GCSE Maths: Speed, distance and time (Foundation)
Part of the GCSE Maths units and compound measures chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Speed, distance and time
A compound measure combines two different units. Speed is the most common one: it tells you how far something travels in one unit of time. If a car goes 60 kilometres in 1 hour its speed is 60 kilometres per hour, written 60 km/h. Speed is measured in km/h, mph (miles per hour) or m/s (metres per second).
Three quantities are linked: distance, speed and time. If you know any two you can find the third. This only works for a constant (steady) speed, or when you are using average speed.
Cover up the letter you want to find. What is left shows the calculation:
- Cover D: you see S next to T, so D = S × T (distance = speed × time)
- Cover S: you see D over T, so S = D ÷ T (speed = distance ÷ time)
- Cover T: you see D over S, so T = D ÷ S (time = distance ÷ speed)
Why does this work? A car at 60 km/h goes 60 km each hour, so in 3 hours it goes 60 × 3 = 180 km. That is distance = speed × time. The other two formulae come from rearranging it.
Matching the units
The units must match. If the speed is in km/h, the time must be in hours and the distance will be in km. If the time is given in minutes, change it to hours first by dividing by 60 (so 30 minutes is 0.5 hours and 45 minutes is 0.75 hours). If you work out a time as a decimal such as 2.5 hours, remember the decimal part is not minutes: 0.5 hours = 30 minutes.
Average speed
In a real journey the speed changes. The average speed is the total distance divided by the total time (including any time standing still). It is the speed you would need to cover the same distance in the same time at a steady pace. Do not find the average of two speeds by adding them and halving, unless the two parts of the journey take the same time.
Changing km/h and m/s
To change 72 km/h into m/s, change the distance to metres (× 1000) and the time to seconds (÷ 3600): 72 × 1000 ÷ 3600 = 20 m/s. The shortcut is: km/h ÷ 3.6 = m/s and m/s × 3.6 = km/h. A speed in m/s is a smaller number than the same speed in km/h.
Key ideas
- S = D ÷ T, D = S × T, T = D ÷ S (use the formula triangle).
- Units must match: km, km/h and hours; m, m/s and seconds.
- Change minutes to hours by dividing by 60. Decimal hours × 60 gives minutes.
- Average speed = total distance ÷ total time.
- km/h ÷ 3.6 = m/s; m/s × 3.6 = km/h.
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (150 km) and the time (2.5 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 150 ÷ 2.5 = 60
- Write the answer with its unit. The unit is km/h.speed = 60 km/h
Check. Work backwards: 60 × 2.5 = 150 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in km per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.45 minutes = 45 ÷ 60 = 0.75 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (48 km/h) and the time (0.75 hours).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 48 × 0.75 = 36
- Write the answer with its unit. The unit is km.distance = 36 km
Check. Work backwards: 36 ÷ 0.75 = 48 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (150 km) and the speed (60 km/h).
We want the time. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.time = distance ÷ speed
- Put the numbers in and calculate. Use the values with matching units.time = 150 ÷ 60 = 2.5
- Change the decimal hours to hours and minutes. The decimal part of an hour is multiplied by 60 to get minutes.0.5 × 60 = 30 minutes
2.5 hours = 2 hours 30 minutes - Write the answer with its unit. The unit is hours.time = 2.5 hours
Check. Work backwards: 60 × 2.5 = 150 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (100 m) and the time (12.5 seconds).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 100 ÷ 12.5 = 8
- Write the answer with its unit. The unit is m/s.speed = 8 m/s
Check. Work backwards: 8 × 12.5 = 100 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Method. Average speed = total distance ÷ total time. Do not average the speeds themselves.
- Add the distances. Average speed uses the total distance.30 + 50 = 80 km
- Add the times. It also uses the total time, including any stops that are part of the journey.20 + 40 = 60 minutes
- Change the total time to hours. We want a speed in per hour, so the time must be in hours: divide minutes by 60.60 ÷ 60 = 1 hour
- Divide total distance by total time. speed = distance ÷ time.80 ÷ 1 = 80
- Write the answer with its unit. average speed = 80 km/h
Check. Work backwards: 80 × 1 = 80 km, which is the total distance. ✓
Watch out. Do not add the speeds and halve them. A slow part that lasts a long time counts for more.
