For: Year 5 and Year 6 | Exams: 11+ (not tested in KS2 SATs) | Calculator: not allowed
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📌 KEY FACTS TO KNOW BY HEART
The formulas: distance = speed × time · speed = distance ÷ time · time = distance ÷ speed
Minutes to hours: 15 min = ¼ h = 0.25 h · 30 min = ½ h = 0.5 h · 45 min = ¾ h = 0.75 h · 20 min = ⅓ h · 6 min = 0.1 h
Average speed = total distance ÷ total time (including stops). It is not the average of the speeds.
Distance–time graphs: steeper line = faster; flat line = stopped.
Level 1 – SATs core: distance = speed × time in whole hours
Foundation for the 11+, using SATs-level arithmetic. All the times here are whole hours.
✏️ WORKED EXAMPLE: A car travels at 45 km/h for 3 hours. How far does it go?
Step 1: 45 km/h means the car goes 45 km in every hour.
Step 2: distance = speed × time = 45 × 3 = 135 km
Common mistake: adding the numbers (45 + 3 = 48) or dividing them. In 3 hours the car goes 45 km three times over.
Your turn
Q1.1 A cyclist rides at 12 km/h for 2 hours. How far does she ride?
Show answer
12 × 2 = 24 km
Q1.2 A bus travels at 30 mph for 4 hours. How far does it travel?
Show answer
30 × 4 = 120 miles
Q1.3 A train travels at 80 km/h for 5 hours. How far does it go?
Show answer
80 × 5 = 400 km
Q1.4 A walker covers 15 km in 3 hours at a steady speed. What is her speed?
Show answer
speed = distance ÷ time = 15 ÷ 3 = 5 km/h
Q1.5 A lorry drives 240 km at a steady 60 km/h. How many hours does the journey take?
Show answer
time = distance ÷ speed = 240 ÷ 60 = 4 hours
Q1.6 A plane flies at 650 km/h for 3 hours. How far does it fly?
Show answer
650 × 3 = 1950 km
Q1.7 A car travels 175 miles in 5 hours. What is its average speed?
Show answer
175 ÷ 5 = 35 mph
Q1.8 A boat sails at 18 km/h. How many hours does it take to sail 126 km?
Show answer
126 ÷ 18 = 7 hours
Q1.9 Mia cycles at 14 km/h for 3 hours, then at 10 km/h for 2 more hours. How far does she cycle altogether?
Show answer
14 × 3 = 42 km and 10 × 2 = 20 km. Total: 42 + 20 = 62 km
Q1.10 Car A travels at 55 mph for 4 hours. Car B travels at 70 mph for 3 hours. Which car goes further, and by how many miles?
Show answer
Car A: 55 × 4 = 220 miles. Car B: 70 × 3 = 210 miles. Car A, by 10 miles
Level 2 – SATs stretch: times in minutes, finding speed and time
Still SATs-level arithmetic, but the times are in minutes. Change the minutes into hours first.
✏️ WORKED EXAMPLE: A train travels at 120 km/h for 45 minutes. How far does it go?
Step 1: 45 minutes = 3/4 of an hour.
Step 2: in 1/4 of an hour the train goes 120 ÷ 4 = 30 km.
Step 3: in 3/4 of an hour it goes 3 × 30 = 90 km
Common mistake: working out 120 × 45 or 120 × 0.45. 45 minutes is 0.75 hours, not 0.45 hours.
Your turn
Q2.1 A car travels at 60 km/h for 30 minutes. How far does it go?
Show answer
30 minutes is half an hour. Half of 60 = 30 km
Q2.2 A cyclist rides at 16 km/h for 1 hour 15 minutes. How far does he ride?
Show answer
1 hour: 16 km. 15 minutes (1/4 hour): 4 km. Total: 20 km
Q2.3 Write 2 hours 45 minutes as a number of hours, using a decimal.
Show answer
45 minutes = 3/4 hour = 0.75 hours, so 2.75 hours
Q2.4 A runner covers 6 km in 30 minutes. What is her speed in km/h?
Show answer
In one hour she would run twice as far: 6 × 2 = 12 km/h
Q2.5 A train travels 90 miles in 1½ hours. What is its average speed?
Show answer
1½ hours is 3 half-hours. 90 ÷ 3 = 30 miles per half-hour, so 30 × 2 = 60 mph
Q2.6 A car travels at 60 km/h. How many minutes does it take to drive 25 km?
