Rate and speed questions appear in almost every E-Math Paper 1, usually as short 2 to 4 mark items on unit conversion or average speed. The same ideas come back in the Sec 3 and Sec 4 distance-time and speed-time graph questions, where average speed is again found by dividing the total distance by the total time.
The whole topic rests on three skills: reading a rate as an amount per one unit of something else, converting units cleanly without losing track of what is being multiplied or divided, and keeping the total distance and the total time as two separate running totals for the whole journey.
What these notes cover
- •Rate and average rate
- •Using a known rate
- •Best-buy comparisons
- •Speed, distance and time
- •Converting between km/h and m/s
- •Converting minutes, hours and seconds
- •Average speed and multi-stage journeys
- •The distance-speed-time table
- •Meeting and overtaking
- •Worked examples
- •Common mistakes and exam technique
- •Answering explain and justify parts
Key definitions
- Rate
- A comparison of two quantities of different kinds, such as dollars and kilograms or litres and minutes, expressed per one unit of the second quantity. Its unit contains the word per or a slash, e.g. $/kg, litres/min, km/h.
- Ratio
- A comparison of quantities of the same kind. Unlike a rate, a ratio has no units.
- Average rate
- The total quantity divided by the total time (or total units). It ignores any changes in the rate and looks only at the totals.
- Average speed
- The total distance travelled divided by the total time taken for the entire journey, including any time spent resting or stopping. It is not the mean of the individual speeds.
- Head-start distance
- How far the object in front is already ahead when the faster object starts. If a head-start time is given instead, head-start distance = speed of the object in front × head-start time.
Key formulas
Average rate
The quantity after the word per goes in the denominator. Rearranged: total quantity = rate × time, and time = total quantity ÷ rate.
Speed, distance and time
Rearranges to distance = speed × time and time = distance ÷ speed. Make units consistent first: km with h gives km/h, m with s gives m/s. Convert minutes to hours by dividing by 60.
Average speed
Total time includes every rest or stop. Never add the stage speeds and divide by the number of stages.
Converting km/h to m/s
Comes from 1 km/h = 1000 m ÷ 3600 s. To go from m/s to km/h, multiply by 18 ÷ 5. For the same speed, the m/s number is always the smaller one.
Time to meet
For two objects moving towards each other from the same starting moment; both close the gap, so it shrinks at the combined rate.
Time to overtake
For two objects moving in the same direction with the slower one ahead. The time is measured from the moment the faster object starts.
Worked example
A van travels 42 km from Tampines to Woodlands at a speed of 63 km/h. (a) Express 63 km/h in m/s. (b) Find the time taken for the journey from Tampines to Woodlands, in minutes. (c) The van then travels a further 26 km in 27 minutes 30 seconds. Find the average speed of the van for the whole journey in km/h, correct to 3 significant figures.
- 1(a) Change kilometres to metres and hours to seconds: 63 km/h = 63 × 1000 m ÷ 3600 s = 63 × (5 ÷ 18) m/s = 17.5 m/s.
- 2(b) The speed is in km/h and the distance is in km, so the time comes out in hours first: time taken = 42 ÷ 63 = 2/3 h.
- 3(b) Change hours to minutes by multiplying by 60: 2/3 × 60 = 40 min.
- 4(c) Total distance = 42 + 26 = 68 km.
- 5(c) 30 s is half a minute, so 27 min 30 s = 27.5 min, and total time = 40 + 27.5 = 67.5 min = 67.5 ÷ 60 h = 1.125 h.
- 6(c) Average speed = total distance ÷ total time = 68 ÷ 1.125 = 60.444… = 60.4 km/h (3 s.f.).
Watch out: Dividing 68 by 67.5 gives a speed in km per minute, not km/h, so the total time must be in hours first. Also, 27 min 30 s is 27.5 min, not 27.30 min, because seconds are out of 60, not 100.
Common mistakes
Averaging the speeds
Wrong: A cyclist rides 30 km at 15 km/h, then 30 km at 10 km/h. Average speed = (15 + 10) ÷ 2 = 12.5 km/h.
Right: Find each stage's time: 30 ÷ 15 = 2 h and 30 ÷ 10 = 3 h. Average speed = (30 + 30) ÷ (2 + 3) = 60 ÷ 5 = 12 km/h.
Confusing decimal hours with hours and minutes
Wrong: 1.35 h is read as 1 h 35 min.
Right: Multiply only the decimal part by 60: 0.35 × 60 = 21 min, so 1.35 h = 1 h 21 min.
Converting km/h to m/s in the wrong direction
Wrong: 72 km/h = 72 × (18 ÷ 5) = 259.2 m/s.
Right: 72 km/h = 72 × 1000 m ÷ 3600 s = 72 × (5 ÷ 18) = 20 m/s. The m/s number is always smaller than the km/h number.
Questions students ask
How do you find the average speed for a journey with two stages?
Find the time for each stage using time = distance ÷ speed, then add the distances and add the times separately. Divide the total distance by the total time. Never add the speeds and divide by the number of stages.
How do you convert km/h to m/s?
Multiply by 5/18, because 1 km/h = 1000 m ÷ 3600 s = 5/18 m/s. For example, 72 km/h = 72 × 5/18 = 20 m/s. To go from m/s back to km/h, multiply by 18/5.
Does rest time count when working out average speed?
Yes. Total time runs from the moment the journey starts to the moment it ends and includes every stop, even though no distance is covered during a rest. A rest adds to the total time but not the total distance, so it always lowers the average speed.
How do you find the time for two objects to meet or for one to overtake the other?
When two objects move towards each other, time to meet = distance apart ÷ sum of speeds. When they move in the same direction, time to overtake = head-start distance ÷ difference of speeds, measured from when the faster object starts. Mixing up sum and difference is the most common error in this type of question.
In the full PDF
- •Two further fully worked exam-style examples: an algebraic multi-stage journey with a show-that part, and a meeting and overtaking pair
- •The step-by-step distance-speed-time table method, including how to add rest rows
- •The remaining three common mistakes: mixing units in one formula, leaving out rest time, and rounding in the middle of the working
- •Unit-rate and best-buy illustrations with the full justify-with-working answer style
- •Exam technique habits and a model answer for explain and justify parts
- •A one-page summary of the five key ideas to carry into every rate and speed question