Method. Change kilometres to metres and hours to seconds. 1 km = 1000 m and 1 hour = 3600 seconds.
- Change the kilometres to metres. Metres are smaller than km, so multiply by 1000.72 km = 72 × 1000 = 72,000 m
so 72 km/h = 72,000 metres per hour - Change the hour to seconds. 1 hour = 60 × 60 = 3600 seconds. In one second the car goes 3600 times less far, so divide by 3600.72,000 ÷ 3600 = 20 metres per second
- Write the answer. 72 km/h = 20 m/s
Check. Shortcut: km/h ÷ 3.6 gives m/s: 72 ÷ 3.6 = 20. ✓
Watch out. You are dividing by 3600 for the time but multiplying by 1000 for the distance. Both parts are needed.
Watch out
Check the units before you calculate. 30 minutes is 0.5 hours, not 30 hours and not 0.30. And average speed is not (speed 1 + speed 2) ÷ 2. Always use total distance ÷ total time.Exercise 16E Speed, distance and time
Calculator allowedKey words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (36 km) and the time (3 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 36 ÷ 3 = 12
- Write the answer with its unit. The unit is km/h.speed = 12 km/h
Check. Work backwards: 12 × 3 = 36 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (120 km) and the time (2 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 120 ÷ 2 = 60
- Write the answer with its unit. The unit is km/h.speed = 60 km/h
Check. Work backwards: 60 × 2 = 120 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (90 miles) and the time (1.5 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 90 ÷ 1.5 = 60
- Write the answer with its unit. The unit is mph.speed = 60 mph
Check. Work backwards: 60 × 1.5 = 90 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (200 m) and the time (25 seconds).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 200 ÷ 25 = 8
- Write the answer with its unit. The unit is m/s.speed = 8 m/s
Check. Work backwards: 8 × 25 = 200 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in km per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.30 minutes = 30 ÷ 60 = 0.5 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (15 km) and the time (0.5 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 15 ÷ 0.5 = 30
- Write the answer with its unit. The unit is km/h.speed = 30 km/h
Check. Work backwards: 30 × 0.5 = 15 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (450 km) and the time (5 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 450 ÷ 5 = 90
- Write the answer with its unit. The unit is km/h.speed = 90 km/h
Check. Work backwards: 90 × 5 = 450 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in miles per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.45 minutes = 45 ÷ 60 = 0.75 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (18 miles) and the time (0.75 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 18 ÷ 0.75 = 24
- Write the answer with its unit. The unit is mph.speed = 24 mph
Check. Work backwards: 24 × 0.75 = 18 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (60 km/h) and the time (3 hours).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 60 × 3 = 180
- Write the answer with its unit. The unit is km.distance = 180 km
Check. Work backwards: 180 ÷ 3 = 60 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (40 mph) and the time (2.5 hours).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 40 × 2.5 = 100
- Write the answer with its unit. The unit is miles.distance = 100 miles
Check. Work backwards: 100 ÷ 2.5 = 40 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (8 m/s) and the time (15 seconds).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 8 × 15 = 120
- Write the answer with its unit. The unit is m.distance = 120 m
Check. Work backwards: 120 ÷ 15 = 8 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in km per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.12 minutes = 12 ÷ 60 = 0.2 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (80 km/h) and the time (0.2 hours).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 80 × 0.2 = 16
- Write the answer with its unit. The unit is km.distance = 16 km
Check. Work backwards: 16 ÷ 0.2 = 80 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in km per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.45 minutes = 45 ÷ 60 = 0.75 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (36 km/h) and the time (0.75 hours).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 36 × 0.75 = 27
- Write the answer with its unit. The unit is km.distance = 27 km
Check. Work backwards: 27 ÷ 0.75 = 36 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in km per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.15 minutes = 15 ÷ 60 = 0.25 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (72 km/h) and the time (0.25 hours).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 72 × 0.25 = 18
- Write the answer with its unit. The unit is km.distance = 18 km
Check. Work backwards: 18 ÷ 0.25 = 72 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (180 km) and the speed (60 km/h).
