Lesson 6.7 · 35 min

Compound Units — Speed, Density and Rates

Speed, density, pressure and other rates; converting compound units such as km/h to m/s and g/cm³ to kg/m³; and average speed for a journey in stages.

In this lesson you will learn to…

Use speed $= \frac{distance}{time}$ , density $= \frac{mass}{volume}$ and pressure $= \frac{force}{area}$ , rearranged as needed
Change between hours and minutes as decimals, e.g. 2 h 15 min $= 2.25$ h
Convert compound units such as km/h ↔ m/s and g/cm³ ↔ kg/m³
Find the average speed of a journey made in several stages

“Per” means divide

A compound unit is made from two units: km/h is kilometres per hour, $\text{g/cm}^3$ is grams per cubic centimetre. The unit tells you the formula — divide the first quantity by the second.

Measure	Formula	Units
Speed	$\dfrac{\text{distance}}{\text{time}}$	km/h, m/s
Density	$\dfrac{\text{mass}}{\text{volume}}$	$\text{g/cm}^3$ , $\text{kg/m}^3$
Pressure	$\dfrac{\text{force}}{\text{area}}$	$\text{N/m}^2$
Other rates	$\dfrac{\text{amount}}{\text{time}}$ , $\dfrac{\text{cost}}{\text{quantity}}$ …	L/min, baht/kg

Time in hours

Calculators need time as a decimal of an hour. There are 60 minutes in an hour, not 100, so divide the minutes by 60.

Converting compound units

Convert the top and the bottom separately. For km/h to m/s: $1$ km $= 1000$ m and $1$ h $= 3600$ s, so

$1\ \text{km/h} = \frac{1000\ \text{m}}{3600\ \text{s}} = \frac{1}{3.6}\ \text{m/s}$

For density, the bottom is cubed: $1\ \text{m}^3 = 1\,000\,000\ \text{cm}^3$ , while $1$ kg $= 1000$ g. Together, $\text{g/cm}^3 \times 1000 = \text{kg/m}^3$ . Water is $1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$ .

Average speed over a journey

For a journey in stages, add up the total distance and the total time (including any stops), then divide. Averaging the speeds gives the wrong answer whenever the stages take different times.

Key rules and formulas

Speed, density, pressure
$speed = \frac{distance}{time} density = \frac{mass}{volume} pressure = \frac{force}{area}$
Each is "something per something". The unit tells you the formula: km/h is km ÷ h; $g/cm^{3}$ is g ÷ $cm^{3}$ ; $N/m^{2}$ is N ÷ $m^{2}$ . Rearrange as usual: distance $=$ speed × time, mass $=$ density × volume.
Time as a decimal
Minutes $\div 60$ gives hours: $2$ h $24$ min $= 2.4$ h. Hours $\times 60$ gives minutes: $0.35$ h $= 21$ min.
Converting compound units
$km/h \times 1000 \div 3600 m/s g/cm^{3} \times 1000 kg/m^{3}$
Convert the top unit and the bottom unit separately, then divide. Take care with squared and cubed units on the bottom.
Average speed
$average speed = \frac{total distance}{total time}$
Include any stops in the total time. Never average the speeds.

Worked examples

Worked example 1

Speed with minutes

A bus travels

51

km in

1

hour

15

minutes. Find its average speed in km/h.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Density, rearranged

Gold has density

19.3 g/cm^{3}

. Find the volume of a

500

g gold bar, to 3 s.f.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

m/s to km/h

A cheetah runs at

30

m/s. Write this in km/h.

Try it yourself first, then reveal the solution one step at a time.

Worked example 4

Pressure

A box weighing

600

N rests on a face measuring

40

cm by

50

cm. Find the pressure in

N/m^{2}

Try it yourself first, then reveal the solution one step at a time.

Practice: compound units

13 questions

Write times as decimals of an hour, and convert the top and bottom units separately.

Question 1 of 13Match the pairs
Match each unit to what it measures.
Pick an answer, then choose the row it belongs to. You can also drag it.
- $m/s$
- $kg/m^{3}$
- $N/m^{2}$
- $L/min$
Question 2 of 13Type the answer
A car travels $174$ km in $2$ hours $24$ minutes. Find its average speed in km/h.
Your answerkm/h
Question 3 of 13Type the answer
A plane flies at $840$ km/h for $2$ hours $45$ minutes. How far does it travel?
Your answerkm
Question 4 of 13Type the answer
How many minutes does it take to walk $3.5$ km at $5$ km/h?
Your answermin
Question 5 of 13Type the answer
Convert $72$ km/h to m/s.
Your answerm/s
Question 6 of 13Type the answer
Convert $15$ m/s to km/h.
Your answerkm/h
Question 7 of 13Type the answer
A block of wood measures $20$ cm by $10$ cm by $5$ cm and has mass $650$ g. Find its density in $g/cm^{3}$ .
Your answerg/cm³
Question 8 of 13Type the answerChallenge
Steel has density $7800 kg/m^{3}$ . Find the mass, in kg, of a steel rod of volume $1500 cm^{3}$ .
Your answerkg
Question 9 of 13Type the answer
A force of $450$ N acts on an area of $1500 cm^{2}$ . Find the pressure in $N/m^{2}$ .
Your answerN/m²
Question 10 of 13Step by stepChallenge
Average speed for a journey in stages.
Tan drives $45$ km at $60$ km/h, stops for $20$ minutes, then drives $105$ km at $90$ km/h.
1. Time for the first stage, in minutes.
  Step 1 answermin
Question 11 of 13Type the answer
A tap fills a $0.24 m^{3}$ tank at $8$ litres per minute. How long does it take, in minutes?
Your answermin
Question 12 of 13Multiple choice
Which is fastest?
Question 13 of 13Type the answerChallenge
Two metals are melted together: $300 cm^{3}$ of copper (density $8.9 g/cm^{3}$ ) and $200 cm^{3}$ of zinc (density $7.1 g/cm^{3}$ ). Find the density of the mixture.
Your answerg/cm³

In this lesson you will learn to…

“Per” means divide

Time in hours

Converting compound units

Average speed over a journey

Key rules and formulas

Speed, density, pressure

Time as a decimal

Converting compound units

Average speed

Worked examples

Practice: compound units