Time, Speed, Distance & Time-and-Work: The Rate Formula That Cracks Both
One idea -- Rate x Time = Total -- powers trains, average speed, relative speed, and pipes & cisterns. Learn the 5/18 shortcut, the average-speed trap, and the add-the-rates method to lock in easy RRB NTPC Maths marks.
By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026
Part of Railway Exams (RRB) prep🧠 The auto-rickshaw meter
Picture an auto meter ticking as you ride. How far you go depends on just two things: how fast the auto moves (speed) and how long you sit in it (time). Speed x Time = Distance. That one sentence is the whole chapter. Time-Speed-Distance (TSD) and Time-and-Work look like two separate topics, but both are really the same idea: RATE x TIME = TOTAL. Master the rate idea once, and trains, boats, pipes and painters all fall in line. Padho, dost -- this is one of the highest-scoring, most predictable areas in RRB NTPC Maths.
In NTPC CBT-1 you face 100 questions in 90 minutes: 40 General Awareness, 30 Maths, 30 Reasoning, with +1 for a correct answer and -1/3 for a wrong one. That is roughly 54 seconds per question. TSD and Time-and-Work together hand you 4 to 6 near-guaranteed Maths marks IF you know the formulas cold and stay calm. Let us build that speed.
Part 1: The core Speed-Time-Distance triangle
📝 Worked example: average speed
Question: A man goes to a town at 60 km/h and returns along the same road at 40 km/h. Find his average speed for the whole trip.
Trap answer: (60 + 40)/2 = 50 km/h. WRONG, because he spends more time on the slower leg.
Correct formula (equal distance both ways): 2xy / (x + y).
= (2 x 60 x 40) / (60 + 40)
= 4800 / 100
= 48 km/h. That is the answer.
Part 2: Relative speed, trains and platforms
When two objects move, we combine their speeds. Same direction: SUBTRACT the speeds (they pull apart slowly). Opposite directions: ADD the speeds (they close in fast). A train question is just a distance-covered question where the distance includes the length of the train itself.
| Situation | Distance covered | Speed to use |
|---|---|---|
| Train crosses a pole / standing man | Length of train | Train's own speed |
| Train crosses a platform / bridge | Train length + platform length | Train's own speed |
| Two trains, opposite directions | Sum of both lengths | Sum of speeds |
| Two trains, same direction | Sum of both lengths | Difference of speeds |
📝 Worked example: train crossing a platform
Question: A 240 m long train runs at 54 km/h. How long does it take to cross a 360 m platform?
Step 1 -- Convert speed: 54 x 5/18 = 15 m/s.
Step 2 -- Total distance = train + platform = 240 + 360 = 600 m.
Step 3 -- Time = Distance / Speed = 600 / 15.
Answer = 40 seconds.
Part 3: Time and Work (and Pipes & Cisterns)
Same rate idea, new costume. If A finishes a job in 10 days, then in ONE day A does 1/10 of the work. That 1/10 is A's rate. To combine workers, simply ADD their one-day rates. Pipes and cisterns work the same way: a filling pipe has a positive rate, and a draining or leaking pipe has a NEGATIVE rate, so you subtract it.
Method: A and B working together
- 1A does the job in 10 days, so A's 1-day work = 1/10.
- 2B does the job in 15 days, so B's 1-day work = 1/15.
- 3Add the rates: 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6.
- 4Together they do 1/6 of the work per day.
- 5Flip it: total time = 6 days. Done.
60-second revision card
- ✓km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5.
- ✓Average speed (equal distance): 2xy/(x+y), never (x+y)/2.
- ✓Same direction: subtract speeds. Opposite direction: add speeds.
- ✓Train + platform = add both lengths for the distance.
- ✓Time & Work: add one-day rates (1/a + 1/b), then flip for the combined time.
- ✓Pipes: filling is +rate, leaking/emptying is -rate. The net rate decides the time.
⚡ Quick check
A can finish a piece of work in 12 days and B can finish the same work in 6 days. Working together, how many days will they take?
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