Unit rates | Year 8 | Ascent Library

01 / LEARN

Understand the idea.

What per means

The word per means for each one. A rate of $3$ dollars per litre connects cost with volume: each litre corresponds to AUD $3$ . To find this rate from a purchase, divide dollars by litres, in that order.

The order of the units tells you what the answer means. Litres per minute measures filling speed; minutes per litre measures the time for one litre. Reversing the division gives a different rate, so always label the result.

Divide, then scale

Dividing both quantities by the same number turns the given comparison into a rate per one. Once that unit rate is known, multiply by the new number of units. This scaling assumes the rate remains unchanged.

Check a unit rate by multiplying it by the original number of units: the original total should return. Keep unrounded values during calculations and round a final money amount to cents when necessary.

unit rate = \frac{total quantity}{number of units}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Cost per litre

Eight litres of paint cost AUD 24. Find the unit price and the cost of 5 L at the same rate.

Worked solution

1
Divide cost by volume.
$Unit price = \frac{24}{8} = 3 dollars/L$
A price per litre has dollars as its numerator and litres as its denominator.
2
Scale the unit price to five litres.
$Cost = 5 \times 3 = 15 dollars$
Each of the five litres costs the same AUD 3.
3
Check against the original purchase.
$8 \times 3 = 24 dollars$
The unit price reproduces the given cost of eight litres.

Answer

AUD 3/L; AUD 15 for 5 L.

EXAMPLE 02

Words per minute

A speaker reads 375 words in 5 minutes. Find the average rate.

Worked solution

1
Form a words-per-minute rate.
$Average rate = \frac{375 words}{5 min}$
The requested rate compares the number of words with the elapsed time.
2
Divide to find the rate per minute.
$375 \div 5 = 75 words/min$
Dividing by five expresses the total over one minute instead of five.
3
Check the five-minute total.
$75 \times 5 = 375 words$
The average rate accounts for the full reading, without claiming every minute was equally fast.

Answer

75 words/minute.

03 / YOUR TURN

Try it yourself.

Check your understanding

A tap fills 28 L in 4 minutes. What is its rate?

Show answer & reasoning

7 L/min.

For litres per minute, divide volume by time: $28 \div 4 = 7 L/min$ . Over four minutes this gives $7 \times 4 = 28$ L, as required.

Keep practising

6 worksheet questions · 10 marks

Open worksheet Worked solutions

Read the questions online

Six kilograms cost AUD 18. Find the cost per kilogram.
1 mark
A car travels 240 km in 3 hours. Find its average speed.
1 mark
A pump moves 42 L in 7 minutes. Find its rate.
1 mark
Three litres of juice cost AUD 8.40. Find the unit price.
2 marks
A machine makes 150 items in 12 seconds. Find the average rate.
2 marks
A 0.6 kg bag costs AUD 5.40. Find the price per kilogram and per 100 g.
3 marks

Read the worked solutions

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

Six kilograms cost AUD 18. Find the cost per kilogram.
1 mark
$18 \div 6 = 3$ .
AUD 3/kg.
A car travels 240 km in 3 hours. Find its average speed.
1 mark
$240 \div 3 = 80$ .
80 km/h.
A pump moves 42 L in 7 minutes. Find its rate.
1 mark
$42 \div 7 = 6$ .
6 L/min.
Three litres of juice cost AUD 8.40. Find the unit price.
2 marks
$8.40 \div 3 = 2.80$ .
AUD 2.80/L.
A machine makes 150 items in 12 seconds. Find the average rate.
2 marks
$150 \div 12 = 12.5$ .
Check: $12.5 \times 12 = 150$ .
12.5 items/s.
A 0.6 kg bag costs AUD 5.40. Find the price per kilogram and per 100 g.
3 marks
$5.40 \div 0.6 = 9$ .
100 g is one tenth of a kilogram, so divide 9 by 10.
AUD 9/kg; AUD 0.90 per 100 g.

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Ratio scaling and division review; unit rates; comparing rates and best buys; speed, distance and time; converting compatible units; direct-proportion tables and graphs; modelling practical rate problems and checking assumptions.

Official curriculum source ↗

In this lesson

Understand the idea.

What per means

Divide, then scale

See the method in action.

Cost per litre

Divide cost by volume.

Scale the unit price to five litres.

Check against the original purchase.

Words per minute

Form a words-per-minute rate.

Divide to find the rate per minute.

Check the five-minute total.

Try it yourself.

Check your understanding

Keep practising