Show answer
60 km in 60 minutes is 1 km every minute, so 25 km takes 25 minutes
Q2.7 A coach travels at 48 km/h for 20 minutes. How far does it go?
Show answer
20 minutes = 1/3 of an hour. 48 ÷ 3 = 16 km
Q2.8 Leo walks 5 km at 4 km/h. How long does the walk take? Give your answer in hours and minutes.
Show answer
5 ÷ 4 = 1¼ hours = 1 hour 15 minutes (75 minutes)
Q2.9 A bus leaves at 09:40. It travels 75 km at a steady 50 km/h. What time does it arrive?
Show answer
Time = 75 ÷ 50 = 1½ hours. 09:40 + 1 hour 30 minutes = 11:10
Q2.10 A cyclist rides 21 km in 1 hour 45 minutes. What is her average speed in km/h?
Show answer
1 hour 45 minutes is 7 quarter-hours. 21 ÷ 7 = 3 km each quarter-hour, so 3 × 4 = 12 km/h
Level 3 – 11+ standard: multi-step journeys, average speed and distance–time graphs
Typical of grammar-school 11+ papers. Average speed = total distance ÷ total time, including any stops.
✏️ WORKED EXAMPLE: Ben goes on a bike ride. The graph shows his distance from home. Work out his average speed for the whole ride.
Step 1: read the end of the graph: total distance = 30 km.
Step 2: total time = 2.5 hours (this includes the half-hour stop, shown by the flat part).
Step 3: average speed = 30 ÷ 2.5 = 12 km/h
Common mistake: leaving out the stop and working out 30 ÷ 2 = 15 km/h. The flat part still counts as time.
Your turn
Q3.1 A car travels 100 km at 50 km/h, then 120 km at 60 km/h. What is its average speed for the whole journey?
Show answer
Times: 100 ÷ 50 = 2 hours and 120 ÷ 60 = 2 hours. Average speed = 220 ÷ 4 = 55 km/h
Q3.2 Priya walks 3 km in 45 minutes, then walks another 5 km in 1 hour 15 minutes. What is her average speed in km/h?
Show answer
Total distance: 8 km. Total time: 45 + 75 = 120 minutes = 2 hours. 8 ÷ 2 = 4 km/h
Q3.3 A train leaves at 10:50 and arrives at 14:20. It travels 280 km. What is its average speed?
Show answer
Journey time: 3 hours 30 minutes = 3.5 hours. 280 ÷ 3.5 = 80 km/h
Q3.4 A car travels at 72 km/h. How far does it travel in 25 minutes?
Show answer
72 km in 60 minutes is 1.2 km each minute. 1.2 × 25 = 30 km
Q3.5 The graph shows Asha’s walk. What was her speed, in km/h, during the first 30 minutes?
Show answer
She walked 3 km in 30 minutes (half an hour), so 3 × 2 = 6 km/h
Q3.6 The graph shows Tom’s bike ride to a lake and back home. What was his speed on the way home?
Show answer
He rode 24 km back home between 3 hours and 4.5 hours, which is 1.5 hours. 24 ÷ 1.5 = 16 km/h
Q3.7 A coach travels 150 miles. It does the first 90 miles at 45 mph and the rest at 60 mph. What is its average speed for the whole journey?
Show answer
First part: 90 ÷ 45 = 2 hours. Second part: 60 miles at 60 mph = 1 hour. Average: 150 ÷ 3 = 50 mph
Q3.8 Change 54 km/h into metres per second.
Show answer
54 km/h = 54000 m in 3600 seconds. 54000 ÷ 3600 = 15 m/s
Q3.9 The graph shows a delivery van’s journey from its depot. During which part of the journey was the van travelling fastest, and what was its speed then?
Show answer
The steepest part is from 1.5 to 2 hours: 40 km in half an hour. Between 1.5 and 2 hours, at 80 km/h
Q3.10 Sophie leaves home at 13:15 and drives 45 km to her gran’s house at 60 km/h. She stays for 1 hour 30 minutes, then drives home the same way at 45 km/h. What time does she get home?
