Speed, Distance and Time

Speed = distance ÷ time: minutes into hours, finding distance and time, arrival times, average speed for a whole journey, and changing units.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 80 real past-paper questions below

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The Speed Formula

Speed tells you how far you go in each hour. Every question here uses one formula, arranged three ways.

Formula

speed = \frac{distance}{time}

Memory trick

Cover what you want in the triangle D over S × T: distance = speed × time, time = distance ÷ speed.

Key idea

Puzzle: two cyclists 50 km apart ride towards each other at 20 km/h and 30 km/h. A bird flies to and fro between them at 40 km/h until they meet. How far does it fly? Answer: 40 km. The gap closes at 50 km/h, so they meet after 1 hour.

Minutes Into Hours

Most marks are lost here. Change minutes into a decimal of an hour before you divide.

Worked example

1
$2 h 24 min$ — The coach takes this long to travel 186 km. The minutes must become hours before we divide.
2
$24 \div 60 = 0.4$ — There are 60 minutes in an hour, so 24 minutes is 24 ÷ 60 of an hour, which is 0.4 h.
3
$2 + 0.4 = 2.4 h$ — Add the whole hours back on to get the time in hours.
4
$speed = \frac{186}{2.4}$ — Speed = distance ÷ time, with the time in hours.
5
$186 \div 2.4 = 77.5$ — Do the division.
6
$77.5 km/h$ — Give the unit. Distance in km and time in hours give km/h.

Common mistake

Using 2.24 hours. 24 minutes is 0.4 of an hour, not 0.24.

Key idea

Write the time in hours on its own line. It earns a mark for working even if the final division slips.

Finding the Distance

Distance = speed × time. Change the time to hours first.

Worked example

1
$8 h 45 min$ — The lorry drives for this long at 64 km/h. Change the minutes to hours first.
2
$45 \div 60 = 0.75$ — 45 minutes is 45 ÷ 60 of an hour.
3
$8 + 0.75 = 8.75 h$ — Add the whole hours back on.
4
$distance = 64 \times 8.75$ — Distance = speed × time.
5
$64 \times 8.75 = 560$ — Do the multiplication.
6
$560 km$ — Give the unit. Speed in km/h and time in hours give km.

Finding the Time, Then the Clock

Some questions want the arrival time. Find the time taken, then add it to the start.

Worked example

1
$time = \frac{100}{80} = 1.25 h$ — Time = distance ÷ speed, for 100 km at 80 km/h.
2
$1.25 = 1 h + 0.25 h$ — Split off the whole hour. The 0.25 is a part of an hour.
3
$0.25 \times 60 = 15 min$ — Change the decimal back to minutes by multiplying by 60.
4
$10 : 50 + 1 h = 11 : 50$ — Start at 10:50 am and add the hour.
5
$11 : 50 + 15 min = 12 : 05 pm$ — 10 minutes reach 12:00, and 5 more minutes make 12:05 pm.

Common mistake

Reading 1.25 hours as 1 hour 25 minutes. The 0.25 is a quarter of an hour: 0.25 × 60 = 15 minutes.

To find when to leave, work backwards: take the travel time away from the arrival time.

Worked example

1
$time = \frac{90}{60} = 1.5 h$ — A 90 km journey at 60 km/h. Time = distance ÷ speed.
2
$1.5 h = 1 h 30 min$ — 0.5 of an hour is 30 minutes.
3
$3 : 30 pm - 1 h = 2 : 30 pm$ — To arrive at 3:30 pm, go back the whole hour.
4
$2 : 30 pm - 30 min = 2 : 00 pm$ — Then go back the 30 minutes. Leave at 2:00 pm.

Average Speed for a Whole Journey

For a journey in parts, average speed is total distance ÷ total time. You cannot average the speeds.

Worked example

1
$180 km in 2.5 h, 90 km in 1.5 h$ — The two parts of the journey.
2
$180 + 90 = 270 km$ — Add the distances to get the total distance.
3
$2.5 + 1.5 = 4 h$ — Add the times to get the total time.
4
$270 \div 4 = 67.5$ — Average speed = total distance ÷ total time.
5
$67.5 km/h$ — Give the unit.

Common mistake

Averaging the two speeds, 72 and 60, to get 66. The slower part takes longer, so it counts for more.

A rest stop counts as time, but no distance is travelled.

