Comparing rates and best buys | Year 8

01 / LEARN

Understand the idea.

Compare the same unit

A pack price pays for a particular amount. To compare unit prices, express all quantities in the same unit and divide each price by its quantity. Then a lower price per kilogram really means less money for the same mass.

Write the common unit beside every rate. AUD $2$ per $100$ g cannot be compared directly with AUD $15$ per kilogram: the first rate is AUD $20$ per kilogram. Larger packs are not automatically better value.

Best value and best purchase

The lowest unit price identifies the better value for equal amounts. The cheapest purchase must also supply the required quantity using available pack sizes. Calculate whole packs and total spending before making a recommendation.

Equal unit prices mean the same cost per unit, but the packs can still suit different needs. Storage, a spending limit and likely waste can matter; state any assumption that changes which purchase you would choose.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Compare different masses

Offer A is 750 g for AUD 6. Offer B is 1 kg for AUD 7.60. Which has the lower unit price?

Worked solution

1
Express both masses in kilograms.
$750 g = 0.75 kg$
The two prices must refer to the same mass unit.
2
Calculate Offer A per kilogram.
$\frac{6}{0.75} = 8 dollars/kg$
Dividing the pack price by its mass gives its unit price.
3
Compare Offer B using the same unit.
$\begin{aligned} B & = \frac{7.60}{1} = 7.60 dollars/kg \\ 7.60 & < 8 \end{aligned}$
Offer B costs less for one kilogram, so it has the lower unit price.

Answer

Offer B has the lower unit price.

EXAMPLE 02

Equal unit prices

Compare 700 mL for AUD 4.20 with 1 L for AUD 6.

Worked solution

1
Convert the smaller bottle to litres.
$700 mL = 0.7 L$
One litre contains 1,000 millilitres.
2
Calculate the smaller bottle unit price.
$\frac{4.20}{0.7} = 6 dollars/L$
A cost per litre allows a like-for-like comparison.
3
Compare the one-litre bottle.
$\frac{6}{1} = 6 dollars/L$
Both offers charge AUD 6 for each litre, so their unit prices are equal.

Answer

They have equal unit prices.

03 / YOUR TURN

Try it yourself.

Check your understanding

Which is cheaper per pen: 4 pens for AUD 6 or 6 pens for AUD 8.40?

Show answer & reasoning

Six pens for AUD 8.40.

The four-pen pack costs $6 \div 4 = 1.50$ dollars per pen. The six-pen pack costs $8.40 \div 6 = 1.40$ dollars per pen, so six pens for AUD 8.40 is cheaper per pen.

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

Compare 500 g for AUD 3.25 with 800 g for AUD 4.80. State the lower price per kilogram.
3 marks
Which is cheaper per pen: 3 pens for AUD 4.20 or 5 pens for AUD 6.50?
2 marks
Compare 2.5 kg for AUD 9 with 1.5 kg for AUD 5.70.
2 marks
Compare 250 mL for AUD 2.10 with 1.5 L for AUD 10.80.
3 marks
You need at least 600 g. A 400 g pack costs AUD 3; a 1 kg pack costs AUD 7. What is the cheapest purchase?
3 marks
A 500 mL bottle costs AUD 3 and a 750 mL bottle costs AUD 4.50. Find the cheapest way to buy exactly 1.5 L.
3 marks

Read the worked solutions

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

Compare 500 g for AUD 3.25 with 800 g for AUD 4.80. State the lower price per kilogram.
3 marks
$3.25 \div 0.5 = 6.50$ .
$4.80 \div 0.8 = 6$ .
800 g offer: AUD 6/kg, compared with AUD 6.50/kg.
Which is cheaper per pen: 3 pens for AUD 4.20 or 5 pens for AUD 6.50?
2 marks
$4.20 \div 3 = 1.40$ .
$6.50 \div 5 = 1.30$ .
Five-pen pack: AUD 1.30/pen.
Compare 2.5 kg for AUD 9 with 1.5 kg for AUD 5.70.
2 marks
$9 \div 2.5 = 3.60$ dollars/kg.
$5.70 \div 1.5 = 3.80$ dollars/kg.
The 2.5 kg offer is cheaper per kilogram.
Compare 250 mL for AUD 2.10 with 1.5 L for AUD 10.80.
3 marks
$2.10 \div 0.25 = 8.40$ .
$10.80 \div 1.5 = 7.20$ .
The 1.5 L offer: AUD 7.20/L, compared with AUD 8.40/L.
You need at least 600 g. A 400 g pack costs AUD 3; a 1 kg pack costs AUD 7. What is the cheapest purchase?
3 marks
Two small packs supply 800 g for AUD 6.
One large pack costs AUD 7.
Both meet the need; the small packs cost AUD 1 less.
Two 400 g packs, costing AUD 6.
A 500 mL bottle costs AUD 3 and a 750 mL bottle costs AUD 4.50. Find the cheapest way to buy exactly 1.5 L.
3 marks
Both unit prices are AUD 6/L.
$3 \times 3 = 9$ and $2 \times 4.50 = 9$ .
Either three 500 mL bottles or two 750 mL bottles; both cost AUD 9.

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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.

Compare the same unit

Best value and best purchase

See the method in action.

Compare different masses

Express both masses in kilograms.

Calculate Offer A per kilogram.

Compare Offer B using the same unit.

Equal unit prices

Convert the smaller bottle to litres.

Calculate the smaller bottle unit price.

Compare the one-litre bottle.

Try it yourself.

Check your understanding

Keep practising