Speed, Distance, Time and Work-Rate Problems
What you must understand
Core relations: distance = speed × time, so speed = distance ÷ time and time = distance ÷ speed. For work-rate, if a worker completes a job in t hours, the rate is 1/t jobs per hour; combined rates add. Average speed over a journey is total distance ÷ total time, not the mean of the speeds.
Method
Keep units consistent: convert minutes to hours or metres to kilometres before dividing. For two-part journeys, compute distance and time for each leg, sum them, then divide. For work problems, express each worker's rate as a fraction of the job per unit time, add rates for workers acting together, and invert to get the time.
Traps
- Averaging speeds arithmetically instead of using total distance over total time.
- Mixing minutes and hours.
- For combined work, adding times instead of adding rates.
- Forgetting that a return journey may have different distances or speeds each way.
- Assuming rate is constant when the question describes changing speed.
Worked example
A cyclist rides 30 km at 15 km/h, then 30 km at 10 km/h. Total distance = 60 km. Times: 2 hours and 3 hours, total 5 hours. Average speed = 60 ÷ 5 = 12 km/h, not the arithmetic mean of 12.5 km/h. The unequal times are why the two differ.