We want the time. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.time = distance ÷ speed
- Put the numbers in and calculate. Use the values with matching units.time = 180 ÷ 60 = 3
- Write the answer with its unit. The unit is hours.time = 3 hours
Check. Work backwards: 60 × 3 = 180 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (150 miles) and the speed (40 mph).
We want the time. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.time = distance ÷ speed
- Put the numbers in and calculate. Use the values with matching units.time = 150 ÷ 40 = 3.75
- Change the decimal hours to hours and minutes. The decimal part of an hour is multiplied by 60 to get minutes.0.75 × 60 = 45 minutes
3.75 hours = 3 hours 45 minutes - Write the answer with its unit. The unit is hours.time = 3.75 hours
Check. Work backwards: 40 × 3.75 = 150 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (100 m) and the speed (8 m/s).
We want the time. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.time = distance ÷ speed
- Put the numbers in and calculate. Use the values with matching units.time = 100 ÷ 8 = 12.5
- Write the answer with its unit. The unit is seconds.time = 12.5 seconds
Check. Work backwards: 8 × 12.5 = 100 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (90 km) and the speed (36 km/h).
We want the time. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.time = distance ÷ speed
- Put the numbers in and calculate. Use the values with matching units.time = 90 ÷ 36 = 2.5
- Change the decimal hours to hours and minutes. The decimal part of an hour is multiplied by 60 to get minutes.0.5 × 60 = 30 minutes
2.5 hours = 2 hours 30 minutes - Write the answer with its unit. The unit is hours.time = 2.5 hours
Check. Work backwards: 36 × 2.5 = 90 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (45 km) and the speed (30 km/h).
We want the time. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.time = distance ÷ speed
- Put the numbers in and calculate. Use the values with matching units.time = 45 ÷ 30 = 1.5
- Change the decimal hours to hours and minutes. The decimal part of an hour is multiplied by 60 to get minutes.0.5 × 60 = 30 minutes
1.5 hours = 1 hours 30 minutes - Write the answer with its unit. The unit is hours.time = 1.5 hours
Check. Work backwards: 30 × 1.5 = 45 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (270 km) and the speed (72 km/h).
We want the time. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.time = distance ÷ speed
- Put the numbers in and calculate. Use the values with matching units.time = 270 ÷ 72 = 3.75
- Change the decimal hours to hours and minutes. The decimal part of an hour is multiplied by 60 to get minutes.0.75 × 60 = 45 minutes
3.75 hours = 3 hours 45 minutes - Write the answer with its unit. The unit is hours.time = 3.75 hours
Check. Work backwards: 72 × 3.75 = 270 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Method. Find the journey time with time = distance ÷ speed, then add it to the start time.
- Find the time. Time = distance ÷ speed.210 ÷ 84 = 2.5 hours
- Change to hours and minutes. 0.5 hours = 30 minutes.2.5 hours = 2 hours 30 minutes
- Add to the start time. Add the hours, then the minutes.09:15 + 2 hours = 11:15, then + 30 minutes = 11:45
Check. Check: 210 km in 2.5 hours is 84 km/h. ✓
Watch out. Do not write 2.5 hours as 2 hours 50 minutes.
Method. Work out the average speed of each car and compare.
- Speed of car A. Speed = distance ÷ time.150 ÷ 2 = 75 km/h
- Speed of car B. Same formula.200 ÷ 2.5 = 80 km/h
- Compare. 80 km/h > 75 km/h, so car B is faster.
Check. In 10 hours car A would go 750 km and car B 800 km. ✓
Method. Try a number example, then explain using time = distance ÷ speed.
- Choose a distance. Keep the distance fixed at 120 km.120 km
- Time at the first speed. Time = distance ÷ speed.120 ÷ 40 = 3 hours
- Time at double the speed. Double 40 km/h is 80 km/h.120 ÷ 80 = 1.5 hours
- Explain. The bottom of the fraction doubled, so the answer halved.Zoe is correct as long as the distance is the same.
Check. 3 ÷ 2 = 1.5 hours. ✓
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (100 m) and the time (9.58 seconds).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 100 ÷ 9.58 = 10.4384
10.4384 = 10.44 to 2 d.p. - Write the answer with its unit. The unit is m/s.speed = 10.44 m/s
Check. Work backwards: 10.44 × 9.58 = 100 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in km per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.144 minutes = 144 ÷ 60 = 2.4 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the speed (15 km/h) and the time (2.4 hours).
