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Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
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Time & Work
CAT'26 QUANT CHEATSHEET
Every time & work formula and CAT shortcut you need for CAT 2026 — on one page.
Time & Work runs on a single conversion: turn every worker, machine, or pipe into a daily rate, and almost every question becomes an addition or subtraction problem. CAT rarely tests the basic together-work formula in isolation — it hides inside efficiency comparisons, leaking pipes, wage splits, and workers who alternate days instead of working side by side. This sheet lays out every formula you need for the CAT quant section: work and rate basics, the together-work and LCM methods, efficiency ratios, pipes and cisterns, wages, and mixed worker-type conversions, 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-work-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Time & Work: every formula you need
1Work = Rate × Time
Work done depends on speed and time spent.
Work = Rate × Time
Example: A works at 4 units/day for 6 days → 4×6 = 24 units.
CAT Hack: If any two of Work, Rate and Time are known, the third can always be found directly.
2Efficiency ↔ Time
Higher efficiency always means less time to finish.
Efficiency ∝ 1/Time
Example: A takes 12 days, B takes 18 → efficiency ratio = 18:12 = 3:2.
CAT Hack: Always invert the time ratio to get the efficiency ratio, and vice versa.
3Individual Work Rate
The fraction of the job one worker completes per day.
Rate = 1 ÷ Time
Example: A finishes a job in 20 days → rate = 1/20 work/day.
CAT Hack: Convert every worker into a rate before combining them — times don't add, rates do.
4Together Work Formula
The combined time for two workers acting together.
T = (xy) ÷ (x+y)
Example: A takes 12 days, B takes 18 → (12×18)/(12+18) = 216/30 = 7.2 days.
5Men × Days Concept
Total work stays fixed, so men and days trade off inversely.
M₁ × D₁ = M₂ × D₂
Example: 20 workers × 15 days = 30 workers × 10 days.
6LCM Method
Set the total work to the LCM of the individual times for clean unit rates.
Total Work = LCM of individual times
Example: 12 & 18 days, LCM=36 → A=3, B=2 units/day, together=5 → 36/5 = 7.2 days.
CAT Favourite: Cross-checks the Together Work Formula exactly — both give 7.2 days here.
7Efficiency Increase
A percentage rise in efficiency shrinks the completion time by less than that percentage.
Time Decrease % = [Increase % ÷ (100+Increase %)] × 100
Example: 25% efficiency increase → (25/125)×100 = 20% time decrease.
Common Mistake: A 25% efficiency increase does not mean a 25% time decrease — always use the ratio formula.
8Remaining Work
What's left is simply the complement of what's done.
Remaining Work = 100% − Completed Work
Example: 70% completed → remaining = 30%.
9Partial Work
The fraction of the job finished after working only part of the total time.
Work Done = Days Worked ÷ Total Days
Example: 5 days out of a 20-day job → 5/20 = 25% completed.
CAT Hack: Convert directly into a percentage whenever possible — it's faster than carrying the fraction through.
10Work & Wages
Wages split in exactly the same ratio as the work contributed.
Wages ∝ Work Done
Example: work ratio 3:2, ₹5000 to split → ₹3000 : ₹2000.
11Men–Women–Children Conversion
Convert every worker type to one common unit before combining.
e.g. 2W = 3M ⇒ 1W = 1.5M
Example: if 2 women = 3 men in output, then 1 woman = 1.5 men for any calculation.
Common Mistake: Never mix worker types before converting — combine only after everyone is in the same unit.
12Pipes & Cisterns
A leak works against the filling pipe, so rates subtract.
Net Rate = Filling Rate − Leakage Rate = (1/x) − (1/y)
Example: fills in 8 hrs, leaks empty in 24 hrs → 1/8−1/24 = 1/12 → tank fills in 12 hrs.
13Alternate Day Problems
Workers take turns on a repeating cycle rather than working together.
cycle: A → Day 1, B → Day 2, repeat
Example: A and B alternate days on a job → track work done per full 2-day cycle, then handle the remainder.
CAT exam shortcuts, traps & revision
14
CAT Exam Shortcuts
- Convert every worker into a rate (1/time) before combining anything
- Invert a time ratio to get the efficiency ratio, and vice versa
- M₁D₁ = M₂D₂ for a fixed total amount of work
- Net pipe rate = filling rate − leakage rate; a leak always subtracts
- Convert partial work directly into a percentage rather than a raw fraction
- Convert mixed worker types (men/women/children) to one common unit before combining
15
Most Common CAT Traps
- Confusing rate with time — rates add when working together, times do not.
- Assuming a percentage efficiency increase gives the same percentage time decrease.
- Forgetting to subtract the leakage rate when a pipe problem includes a leak.
- Mixing worker types (men, women, children) before converting to a common unit.
- Treating an alternate-day (cyclical) problem as if both workers worked every day together.
16
30-Second Revision Box
- Work = Rate × Time; Rate = 1/Time
- Efficiency ∝ 1/Time
- Time Together = (x×y)/(x+y)
- M₁D₁ = M₂D₂
- Wages ∝ Work done
- Net Pipe Rate = (1/x) − (1/y)
This topic rewards converting to rates early over juggling raw times — once every worker and pipe is a daily fraction, combining them is just arithmetic. Drill this sheet until the rate conversion and the efficiency-ratio inversion are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Ratio, Proportion & Variation cheat sheet, since inverse variation is exactly the efficiency-time relationship here. 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.
