GCSE · Maths · AQA · Spec 8300 · Foundation

Converting between standard and compound units

A car travels 17 metres every second. How far is that in a whole hour, and how fast is that in kilometres per hour?

Maths · Units and proportion

One journey, written two ways
0 min10 min20 min30 min40 min50 min60 mindrag the bar to change how long the journey lasts →

Journey time: 1/60. Time 1 min. Same time in seconds 60 s. Distance in metres 1020 m. Same distance in km 1.02 km. Speed 17 m/s

Time1 minSame time in seconds60 sDistance in metres1020 mSame distance in km1.02 kmSpeed17 m/s

A car travels at a steady 17 metres every second. Drag the bar to choose how long the journey lasts, from 0 to 60 minutes. Look at what the readouts do: when the time is multiplied by something, the distance is multiplied by exactly the same thing, and the speed never moves.

Watch out: The 60 belongs to time (seconds, minutes, hours). Turning metres into kilometres is a separate fact: divide the metres by 1000.

Maths · Standard units

Which multiplier does each pair of units use?

Pick a pair of units, then choose the multiplier that turns the bigger unit into the smaller one.

Still to sort

× 10 (0)

10 of the smaller unit make 1 of the bigger.

Where the line is: Only one of these pairs steps by 10.

× 60 (0)

60 of the smaller unit make 1 of the bigger.

Where the line is: Hours, minutes and seconds are the odd ones out: each step is 60, so they never follow the metric pattern.

× 100 (0)

100 of the smaller unit make 1 of the bigger.

× 1000 (0)

1000 of the smaller unit make 1 of the bigger.

Where the line is: The kilo prefix means 1000, and a millilitre is a thousandth of a litre, so these families share one multiplier.

7 of 7 still to sort.

Before you can scale anything, you need the fact that links the two units. Metric units are built around the metre, the gram and the litre.

Watch out: Time breaks the pattern. Hours to minutes and minutes to seconds use 60, not 100 or 1000.

Maths · Area units

How many square centimetres in one square metre?

A square metre is a square that is 1 metre long and 1 metre wide, and 1 metre is 100 centimetres. How many square centimetres do you think make 1 square metre? Slide to your estimate before you look.

Your estimate

6000 cm²

0 cm²12000 cm²

Watch out: For area, square the length multiplier. For volume, cube it. Using the length multiplier on its own is the classic slip.

Density, kg/m³ to g/cm³

Problem

A material has a density of 240 kilograms per cubic metre (kg/m³). Write its density in grams per cubic centimetre (g/cm³).

Maths · Pressure

Now you choose the steps

Pressure is the force pushing straight onto a surface for each unit of area, so its unit has two parts: newtons (force) per square centimetre or per square metre. A pressure is 0.5 N/cm². Write it in N/m².

  1. The newtons stay the same. Only the area unit changes, from cm² to m², so we need the fact that links those two areas.
  2. missing step
Which line is step 2?

One question, two routes

Convert firstvsCalculate first

A force of 4000 N pushes on a face that measures 50 cm by 40 cm. What is the pressure in N/m²?

Focus

First move

Convert first

Change both lengths to metres: 50 cm = 0.5 m and 40 cm = 0.4 m.

Calculate first

Leave the lengths alone and find the area in cm²: 50 × 40 = 2000 cm².

The insight

Same face, same force. The only choice is whether the units change before the calculation or after it.

Pressure

Convert first

Area = 0.5 × 0.4 = 0.2 m², so pressure = 4000 ÷ 0.2 = 20,000 N/m².

Calculate first

Pressure = 4000 ÷ 2000 = 2 N/cm².

Last move

Convert first

Nothing left to do. The answer is already in N/m².

Calculate first

Convert with 1 m² = 10,000 cm²: 2 × 10,000 = 20,000 N/m².

Answer

Convert first

20,000 N/m²

Calculate first

20,000 N/m²

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Change one part, change its partner by the same multiplier.

What you need to know

  • A unit conversion is direct proportion: scale one part by a multiplier and its partner by the same multiplier.
  • Recall the standard facts first: kilo means 1000, 1 m is 100 cm, 1 cm is 10 mm, and time steps by 60.
  • For area, square the length multiplier; for volume, cube it.
  • A compound unit has two parts (speed, density, pressure), and each part gets its own conversion fact.