We want the distance. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.distance = speed × time
- Put the numbers in and calculate. Use the values with matching units.distance = 15 × 2.4 = 36
- Write the answer with its unit. The unit is km.distance = 36 km
Check. Work backwards: 36 ÷ 2.4 = 15 (the speed). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Method. Use a distance and time you can picture to test each speed.
- Test 5 km/h. That is 5 km in an hour, a brisk walking pace, much too slow for cycling.5 km in 60 minutes
- Test 200 km/h. That is faster than a racing car.200 km in 60 minutes
- Test 20 km/h. 20 km/h is 1 km every 3 minutes: a normal cycling speed.60 ÷ 20 = 3 minutes per km
Check. A fit cyclist covers about 1 km in 2 to 3 minutes, so 20 km/h is realistic. ✓
Method. Find the time in hours, then change to minutes.
- Find the time in hours. Time = distance ÷ speed.12 ÷ 30 = 0.4 hours
- Change to minutes. Hours to minutes: multiply by 60.0.4 × 60 = 24 minutes
Check. At 30 mph the bus goes 0.5 miles each minute, so 12 miles takes 24 minutes. ✓
Method. Find the journey time first, then use speed = distance ÷ time.
- Find the time taken. Count on from 13:40 to 14:00 (20 min), then to 15:00 (1 hour), then to 15:10 (10 min).20 min + 1 hour + 10 min = 1 hour 30 minutes
- Change to hours. 30 minutes = 0.5 hours.1 hour 30 minutes = 1.5 hours
- Divide. Speed = distance ÷ time.36 ÷ 1.5 = 24 mph
Check. 24 mph for 1.5 hours is 36 miles. ✓
Method. Together they close the gap at a combined speed, so add the speeds.
- Add the speeds. Each hour the gap shrinks by 18 km (Amy) plus 12 km (Ben).18 + 12 = 30 km/h
- Divide the gap by the closing speed. Time = distance ÷ speed.90 ÷ 30 = 3 hours
Check. In 3 hours Amy rides 54 km and Ben rides 36 km: 54 + 36 = 90. ✓
Method. Use distance = speed × time with the time you found.
- Write the formula. Distance = speed × time.D = 18 × 3
- Calculate. 18 × 3 = 54 km
Check. 90 − 54 = 36 km, which is how far Ben rode (12 × 3). ✓
Exercise 16F Average speed and changing speed units
Calculator allowedMethod. Change kilometres to metres and hours to seconds. 1 km = 1000 m and 1 hour = 3600 seconds.
- Change the kilometres to metres. Metres are smaller than km, so multiply by 1000.18 km = 18 × 1000 = 18,000 m
so 18 km/h = 18,000 metres per hour - Change the hour to seconds. 1 hour = 60 × 60 = 3600 seconds. In one second the car goes 3600 times less far, so divide by 3600.18,000 ÷ 3600 = 5 metres per second
- Write the answer. 18 km/h = 5 m/s
Check. Shortcut: km/h ÷ 3.6 gives m/s: 18 ÷ 3.6 = 5. ✓
Watch out. You are dividing by 3600 for the time but multiplying by 1000 for the distance. Both parts are needed.
Method. Change kilometres to metres and hours to seconds. 1 km = 1000 m and 1 hour = 3600 seconds.
- Change the kilometres to metres. Metres are smaller than km, so multiply by 1000.36 km = 36 × 1000 = 36,000 m
so 36 km/h = 36,000 metres per hour - Change the hour to seconds. 1 hour = 60 × 60 = 3600 seconds. In one second the car goes 3600 times less far, so divide by 3600.36,000 ÷ 3600 = 10 metres per second
- Write the answer. 36 km/h = 10 m/s
Check. Shortcut: km/h ÷ 3.6 gives m/s: 36 ÷ 3.6 = 10. ✓
Watch out. You are dividing by 3600 for the time but multiplying by 1000 for the distance. Both parts are needed.
Method. Change kilometres to metres and hours to seconds. 1 km = 1000 m and 1 hour = 3600 seconds.
