Time-Speed-Distance & Time-Work

Snapshot

• Time-Speed-Distance (TSD) and Time & Work are two faces of the same idea: a rate acting over time produces an output (distance, or work done). Master the rate idea and trains, boats, pipes and "A and B together" all become one method.
• TSD's core is the triangle Distance = Speed × Time; Work's core is Work = Rate × Time, where one person's rate is 1 ÷ (days they take alone).
• This guide covers unit conversion, average and relative speed, trains, boats & streams, and time–work including pipes & cisterns — with worked examples for each.
• Exam reality: +5 / −1. Set the rate, watch the units, and these are quick marks.

Part 1 — Speed, Distance, Time

Distance = Speed × Time. Rearrange as needed: Speed = D ÷ T, Time = D ÷ S. The one conversion you must own:

km/h → m/s: multiply by 5/18. m/s → km/h: multiply by 18/5. (72 km/h = 72 × 5/18 = 20 m/s.)

When distance is constant, time and speed are inversely proportional — double the speed, halve the time. So "if speed increases by 25%, time falls by 20%" (the base-flip from the percentage guide).

Part 2 — Average & Relative Speed

Average speed = total distance ÷ total time — never the simple average of the speeds. For a journey covering equal distances at speeds a and b, average speed is the harmonic mean 2ab ÷ (a + b).

Relative speed is how fast one object closes on another:

Moving toward each other, speeds add; moving in the same direction, speeds subtract. Time to meet = (gap between them) ÷ (relative speed).

Part 3 — Trains, Boats & Streams

Trains. A train crossing a pole travels its own length; crossing a platform travels (length of train + length of platform). Two trains crossing each other travel the sum of their lengths at their relative speed.

Boats & streams. With stream speed s and boat speed b in still water: downstream speed = b + s, upstream speed = b − s. Reverse-engineering: b = ½(downstream + upstream), s = ½(downstream − upstream).

Part 4 — Time & Work

One worker who finishes a job in a days has a rate of 1/a per day. Working together, rates add: A (1/a) and B (1/b) together do 1/a + 1/b per day, finishing in ab ÷ (a + b) days.

Pipes & cisterns is the same idea — a filling pipe has a positive rate, a leak/outlet a negative one; add them. Men–days–hours: for two situations doing the same kind of work, M₁ D₁ H₁ ÷ W₁ = M₂ D₂ H₂ ÷ W₂ (the compound-proportion formula).

Part 5 — Speed techniques

1. Convert with 5/18 (or 18/5) on sight — most train/speed sums hide a unit change.
2. Use inverse proportion for constant distance — set up speed₁ × time₁ = speed₂ × time₂.
3. For "together", add rates, not days — the classic error is averaging the days.
4. Take total work as the LCM of the days to avoid fractions (job = LCM(a, b) units).
5. Average speed for equal distances is harmonic — 2ab/(a+b), never (a+b)/2.
6. For a leak, subtract its rate — a tank "filled in x but leak empties in y" fills in xy/(y−x).

Part 6 — Worked examples

1. Conversion. A train at 90 km/h crosses a 200 m pole-to-platform... first, 90 km/h = 25 m/s.

2. Average speed. Goes at 40, returns at 60 (equal distance). Average = 2×40×60/(100) = 48 km/h.

3. Relative (opposite). Two trains 120 m and 130 m long run toward each other at 40 and 50 km/h. Time to cross? rel speed = 90 km/h = 25 m/s; total length 250 m; time = 250 ÷ 25 = 10 s.

4. Train + platform. A 150 m train at 20 m/s crosses a 250 m platform in (150+250)/20 = 20 s.

5. Boats. A boat does 16 km downstream and 8 km upstream, each in 2 h. Speeds: down 8, up 4 → boat = 6 km/h, stream = 2 km/h.

6. Work together. A does a job in 12 days, B in 18. Together? 12×18/(12+18) = 216/30 = 7.2 days.

7. Work with LCM. A (10 days) and B (15 days): job = LCM 30 units; A = 3/day, B = 2/day; together 5/day → 6 days.

8. Pipes. Pipe fills a tank in 6 h, leak empties in 9 h. Net fill = 6×9/(9−6) = 54/3 = 18 h.

Part 7 — Common traps

• Unit mismatch — convert km/h ↔ m/s before using a length in metres.
• Average speed is harmonic for equal distances — not the mean of the speeds.
• "Together" adds rates, not days.
• Pole vs platform — a pole adds only the train's length; a platform adds both lengths.
• Upstream is b − s, not s − b — the boat must beat the current.

Part 8 — How to use this page

Lock the 5/18 conversion, the harmonic-mean rule, and "add rates for together", re-solve the eight examples closed-book, then attempt the practice set and the timed test.

One-line revision: D = S × T, convert with 5/18, average speed for equal distances is 2ab/(a+b), relative speed adds when opposite and subtracts when same-way, and for work add rates (or use the LCM of the days).