Show answer
There: 45 ÷ 60 = 3/4 hour = 45 minutes. Stay: 90 minutes. Home: 45 ÷ 45 = 1 hour. Total: 3 hours 15 minutes. 13:15 + 3:15 = 16:30
Level 4 – 11+ selective: meeting, catching up, trains and tunnels
The style of selective independent-school papers. Ask yourself: how fast is the gap between them changing?
✏️ WORKED EXAMPLE: Two towns are 150 km apart. At 09:00 one car leaves the first town at 70 km/h, and another leaves the second town at 50 km/h. They drive towards each other on the same road. How long is it until they meet?
Step 1: they are driving towards each other, so the gap closes by 70 + 50 = 120 km every hour.
Step 2: time = 150 ÷ 120 = 1¼ hours = 1 hour 15 minutes (they meet at 10:15)
Check: in 1¼ hours the cars go 87.5 km and 62.5 km, and 87.5 + 62.5 = 150 ✓
Common mistake: subtracting the speeds (70 − 50 = 20 km/h). You subtract only when one is chasing the other in the same direction.
Your turn
Q4.1 Two cyclists start 54 km apart and ride towards each other, one at 10 km/h and the other at 8 km/h. How long is it before they meet?
Show answer
The gap closes at 10 + 8 = 18 km/h. 54 ÷ 18 = 3 hours
Q4.2 A car drives 60 km to the coast at 60 km/h and comes back the same way at 40 km/h. What is its average speed for the whole round trip?
Show answer
There: 1 hour. Back: 60 ÷ 40 = 1.5 hours. Average: 120 ÷ 2.5 = 48 km/h (not 50 km/h: the car spends longer at the slower speed)
Q4.3 A train 200 m long travels at 20 m/s. It goes through a tunnel 600 m long. How many seconds is it from the front of the train entering the tunnel to the back of the train leaving it?
Show answer
The front must travel the tunnel plus the train’s own length: 600 + 200 = 800 m. 800 ÷ 20 = 40 seconds
Q4.4 Dan sets off walking at 5 km/h. 30 minutes later his sister sets off from the same place on a bike at 15 km/h, along the same path. How many minutes after she sets off does she catch up with him?
Show answer
Dan’s head start: 5 × ½ = 2.5 km. She gains 15 − 5 = 10 km every hour. 2.5 ÷ 10 = ¼ hour = 15 minutes
Q4.5 A train 150 m long passes a signal post in 6 seconds. What is its speed in km/h?
Show answer
150 ÷ 6 = 25 m/s. In one hour: 25 × 3600 = 90000 m = 90 km/h
Q4.6 A lorry arrives at 15:10 after a 210 km journey. It drove at an average of 60 km/h and also stopped for a 20-minute break. What time did it leave?
Show answer
Driving time: 210 ÷ 60 = 3½ hours. Add the break: 3 hours 50 minutes. Work back from 15:10: 11:20
Q4.7 A car travels for 2 hours at 50 km/h. How fast must it go for the next hour so that its average speed for all 3 hours is 60 km/h?
Show answer
For an average of 60 km/h over 3 hours it needs 180 km. So far: 100 km. It needs 80 km in the last hour: 80 km/h
Q4.8 The graph shows Zara’s run along a path from home. 10 minutes after Zara sets off, her brother leaves home and cycles after her along the same path at 18 km/h. How far from home are they when he catches up with her?
Show answer
Zara runs 6 km in 40 minutes, which is 9 km/h. After 10 minutes she is 1.5 km ahead. He gains 18 − 9 = 9 km/h, so he catches her after 1.5 ÷ 9 = 1/6 hour (10 minutes). He has cycled 18 × 1/6 = 3 km from home (Zara is still running then).
Q4.9 Two trains, each 150 m long, travel towards each other on parallel tracks at 20 m/s and 30 m/s. How many seconds is it from when their fronts meet until their backs have passed each other?
Show answer
Together they must cover both lengths: 150 + 150 = 300 m. They close at 20 + 30 = 50 m/s. 300 ÷ 50 = 6 seconds
Q4.10 Kai cycles from home to the park at 12 km/h and walks back the same way at 4 km/h. The two trips take 2 hours altogether. How far is it from his home to the park?
Show answer
Each km takes 5 minutes there (1/12 hour) and 15 minutes back (1/4 hour), so 20 minutes for each km of the route. 2 hours = 120 minutes, and 120 ÷ 20 = 6 km. Check: ½ hour + 1½ hours = 2 hours ✓