- Change the kilometres to metres. Metres are smaller than km, so multiply by 1000.54 km = 54 × 1000 = 54,000 m
so 54 km/h = 54,000 metres per hour - Change the hour to seconds. 1 hour = 60 × 60 = 3600 seconds. In one second the car goes 3600 times less far, so divide by 3600.54,000 ÷ 3600 = 15 metres per second
- Write the answer. 54 km/h = 15 m/s
Check. Shortcut: km/h ÷ 3.6 gives m/s: 54 ÷ 3.6 = 15. ✓
Watch out. You are dividing by 3600 for the time but multiplying by 1000 for the distance. Both parts are needed.
Method. Change kilometres to metres and hours to seconds. 1 km = 1000 m and 1 hour = 3600 seconds.
- Change the kilometres to metres. Metres are smaller than km, so multiply by 1000.90 km = 90 × 1000 = 90,000 m
so 90 km/h = 90,000 metres per hour - Change the hour to seconds. 1 hour = 60 × 60 = 3600 seconds. In one second the car goes 3600 times less far, so divide by 3600.90,000 ÷ 3600 = 25 metres per second
- Write the answer. 90 km/h = 25 m/s
Check. Shortcut: km/h ÷ 3.6 gives m/s: 90 ÷ 3.6 = 25. ✓
Watch out. You are dividing by 3600 for the time but multiplying by 1000 for the distance. Both parts are needed.
Method. Change kilometres to metres and hours to seconds. 1 km = 1000 m and 1 hour = 3600 seconds.
- Change the kilometres to metres. Metres are smaller than km, so multiply by 1000.144 km = 144 × 1000 = 144,000 m
so 144 km/h = 144,000 metres per hour - Change the hour to seconds. 1 hour = 60 × 60 = 3600 seconds. In one second the car goes 3600 times less far, so divide by 3600.144,000 ÷ 3600 = 40 metres per second
- Write the answer. 144 km/h = 40 m/s
Check. Shortcut: km/h ÷ 3.6 gives m/s: 144 ÷ 3.6 = 40. ✓
Watch out. You are dividing by 3600 for the time but multiplying by 1000 for the distance. Both parts are needed.
Method. Change metres to kilometres and seconds to hours. In one hour there are 3600 seconds, so the object travels 3600 times as far.
- Find the distance in one hour in metres. There are 3600 seconds in 1 hour, so multiply by 3600.5 m/s = 5 × 3600 = 18,000 metres per hour
- Change metres to kilometres. 1000 m = 1 km, so divide by 1000.18,000 ÷ 1000 = 18 km per hour
- Write the answer. 5 m/s = 18 km/h
Check. Shortcut: m/s × 3.6 gives km/h: 5 × 3.6 = 18. ✓
Watch out. m/s to km/h makes the number bigger. A km/h number is always bigger than the m/s number for the same speed.
Method. Change metres to kilometres and seconds to hours. In one hour there are 3600 seconds, so the object travels 3600 times as far.
- Find the distance in one hour in metres. There are 3600 seconds in 1 hour, so multiply by 3600.12 m/s = 12 × 3600 = 43,200 metres per hour
- Change metres to kilometres. 1000 m = 1 km, so divide by 1000.43,200 ÷ 1000 = 43.2 km per hour
- Write the answer. 12 m/s = 43.2 km/h
Check. Shortcut: m/s × 3.6 gives km/h: 12 × 3.6 = 43.2. ✓
Watch out. m/s to km/h makes the number bigger. A km/h number is always bigger than the m/s number for the same speed.
Method. Change metres to kilometres and seconds to hours. In one hour there are 3600 seconds, so the object travels 3600 times as far.
- Find the distance in one hour in metres. There are 3600 seconds in 1 hour, so multiply by 3600.25 m/s = 25 × 3600 = 90,000 metres per hour
- Change metres to kilometres. 1000 m = 1 km, so divide by 1000.90,000 ÷ 1000 = 90 km per hour
- Write the answer. 25 m/s = 90 km/h
Check. Shortcut: m/s × 3.6 gives km/h: 25 × 3.6 = 90. ✓
Watch out. m/s to km/h makes the number bigger. A km/h number is always bigger than the m/s number for the same speed.
Method. Change metres to kilometres and seconds to hours. In one hour there are 3600 seconds, so the object travels 3600 times as far.
