Rates, Speed, Distance and Time
Rate questions combine three quantities - rate, quantity and time - and the difficulty in EPSO items is almost always unit conversion or averaging rather than the formula itself.
Core rules
- speed = distance / time; rearranged, distance = speed × time and time = distance / speed.
- Average speed for a whole journey = total distance / total time. It is not the mean of the two speeds unless the times, or the distances, are equal.
- Opposing motion (approaching, or two pipes filling one tank): the rates add.
- Same-direction motion (catching up): the rates subtract.
- Unit conversion: km/h to m/s divide by 3.6; m/s to km/h multiply by 3.6.
- Time arithmetic: 1.2 hours is 1 h 12 min, not 1 h 20 min. Convert the decimal part by multiplying by 60.
Worked example
Two cyclists are 60 km apart and ride towards each other at 20 km/h and 30 km/h. When do they meet, and how far has each ridden?
Closing speed = 20 + 30 = 50 km/h. Time = 60/50 = 1.2 h = 1 h 12 min. The first covers 20 × 1.2 = 24 km; the second 30 × 1.2 = 36 km. Together 60 km, which checks.
Contrast the averaging trap: a train runs 120 km at 80 km/h and returns at 120 km/h. Total distance 240 km, total time 1.5 + 1.0 = 2.5 h, so average speed = 96 km/h. The naive (80 + 120)/2 = 100 km/h is wrong because the slower leg occupies more time.
Production and consumption rates
The same structure governs units per hour, litres per minute and pages per day. Multiply the rate by the number of parallel units - machines, workers, pumps - before multiplying by time.
Traps
- Averaging speeds instead of dividing total distance by total time.
- Adding rates when the objects move in the same direction.
- Converting minutes to decimal hours incorrectly (45 min = 0.75 h, not 0.45 h).
- Ignoring a head start, which has already consumed distance before the closing rate applies.