Edexcel GCSE Maths (1MA1), Higher tier · Ratio, proportion and rates of change
Practise Speed, density, pressure and best buys. 4 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Speed, density and pressure are compound measures: each one combines two other measures, such as distance and time. Questions ask you to use the three formulas (and other rates such as pay per hour), change times and units first, and often link to the area or volume of a shape. Working out the best buy is part of this subtopic too.
Best buy: work out the price per 100 g (or per unit) for each option; the lowest is the best buy
Notes
The three formulas
diagram
Speed = distance ÷ time, so distance = speed × time and time = distance ÷ speed.
Density = mass ÷ volume, so mass = density × volume and volume = mass ÷ density.
Pressure = force ÷ area, so force = pressure × area and area = force ÷ pressure.
Cover the one you want: side by side means multiply, one above the other means divide (e.g. S = D ÷ T).
The units come from the formula: km ÷ h gives km/h, g ÷ cm3 gives g/cm3, and N ÷ m2 gives N/m2.
Other rates work the same way: pay per hour, cost per kilogram, litres per minute.
Getting the units right
Match the units before you calculate: a speed in km/h needs a time in hours.
Minutes to hours: divide by 60. 45 minutes = 0.75 hours, but 20 minutes = \(\dfrac{1}{3}\) hour (not 0.2 hours).
Decimal hours to minutes: multiply the decimal part by 60. 2.3 hours = 2 hours 18 minutes.
km/h to m/s: multiply by 1000, then divide by 3600. 72 km/h = 72 000 m per hour = 20 m/s.
For a pressure in N/m2, the area must be in m2: 1 m2 = 10 000 cm2.
Average speed and multi-step problems
Average speed = total distance ÷ total time. Do not find the mean of the speeds.
For a mass from a density, find the volume of the shape first (e.g. \(\pi r^2 h\) for a cylinder), then mass = density × volume.
Mixtures and alloys: density = total mass ÷ total volume. Find the mass or volume of each part first. Grade 6+ only
Best buys and rates of pay
A best buy is the option that gives the most for the money. Work out the price of the same amount for each option, then compare.
Use one unit for every option, e.g. pence per 100 g. The lower price per 100 g is the better buy.
Example: 500 g for £2.40 is 48p per 100 g (240 ÷ 5). 800 g for £3.60 is 45p per 100 g (360 ÷ 8). The larger box is the better buy.
With an offer such as 'buy 2, get the third half price', work out the total cost and the total amount first. Three boxes of 500 g at £2.40 with the third at half price cost £6.00 for 1500 g, which is 40p per 100 g.
A rate of pay is money per hour: pay = rate × hours.
How to answer each type of question
M = method mark, A = answer mark, B = a mark on its own.
Work out speed, distance or time
3 marksGrade 4
Write down the formula you need.
Change the time into hours (or the units the answer needs), e.g. 20 minutes = \(\dfrac{1}{3}\) hour.
Substitute and calculate, then give the units.
Example. A train travels 210 km in 2 hours 20 minutes. Work out the average speed of the train in km/h.
Show the model answerHide the model answer
2 hours 20 minutes = 140 minutes, or \(2\dfrac{1}{3}\) hours (M1) 210 ÷ 140 × 60, or \(210 \div 2\dfrac{1}{3}\) (M1) = 90 km/h (A1)
Use density with the volume of a solid
3 marksGrade 5
Work out the volume of the solid (cuboid, prism or cylinder).
Use mass = density × volume.
Convert to the units asked for, e.g. grams to kilograms.
Example. A solid metal bar is a cuboid measuring 12 cm by 5 cm by 2.5 cm. The metal has a density of 8.4 g/cm3. Work out the mass of the bar in kilograms.
Show the model answerHide the model answer
Volume = 12 × 5 × 2.5 = 150 cm3 (M1) Mass = 8.4 × 150 = 1260 g (M1) = 1.26 kg (A1)
Work out pressure with a change of units
3 marksGrade 5
Change the lengths to metres before finding the area (or change cm2 to m2 by dividing by 10 000).
Use pressure = force ÷ area, rearranged if you need the force or the area.
Give the units, N/m2.
Example. A box rests on a floor. The base of the box is a rectangle measuring 40 cm by 30 cm. The box exerts a force of 180 newtons on the floor. pressure = \(\dfrac{\text{force}}{\text{area}}\) Work out the pressure on the floor in N/m2.
Deal with one unit at a time: first the time, then the distance (or the other way round).
Per second to per hour: multiply by 3600. Metres to kilometres: divide by 1000.
Sense check: the same speed is a bigger number in km/h than in m/s.
Example. A cheetah can run at 30 m/s. Change 30 m/s into km/h.
Show the model answerHide the model answer
30 × 3600 = 108 000 metres per hour (M1) 108 000 ÷ 1000 = 108 km/h (A1)
Find the density of a mixture or alloy
3 marksGrade 6
Find the mass and the volume of each part separately, using mass = density × volume or volume = mass ÷ density.
Add to get the total mass and the total volume.
Density of the mixture = total mass ÷ total volume (never the mean of the two densities).
