speed doubles → time becomes half, for the same distance covered.
CAT Hack: Faster speed always means proportionately less time, for a fixed distance.
14Delay Problems
The time difference between arriving early or late at two speeds.
Difference in Time = Distance × (1/S₁ − 1/S₂)
Example: 60km at 30 km/hr vs 40 km/hr → 60×(1/30−1/40) = 60×(1/120) = 0.5 hr (30 min).
Common Mistake: Convert every speed and distance to the same units before substituting — mixing km/hr with metres is a common slip.
15Speed Increase
A percentage rise in speed cuts travel time by a smaller percentage.
% Decrease in Time = [x ÷ (100+x)] × 100
Example: speed up 25% → 25÷125×100 = 20% time decrease.
CAT exam shortcuts, traps & revision
16
CAT Exam Shortcuts
- Write the Speed-Distance-Time triangle and cover the unknown
- km/hr → m/s: ×5/18; m/s → km/hr: ×18/5
- Equal-distance average speed = 2ab ÷ (a+b)
- Opposite directions: add speeds; same direction (overtaking): subtract speeds
- Pole crossing = train length ÷ speed; platform crossing = (train+platform) ÷ speed
- Downstream = B+S; Upstream = B−S
17
Most Common CAT Traps
- Averaging speeds directly instead of using total distance ÷ total time.
- Using the wrong relative speed (adding when the objects move the same way).
- Forgetting to convert units before substituting into a formula.
- Adding speeds instead of subtracting them in overtaking problems.
- Ignoring the platform length when a train crosses a platform.
- Mixing km/hr with metres without converting first.
18
30-Second Revision Box
- Speed = Distance ÷ Time
- Average Speed = Total Distance ÷ Total Time
- Equal-distance average speed = 2ab ÷ (a+b)
- Relative speed: opposite = a+b, same direction = a−b
- Pole crossing = Length ÷ Speed; platform crossing = Total Length ÷ Speed
- Downstream = B+S; Upstream = B−S
This topic rewards recognising the setup over grinding through algebra — spot whether the objects move toward, away, or alongside each other, and the relative speed falls out immediately. Drill this sheet until the equal-distance shortcut and the train-crossing formulas are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Time & Work cheat sheet, since both lean on the same rate-and-ratio reasoning. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
