R1 - Converting standard and compound units | AQA GCSE Mathematics

A unit change can turn a small number into a large one without changing the quantity. In R1 you will use standard conversions in calculations, then convert speed, pay, prices, density and pressure. You will also write conversions using letters, so the same reasoning works for any value.

Standard units in calculations

A runner travels at $18\,\mathrm{km/h}$ . A cyclist travels at $6\,\mathrm{m/s}$ . Who is faster? Comparing $18$ with $6$ is no help until the units agree. Keep that race in mind while we build a reliable conversion route.

You already know

From N13, $1\,\mathrm{km}=1000\,\mathrm{m}$ , $1\,\mathrm{kg}=1000\,\mathrm{g}$ , and an hour contains $60$ minutes. Smaller units give a larger count of the same quantity.

Write the starting unit and target unit before doing arithmetic. For example, $1.2\,\mathrm{km}+350\,\mathrm{m}=1200\,\mathrm{m}+350\,\mathrm{m}=1550\,\mathrm{m}$ . Similarly, $2.4\,\mathrm{kg}-750\,\mathrm{g}=2400\,\mathrm{g}-750\,\mathrm{g}=1650\,\mathrm{g}$ . Adding the bare numbers first would mix different-sized units.

Time needs particular care because the conversion factor is $60$ , rather than $100$ . A decimal fraction of an hour is a fraction of $60$ minutes.

Worked example 1: a duration in seconds

Question. Write $2$ hours $18$ minutes in seconds. Non-calculator.

Plan. Change the hours to minutes, combine the minutes, then change to seconds.

\begin{aligned} 2\times60+18&=138 &&\text{total minutes}\\ 138\times60&=8280 &&\text{seconds in those minutes} \end{aligned}

Why combine in minutes first? The two parts must use the same unit before they can be added.

Answer. $8280$ seconds.

Check. $8280\div60=138$ minutes, which is $120+18$ minutes.

The next example changes only the target unit. This time the minutes become a fraction of an hour.

Worked example 2: the same duration in decimal hours

Question. Write $2$ hours $18$ minutes as a decimal number of hours. Non-calculator.

Plan. Divide the extra minutes by $60$ , then add the whole hours.

\begin{aligned} 18\div60&=\frac3{10}=0.3 &&\text{fraction of an hour}\\ 2+0.3&=2.3 &&\text{total hours} \end{aligned}

Why divide by $60$ ? Eighteen minutes is $18$ out of the $60$ minutes making one hour.

Answer. $2.3$ hours.

Check. $0.3\times60=18$ minutes.

A decimal time can look like a clock reading, but the digits do different jobs.

Watch out

Wrong: $2$ hours $18$ minutes is $2.18$ hours. Right: $18\div60=0.3$ , so it is $2.3$ hours. In fact, $0.18$ hours is only $10.8$ minutes.

Use this route whenever a rate is quoted per hour but a duration is given in minutes. First practise the time conversion alone.

Area, volume and capacity

The small power on a unit tells you how many times to use the length conversion factor.

For $\mathrm{m}^2$ to $\mathrm{cm}^2$ , use $100^2=10\,000$ . For $\mathrm{m}^3$ to $\mathrm{cm}^3$ , use $100^3=1\,000\,000$ . A litre is $1000\,\mathrm{cm}^3$ , so $1\,\mathrm{m}^3=1000$ litres. These N13 conversions are not on the formulae sheet: learn them.

Picture a $1\,\mathrm{m}$ cube packed with $1\,\mathrm{cm}$ cubes: $100$ along, $100$ across and $100$ layers. There are a million little cubes inside it. That is why a volume conversion can change six digits at once without changing the space being measured.

Worked example 3: an area conversion

Question. A panel has area $0.045\,\mathrm{m}^2$ . Write its area in square centimetres. Non-calculator.

Plan. Multiply by the square of the metre-to-centimetre factor.

\begin{aligned} 100^2&=10\,000 &&\text{area factor}\\ 0.045\times10\,000&=450 &&\text{square centimetres} \end{aligned}

Why square $100$ ? Both length directions change unit.

Answer. $450\,\mathrm{cm}^2$ .

Check. $450\div10\,000=0.045$ , reversing the conversion.

Keep the number the same but change the dimension from area to volume. The factor must change too.

Worked example 4: a volume in litres

Question. A container holds $0.045\,\mathrm{m}^3$ . Write its capacity in litres. Non-calculator.

Plan. Convert to cubic centimetres, then group them into litres.

\begin{aligned} 0.045\times1\,000\,000&=45\,000 &&\text{cubic centimetres}\\ 45\,000\div1000&=45 &&\text{litres} \end{aligned}

Why divide on the second line? A litre contains $1000$ cubic centimetres. Together the two steps multiply the original volume by $1000$ .

Answer. $45$ litres.

Check. $45\div1000=0.045\,\mathrm{m}^3$ .

The number of directions matters more than the size of the written number.

Watch out

Wrong: $0.045\,\mathrm{m}^3=4.5\,\mathrm{cm}^3$ , using the length factor $100$ . Right: cube that factor, giving $45\,000\,\mathrm{cm}^3$ .

You can use the direct cubic-metre-to-litre route once you know why its factor is $1000$ .

Speed, pay and prices

A compound unit joins units together: $\mathrm{m/s}$ means metres per second. Read “per” as “for each one”.

A speed of $54\,\mathrm{km/h}$ means $54\,\mathrm{km}$ for each hour. Change the distance to metres and the hour to seconds. At a constant speed, the distance in one second is the distance in an hour divided by $3600$ .

Changing speed units

For speed values,

\begin{aligned} \mathrm{km/h}\text{ to }\mathrm{m/s}:&\quad \times\frac{1000}{3600}=\div3.6\\ \mathrm{m/s}\text{ to }\mathrm{km/h}:&\quad \times\frac{3600}{1000}=\times3.6 \end{aligned}

Convert the quantity above “per”, then adjust the time unit below it.

Not on the formulae sheet: learn it.

Writing both changes is safer than memorising a direction for $3.6$ .

Worked example 5: kilometres per hour to metres per second

Question. Convert $54\,\mathrm{km/h}$ to metres per second. Non-calculator.

Plan. Work out the metres travelled in an hour, then divide by the seconds in an hour.

\begin{aligned} 54\times1000&=54\,000 &&\text{metres per hour}\\ 54\,000\div3600&=15 &&\text{metres per second} \end{aligned}

Why divide by $3600$ ? One second is much shorter than one hour, so the distance per second is smaller.

Answer. $15\,\mathrm{m/s}$ .

Check. $15\times3600=54\,000\,\mathrm{m}=54\,\mathrm{km}$ .

Now reverse the conversion. The distance unit becomes larger, but the time period also becomes longer.

Worked example 6: metres per second to kilometres per hour

Question. Convert $7.5\,\mathrm{m/s}$ to kilometres per hour. Non-calculator.

Plan. Multiply by $3600$ to get metres per hour, then divide by $1000$ .

\begin{aligned} 7.5\times3600&=27\,000 &&\text{metres per hour}\\ 27\,000\div1000&=27 &&\text{kilometres per hour} \end{aligned}

Why multiply first? One hour contains $3600$ one-second distances.

Answer. $27\,\mathrm{km/h}$ .

Check. $27\div3.6=7.5$ , returning to the starting speed.

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Number

Algebra

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Probability

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R1 - Converting standard and compound units

Standard units in calculations

Area, volume and capacity

Speed, pay and prices

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