- Find the distance in one hour in metres. There are 3600 seconds in 1 hour, so multiply by 3600.40 m/s = 40 × 3600 = 144,000 metres per hour
- Change metres to kilometres. 1000 m = 1 km, so divide by 1000.144,000 ÷ 1000 = 144 km per hour
- Write the answer. 40 m/s = 144 km/h
Check. Shortcut: m/s × 3.6 gives km/h: 40 × 3.6 = 144. ✓
Watch out. m/s to km/h makes the number bigger. A km/h number is always bigger than the m/s number for the same speed.
Method. Average speed = total distance ÷ total time. Do not average the speeds themselves.
- Add the distances. Average speed uses the total distance.24 + 36 = 60 km
- Add the times. It also uses the total time, including any stops that are part of the journey.30 + 30 = 60 minutes
- Change the total time to hours. We want a speed in per hour, so the time must be in hours: divide minutes by 60.60 ÷ 60 = 1 hour
- Divide total distance by total time. speed = distance ÷ time.60 ÷ 1 = 60
- Write the answer with its unit. average speed = 60 km/h
Check. Work backwards: 60 × 1 = 60 km, which is the total distance. ✓
Watch out. Do not add the speeds and halve them. A slow part that lasts a long time counts for more.
Method. Average speed = total distance ÷ total time. Do not average the speeds themselves.
- Add the distances. Average speed uses the total distance.20 + 30 = 50 km
- Add the times. It also uses the total time, including any stops that are part of the journey.24 + 36 = 60 minutes
- Change the total time to hours. We want a speed in per hour, so the time must be in hours: divide minutes by 60.60 ÷ 60 = 1 hour
- Divide total distance by total time. speed = distance ÷ time.50 ÷ 1 = 50
- Write the answer with its unit. average speed = 50 km/h
Check. Work backwards: 50 × 1 = 50 km, which is the total distance. ✓
Watch out. Do not add the speeds and halve them. A slow part that lasts a long time counts for more.
Method. Average speed = total distance ÷ total time. Do not average the speeds themselves.
- Add the distances. Average speed uses the total distance.30 + 10 = 40 km
- Add the times. It also uses the total time, including any stops that are part of the journey.45 + 15 = 60 minutes
- Change the total time to hours. We want a speed in per hour, so the time must be in hours: divide minutes by 60.60 ÷ 60 = 1 hour
- Divide total distance by total time. speed = distance ÷ time.40 ÷ 1 = 40
- Write the answer with its unit. average speed = 40 km/h
Check. Work backwards: 40 × 1 = 40 km, which is the total distance. ✓
Watch out. Do not add the speeds and halve them. A slow part that lasts a long time counts for more.
Method. Average speed = total distance ÷ total time. Do not average the speeds themselves.
- Add the distances. Average speed uses the total distance.45 + 25 = 70 km
- Add the times. It also uses the total time, including any stops that are part of the journey.30 + 30 = 60 minutes
- Change the total time to hours. We want a speed in per hour, so the time must be in hours: divide minutes by 60.60 ÷ 60 = 1 hour
- Divide total distance by total time. speed = distance ÷ time.70 ÷ 1 = 70
- Write the answer with its unit. average speed = 70 km/h
Check. Work backwards: 70 × 1 = 70 km, which is the total distance. ✓
Watch out. Do not add the speeds and halve them. A slow part that lasts a long time counts for more.
Method. Use the whole time (including the stop): 09:00 to 11:30 is 2.5 hours.
- Find the total time. 09:00 to 11:30 is 2 hours 30 minutes.2 hours 30 minutes = 2.5 hours
- Divide. Speed = distance ÷ time.135 ÷ 2.5 = 54 mph
Check. 54 × 2.5 = 135. ✓
Method. Take the 30 minute stop away from the time first.
- Find the moving time. The coach is stopped for 0.5 hours.2.5 − 0.5 = 2 hours
- Divide. Speed = distance ÷ time.135 ÷ 2 = 67.5 mph
Check. 67.5 × 2 = 135. ✓
Watch out. Average speed for the whole journey includes stops; the speed while moving does not.
Method. Find each time, add them, then divide the total distance by the total time.
