YEAR 9 · RATIO, RATES AND SIMILARITY · 15 MIN

Rates and unit conversions

Convert units before comparing rates, and use the relationship between speed, distance and time.

In this lesson

Convert common compound units correctly
Compare unit rates in matching units
Solve constant-speed problems and state units

Before you start

Metric length units
Fractions of an hour

01 / LEARN

Understand the idea.

A rate has two units

A rate tells how much of one quantity corresponds to one unit of another. Before comparing prices or speeds, choose matching units for both quantities; dollars per kilogram cannot be compared directly with dollars per gram.

Convert both parts of a compound unit. Since $1 km = 1000 m$ and $1 h = 3600 s$ , $1 km/h = \frac{1000}{3600} m/s$ . Therefore divide kilometres per hour by $3.6$ to obtain metres per second.

speed = \frac{distance}{time}

Use consistent time and distance

Choose units before substituting into $d = v t$ , $v = \frac{d}{t}$ or $t = \frac{d}{v}$ . With speed in kilometres per hour, express time in hours. For instance, minutes become hours by division by $60$ .

For an entire journey, average speed is total distance divided by total elapsed time, including stops when requested. Simply averaging two speeds works only when equal amounts of time were spent at those speeds.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Convert a speed

Convert 72 km/h to metres per second.

Worked solution

1
Express the hourly distance in metres.
$72 km = 72 000 m$
Each kilometre contains one thousand metres.
2
Express one hour in seconds.
$1 h = 60 \times 60 = 3600 s$
A speed in metres per second must use seconds in the denominator.
3
Divide distance by time.
$v = \frac{72 000}{3600} = 20 m/s$
Twenty metres are covered each second at this speed.

Answer

20 m/s

EXAMPLE 02

Time in a rate problem

A pump transfers 135 L in 7.5 min at a constant rate. How long for 216 L?

Worked solution

1
Find the volume transferred each minute.
$r = \frac{135}{7.5} = 18 L/min$
The constant pumping rate is volume divided by time.
2
Divide the target volume by the rate.
$t = \frac{216}{18} = 12 min$
Each minute accounts for eighteen litres of the required volume.
3
Check the volume at this time.
$18 \times 12 = 216 L$
The calculated time gives the required transfer at the stated constant rate.

Answer

12 minutes

03 / YOUR TURN

Try it yourself.

Check your understanding

A cyclist travels at 15 km/h for 24 minutes. How far does the cyclist travel?

Show answer & reasoning

6 km

Convert the time first: $24 min = \frac{24}{60} h = 0.4 h$ . Then $d = v t = 15 \times 0.4 = 6 km$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Convert 54 km/h to m/s.
2 marks
Convert 8 m/s to km/h.
2 marks
A car travels 126 km in 1.5 h. Find its average speed.
2 marks
A tap fills 42 L in 3.5 min. Find its rate in L/min.
2 marks
Which is cheaper: 750 g of cereal for 4.50 dollars, or 1.2 kg for 6.60 dollars?
3 marks
A bus travels 40 km in 50 minutes, then stops for 10 minutes. Find the average speed for the entire hour.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Convert 54 km/h to m/s.
2 marks
$54 \div 3.6 = 15$ .
$15$ m/s
Convert 8 m/s to km/h.
2 marks
$8 \times 3.6 = 28.8$ .
$28.8$ km/h
A car travels 126 km in 1.5 h. Find its average speed.
2 marks
$v = 126 \div 1.5 = 84$ km/h.
$84$ km/h
A tap fills 42 L in 3.5 min. Find its rate in L/min.
2 marks
$42 \div 3.5 = 12$ L/min.
$12$ L/min
Which is cheaper: 750 g of cereal for 4.50 dollars, or 1.2 kg for 6.60 dollars?
3 marks
$750$ g $= 0.75$ kg.
First rate: $\frac{4.50}{0.75} = 6.00$ dollars/kg.
Second rate: $\frac{6.60}{1.2} = 5.50$ dollars/kg.
The 1.2 kg pack.
A bus travels 40 km in 50 minutes, then stops for 10 minutes. Find the average speed for the entire hour.
2 marks
Total time is $50 + 10 = 60$ min $= 1$ h.
Average speed $= \frac{40}{1} = 40$ km/h.
$40$ km/h

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 proportional modelling, rates, scale, similarity and enlargement. Assumptions, interpretation and model limitations are taught alongside calculation.

Official curriculum source ↗

In this lesson

Understand the idea.

A rate has two units

Use consistent time and distance

See the method in action.

Convert a speed

Express the hourly distance in metres.

Express one hour in seconds.

Divide distance by time.

Time in a rate problem

Find the volume transferred each minute.

Divide the target volume by the rate.

Check the volume at this time.

Try it yourself.

Check your understanding

Keep practising