TSD, trains, boats and streams, time and work, and pipes and cisterns, with worked examples and practice
Two classic word-problem families that share one habit of mind: convert the rate to a per-unit basis, then add or compare rates. Build on the ratio and average skills in percentage ratio and average.
| Concept | Formula |
|---|---|
| Basic relation | Distance = Speed times Time |
| Speed | Distance / Time |
| Time | Distance / Speed |
| km/h to m/s | multiply by 5/18 |
| m/s to km/h | multiply by 18/5 |
| Average speed (equal distance) | 2ab / (a + b) |
| Relative speed (same direction) | difference of speeds |
| Relative speed (opposite direction) | sum of speeds |
| Situation | Rule |
|---|---|
| Train passing a pole or man | distance = length of train |
| Train passing a platform or bridge | distance = length of train + length of platform |
| Two trains crossing (opposite) | distance = sum of lengths, speed = sum of speeds |
| Two trains crossing (same direction) | distance = sum of lengths, speed = difference of speeds |
| Term | Formula |
|---|---|
| Downstream speed | boat speed + stream speed |
| Upstream speed | boat speed - stream speed |
| Boat speed (still water) | (downstream + upstream) / 2 |
| Stream speed | (downstream - upstream) / 2 |
| Concept | Rule |
|---|---|
| If A does a job in n days | A's one-day work = 1/n |
| Combined rate | add the one-day works |
| Time for A and B together | 1 / (1/a + 1/b) = ab / (a + b) |
| Work, men, days link | (M1 times D1 times H1) / W1 = (M2 times D2 times H2) / W2 |
| Pipes filling | inflow positive, outflow (leak) negative |
The fastest method is the LCM (units of work) method: take the total work as the LCM of the given times, then each worker's daily output is a whole number.
A patrol vehicle moves at 72 km/h. Express this in m/s.
72 times 5/18 = 20 m/s.
A 150 m long train running at 54 km/h crosses a 350 m platform. How long does it take?
Speed = 54 times 5/18 = 15 m/s. Distance = 150 + 350 = 500 m. Time = 500 / 15 = 33.33 seconds (one third of a minute).
Two trains 120 m and 180 m long run towards each other at 40 km/h and 50 km/h. How long to cross?
Relative speed = 40 + 50 = 90 km/h = 90 times 5/18 = 25 m/s. Total distance = 120 + 180 = 300 m. Time = 300 / 25 = 12 seconds.
A boat goes 30 km downstream in 2 hours and the same distance upstream in 3 hours. Find the boat speed in still water and the stream speed.
Downstream speed = 30/2 = 15 km/h. Upstream speed = 30/3 = 10 km/h. Boat speed = (15 + 10)/2 = 12.5 km/h. Stream speed = (15 - 10)/2 = 2.5 km/h.
A can do a job in 12 days and B in 18 days. Working together, how long do they take?
Total work = LCM(12, 18) = 36 units. A does 36/12 = 3 units a day, B does 36/18 = 2 units a day. Together = 5 units a day. Time = 36 / 5 = 7.2 days.
A pipe fills a tank in 6 hours; a drain empties it in 9 hours. If both are open, how long to fill?
Take tank = LCM(6, 9) = 18 units. Fill pipe = 18/6 = 3 units/h. Drain = 18/9 = 2 units/h (negative). Net = 3 - 2 = 1 unit/h. Time = 18 / 1 = 18 hours.