- Time for the first part. Time = distance ÷ speed.40 ÷ 50 = 0.8 hours
- Time for the second part. Same formula.60 ÷ 120 = 0.5 hours
- Total time and total distance. Add them.time: 0.8 + 0.5 = 1.3 hours
distance: 40 + 60 = 100 km - Divide. Average speed = total distance ÷ total time.100 ÷ 1.3 = 76.9231
= 76.9 km/h to 1 d.p.
Check. The simple average (50 + 120) ÷ 2 = 85 km/h is wrong because more time is spent at the slow speed. Our answer 76.9 is closer to 50. ✓
Watch out. Never average two speeds by adding and halving unless the two parts take the same time.
Method. Pick a distance for the one-way trip (any number works), find both times, then use total distance ÷ total time.
- Choose a distance. Choose 120 km one way because it divides by both 60 and 40.One way = 120 km, round trip = 240 km
- Find both times. Time = distance ÷ speed.Out: 120 ÷ 60 = 2 hours
Back: 120 ÷ 40 = 3 hours - Find the average speed. Total distance ÷ total time.total time = 2 + 3 = 5 hours
240 ÷ 5 = 48 km/h - Conclude. 48 km/h is not 50 km/h, so Ravi is wrong.
Check. Any other distance gives the same answer, because the distance cancels. ✓
Method. Change both speeds to the same unit. Using km/h avoids a recurring decimal.
- Change 10 m/s to km/h. m/s to km/h: multiply by 3.6.10 × 3.6 = 36 km/h
- Compare. Both are now in km/h.36 km/h < 40 km/h, so the car is faster.
Check. In m/s: 40 ÷ 3.6 = 11.1 m/s, which is more than 10 m/s. ✓
Method. Use distance = speed × time, then change metres to kilometres.
- Calculate the distance in metres. Distance = speed × time. Light is so fast we can ignore its time.340 × 6 = 2040 m
- Change to km. Metres to kilometres: divide by 1000.2040 ÷ 1000 = 2.04 km
Check. In 1 second sound goes 340 m, so about 3 seconds per km. 6 seconds is about 2 km. ✓
Method. Find the speed in m/s, change to km/h, and compare with the limit.
- Find the speed in m/s. Speed = distance ÷ time.600 ÷ 40 = 15 m/s
- Change to km/h. m/s to km/h: multiply by 3.6.15 × 3.6 = 54 km/h
- Compare. 54 km/h > 50 km/h, so the car is over the limit.
Check. 50 km/h is 13.9 m/s and 15 m/s is faster. ✓
Method. Change metres to kilometres and seconds to hours. In one hour there are 3600 seconds, so the object travels 3600 times as far.
- Find the distance in one hour in metres. There are 3600 seconds in 1 hour, so multiply by 3600.4 m/s = 4 × 3600 = 14,400 metres per hour
- Change metres to kilometres. 1000 m = 1 km, so divide by 1000.14,400 ÷ 1000 = 14.4 km per hour
- Write the answer. 4 m/s = 14.4 km/h
Check. Shortcut: m/s × 3.6 gives km/h: 4 × 3.6 = 14.4. ✓
Watch out. m/s to km/h makes the number bigger. A km/h number is always bigger than the m/s number for the same speed.
Key words. Speed tells you how far something travels in one unit of time. Average speed is the total distance divided by the total time.
Method. Use the formula triangle for distance, speed and time: distance = speed × time. Cover what you want to find to see the calculation.
- Change the time to hours. The speed is in km per hour, so the time must be in hours. There are 60 minutes in 1 hour, so divide by 60.24 minutes = 24 ÷ 60 = 0.4 hours
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the distance (5 km) and the time (0.4 hours).
We want the speed. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.speed = distance ÷ time
- Put the numbers in and calculate. Use the values with matching units.speed = 5 ÷ 0.4 = 12.5
- Write the answer with its unit. The unit is km/h.speed = 12.5 km/h
Check. Work backwards: 12.5 × 0.4 = 5 (the distance). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Method. Time = distance ÷ speed, then change to minutes.
- Divide. Use the speed from part (a).10 ÷ 12.5 = 0.8 hours
- Change to minutes. Hours to minutes: multiply by 60.0.8 × 60 = 48 minutes
Check. 10 km is double 5 km, so the time is double 24 minutes = 48 minutes. ✓