Example. Metal A has a density of 6 g/cm3. Metal B has a density of 9 g/cm3. 240 g of metal A and 540 g of metal B are mixed to make an alloy. Assume that the volume of the alloy is the total volume of the two metals. Work out the density of the alloy.
Show the model answerHide the model answer
Volume of A = 240 ÷ 6 = 40 cm3 or volume of B = 540 ÷ 9 = 60 cm3 (M1) Total mass = 780 g and total volume = 100 cm3, so density = 780 ÷ 100 (M1) = 7.8 g/cm3 (A1)
Don’t lose marks
Writing 2 hours 20 minutes as 2.2 hours: it is \(2\dfrac{1}{3}\) hours, about 2.33 hours.
Finding the mean of two speeds instead of total distance ÷ total time.
Mixing units, e.g. an area in cm2 with a pressure in N/m2, or grams with kg/m3.
Turning a formula upside down, e.g. volume ÷ mass for density.
Leaving a time as a decimal (3.6 hours) when the question asks for hours and minutes (3 hours 36 minutes).
More tips
Memory tricks
Formula triangles: D over S × T, M over D × V, F over P × A. Cover the one you want to find.
The units tell you the formula: g/cm3 means grams ÷ cm3, so density = mass ÷ volume.
m/s to km/h: multiply by 3.6; km/h to m/s: divide by 3.6.
Sense check: walking is about 5 km/h, a car on a motorway about 110 km/h, and water has a density of 1 g/cm3.
Exam technique
Write the formula before you substitute: it shows your method.
Check the units the answer needs (km/h or m/s, g or kg) and convert at the start or the end.
If the question gives a formula, such as pressure = force ÷ area, use it; learn all three formulas anyway.
Give time answers in the form the question asks for.
What each grade needs
What you need to be able to do, from the first marks up to the top grade.
Grade 3
Use speed = distance ÷ timeFor example, 150 km in 2.5 hours is an average speed of 60 km/h.
Grade 4
Convert between minutes and decimal hoursDivide the minutes by 60, e.g. 1 hour 45 minutes = 1.75 hours.
Grade 4
Calculate density, mass or volumeUse density = mass ÷ volume and its rearrangements, with matching units.
Grade 5
Calculate pressure, force or areaUse pressure = force ÷ area, with the area in m2 for a pressure in N/m2.
Grade 5
Convert compound units such as km/h to m/sConvert one unit at a time, e.g. 72 km/h = 72 000 m per hour = 72 000 ÷ 3600 = 20 m/s.
Grade 5
Find the average speed for a whole journeyDivide the total distance by the total time, including any stops.
Grade 5
Use density with volumes of 3D shapesFind the volume of the solid first, then multiply by the density to get the mass.
Grade 6
Find the density of a mixtureDivide the total mass by the total volume, working out each part separately first.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
Ella runs 5 km in 24 minutes. Jon runs at an average speed of 12 km/h.
Who runs at the greater average speed? You must show your working.[3]
Show the answer and mark scheme
Answer: Ella: her speed is 12.5 km/h, which is more than 12 km/h.
M1 for 24 minutes = 0.4 hours, or \(5 \div 24\) km per minute
A1 for 12.5 km/h (or Jon runs 4.8 km in 24 minutes, or Jon takes 25 minutes for 5 km)
C1 for Ella, from a correct comparison
Worked solution: 24 minutes \(= \frac{24}{60} = 0.4\) hours. Ella’s speed \(= 5 \div 0.4 = 12.5\) km/h. 12.5 km/h > 12 km/h, so Ella is faster.
Question 2Medium3 marks
A car travels 60 km at an average speed of 40 km/h. It then travels a further 60 km at an average speed of 60 km/h. Ali says, ‘The average speed for the whole journey is 50 km/h.’
Show that Ali is wrong.[3]
Show the answer and mark scheme
Answer: Times: 1.5 h and 1 h. Average speed \(= \frac{120}{2.5} = 48\) km/h, not 50 km/h.
M1 for the time of one section: \(60 \div 40 = 1.5\) h or \(60 \div 60 = 1\) h
M1 for total distance ÷ total time: \(120 \div 2.5\)
C1 for 48 km/h with the conclusion that Ali is wrong
Worked solution: First part: \(60 \div 40 = 1.5\) h. Second part: \(60 \div 60 = 1\) h. Average speed \(= \frac{120}{2.5} = 48\) km/h. The car spends longer at the slower speed, so the average is less than 50 km/h.
Question 3Hard4 marks
220 g of liquid X is mixed with 80 g of liquid Y. The density of liquid X is 1.1 g/cm³ and the density of liquid Y is 0.8 g/cm³. Assume that the volume of the mixture is the total volume of the two liquids. Work out the density of the mixture.[4]
Show the answer and mark scheme
Answer: 1 g/cm³
M1 for the volume of X, \(220 \div 1.1 = 200\)
M1 for the volume of Y, \(80 \div 0.8 = 100\)
M1 for \((220 + 80) \div (200 + 100)\)
A1 for 1 cao
Worked solution: Volume of X \(= 220 \div 1.1 = 200\) cm³. Volume of Y \(= 80 \div 0.8 = 100\) cm³. Density \(= \frac{220 + 80}{200 + 100} = \frac{300}{300} = 1\) g/cm³.