Worked example

1
$24 km in 1.5 h, stop 0.5 h, 16 km in 2 h$ — A cycle ride with a rest in the middle.
2
$24 + 16 = 40 km$ — Add the distances. The stop adds none.
3
$1.5 + 0.5 + 2 = 4 h$ — Add all the times, including the stop.
4
$40 \div 4 = 10 km/h$ — Average speed = total distance ÷ total time.

Changing Units

If the answer must be in seconds or metres, work in the given units, then convert at the end.

Worked example

1
$90 \div 36000 = 0.0025 h$ — Time = distance ÷ speed, for 90 km at 36 000 km/h.
2
$60 \times 60 = 3600 s$ — One hour has 60 minutes and each minute has 60 seconds.
3
$0.0025 \times 3600 = 9 s$ — Change hours into seconds by multiplying by 3600.

To change m/s into km/h, find how far you go in one hour.

Worked example

1
$100 \div 20 = 5 m/s$ — A runner covers 100 m in 20 s. Speed = distance ÷ time.
2
$5 \times 3600 = 18000 m$ — In one hour (3600 s) the runner goes 3600 times as far.
3
$18000 \div 1000 = 18 km$ — 1000 m make 1 km.
4
$18 km/h$ — That is the distance in one hour.

Key idea

Shortcut: m/s to km/h, multiply by 3.6. km/h to m/s, divide by 3.6. For example 72 km/h = 20 m/s, because 72 ÷ 3.6 = 20.

Trains and Bridges (Extended)

A train passing right through a bridge or tunnel must travel its own length as well as the length of the bridge. The back of the train has to clear the far end.

Worked example

1
$200 + 400 = 600 m$ — A 200 m train goes through a 400 m tunnel. The distance is the train plus the tunnel.
2
$72 \times 1000 \div 3600 = 20 m/s$ — The speed is 72 km/h. Change it to m/s so it matches the metres.
3
$time = \frac{600}{20}$ — Time = distance ÷ speed.
4
$600 \div 20 = 30 s$ — Do the division. The answer is in seconds because we used m and m/s.

To find the length of the train, work the other way.

Worked example

1
$54 \times 1000 \div 3600 = 15 m/s$ — A train at 54 km/h takes 20 s to pass completely through a 150 m station. Change the speed to m/s.
2
$15 \times 20 = 300 m$ — Distance = speed × time. This is the whole distance the train travels.
3
$300 - 150 = 150 m$ — The distance is train plus station, so take off the station to leave the train.

Time as an Expression (Extended)

Extended questions give a speed as x and ask for an equation. The formula is the same: time = distance ÷ speed. This uses the algebra from the quadratics lessons.

Worked example

1
$Sam’s time = \frac{24}{x}$ — Sam walks 24 km at x km/h. Time = distance ÷ speed.
2
$Tom’s time = \frac{24}{x - 2}$ — Tom walks the same 24 km at (x − 2) km/h and takes 1 hour longer.
3
$\frac{24}{x - 2} - \frac{24}{x} = 1$ — Tom's time minus Sam's time is 1 hour.
4
$24 x - 24 (x - 2) = x (x - 2)$ — Multiply every term by x(x − 2) to clear the fractions.
5
$24 x - 24 x + 48 = x^{2} - 2 x$ — Expand. The −24 times −2 gives +48.
6
$x^{2} - 2 x - 48 = 0$ — The 24x terms cancel. Move everything to one side.
7
$(x - 8) (x + 6) = 0$ — Factorise. −8 × 6 = −48 and −8 + 6 = −2.
8
$x = 8 or x = - 6$ — Each bracket can be zero.
9
$x = 8$ — A speed cannot be negative, so reject −6. Sam walks at 8 km/h, and Tom at 6 km/h.

Key idea

Check by finding both times: 24 ÷ 8 = 3 hours and 24 ÷ 6 = 4 hours, which is 1 hour longer. If the difference is in minutes, change it to hours first, for example 30 minutes = 0.5 h.

Which Method? Read the Question

Find what is missing, then check the units.

You see	You do
Speed missing	distance ÷ time
Distance missing	speed × time
Time or arrival time	distance ÷ speed, then minutes
Hours and minutes	minutes ÷ 60 first
A journey in parts	total distance ÷ total time
A train through a bridge	distance = train length + bridge length
Speed given as x	time = distance ÷ x, then form an equation

Key idea

Check the unit the question asks for: km/h, m/s, minutes or seconds.

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