The big picture

Every unit conversion is direct proportion: whatever you multiply one part by, you multiply its partner by the same amount. That works for lengths, mass, capacity and time, and for compound units like speed, density, pressure and price. Area and volume use the length multiplier squared and cubed.

Key points

1Converting between units of measure is an example of direct proportion, so a ratio table works: scale both rows by one multiplier.
2Metric units are built on the metre, gram and litre. 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. Grams to kilograms and millilitres to litres divide by 1000. Hours to minutes and minutes to seconds multiply by 60.
3Area units convert by the square of the length multiplier: 1 m² = 10,000 cm², 1 cm² = 100 mm², 1 m² = 1,000,000 mm². Volume units use the cube: 1 m³ = 1,000,000 cm³, 1 cm³ = 1000 mm³, 1 m³ = 1 billion mm³.
4Speed is a compound measure of distance and time. If the time is multiplied or divided by 60, the distance is multiplied or divided by 60 too: 17 m/s = 1020 m/min = 61,200 m/h = 61.2 km/h.
5Metric and imperial speeds use a conversion fact in a ratio table: with 5 miles = 8 km, 70 mph = 112 km/h; with 1 mile = 1.6 km, 45 mph = 72 km/h.
6Density (mass per unit of volume) and pressure (force per unit area) convert one part at a time. 240 kg/m³ = 0.24 g/cm³, and a pressure in N/cm² becomes N/m² using 10,000 cm² = 1 m².
7When lengths are given in a compound-measure question, convert first then calculate, or calculate then convert. Both give the same answer.

Worked example

Problem

Using 5 miles = 8 kilometres, write a speed of 70 miles per hour in kilometres per hour.

⚠ Watch out

Using the length multiplier for area or volume, for example thinking a square metre is 100 square centimetres. It is 100 × 100 = 10,000, and a cubic metre is 100 × 100 × 100 = 1,000,000 cubic centimetres.

🧠

Memory hook

One multiplier, both parts. Area uses it twice (squared); volume uses it three times (cubed).

✓

Check yourself

Without looking back: how many square millimetres make 1 square centimetre, and why is it not 10?

Flashcards

(15)
Which kind of relationship is every unit conversion an example of?
Direct proportion. Two variables are in direct proportion when they have a constant multiplicative relationship, so a ratio table can be used.
Which standard units are metric units built around?
The metre, the gram and the litre.
Metric length facts to recall?
1 km = 1000 m, 1 m = 100 cm and 1 cm = 10 mm. Metres change to kilometres by dividing by 1000.
How do you convert grams to kilograms, and millilitres to litres?
Divide by 1000 each time, because kilo means 1000 and a millilitre is a thousandth of a litre. For example 840 g = 0.84 kg and 6050 ml = 6.05 l.
What multiplier turns hours into minutes, or minutes into seconds?
60 each time: 60 minutes in an hour and 60 seconds in a minute.
How many cm² are in 1 m²? Why?
10,000. 1 m = 100 cm, and area uses the square of the length multiplier: 100 × 100. Also 1 cm² = 100 mm² and 1 m² = 1,000,000 mm².
How many cm³ are in 1 m³?
1,000,000, the cube of 100. Also 1 cm³ = 1000 mm³ and 1 m³ = 1 billion mm³.
Why is speed called a compound measure?
It is made of two units, distance and time. Speed is the distance travelled per unit of time.
The time in a speed is multiplied by 60. What happens to the distance?
It is multiplied by 60 as well. For example 17 m/s is 1020 m/min and 61,200 m/h, which is 61.2 km/h after dividing the metres by 1000.
Convert a speed from miles per hour to kilometres per hour?
Use a conversion fact in a ratio table. With 5 miles = 8 km, 70 mph = 112 km/h. With 1 mile = 1.6 km, 45 mph = 72 km/h.
What is density, and which parts of its unit need converting?
Mass per unit of volume. Convert the mass part and the volume part separately. For example 2.4 g/cm³ = 2400 kg/m³.
What is pressure?
The perpendicular force on a surface per unit area. It can be given in newtons per square centimetre or newtons per square metre.
How do you change N/cm² into N/m²?
Use 10,000 cm² = 1 m², and scale the newtons by the same multiplier as the area.
Lengths are given in a compound-measure question. What are your two routes?
Convert the lengths first and then calculate, or calculate in the given units and then convert. Both give the same answer.
A price is directly proportional to the quantity bought. How does the price change when the quantity does?
It scales by the same multiplier as the quantity.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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