Comparing Rates & Best Buys
Find the best buy for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-05). To compare prices fairly, turn each one into a unit price — the cost of the same amount, such as 1 kg, 1 L or 100 g — and the lowest unit price is the better value.
You will calculate unit prices, use them to pick the better value, work out the saving on a purchase, and compare rates in real situations such as petrol prices and mobile data plans — skills that save real money at the supermarket.
Theory
To compare prices fairly, change each one into a unit price, the cost of the same amount such as \(1\text{ kg}\), \(1\text{ L}\) or \(100\text{ g}\). This Year 12 Standard 1 (NSW) guide shows how to find unit prices, pick the best buy and work out the saving.
A unit price is the cost of one unit of a product, such as \(1\text{ kg}\), \(1\text{ L}\) or \(100\text{ g}\). Find it with unit price \(=\) total cost \(\div\) amount, with every amount written in the same unit (change \(750\text{ mL}\) to \(0.75\text{ L}\) for a price per litre).
The option with the smallest unit price is the best buy, or better value. The bigger pack is often, but not always, cheaper per unit. For a price per \(100\text{ g}\), divide the cost by the number of hundreds: \(400\text{ g}\) is \(4\) lots of \(100\text{ g}\).
The saving on a purchase \(=\) difference in unit price \(\times\) amount bought. Any rate can be compared this way: price per litre of petrol, per egg, or per gigabyte of data. For example, Plan A: \(15\text{ GB}\) for \(\$27\) is \(\$27 \div 15 = \$1.80\) per GB; Plan B: \(40\text{ GB}\) for \(\$64\) is \(\$64 \div 40 = \$1.60\) per GB, so Plan B is better value. This is a core Year 12 Standard 1 (NSW) Ratios and rates skill.
On a graph of cost against amount, a fixed unit price gives a straight line through the origin; its gradient (rise \(\div\) run) is the unit price, e.g. \(\$3\) per kg rises \(\$3\) for every \(1\text{ kg}\) (see the graph in Diagrams).
The unit price is the cost of one unit:
Price per \(100\text{ g}\) (or per \(100\text{ mL}\)):
The saving from choosing the cheaper option:
How to find the best buy
- Choose a common unit (per kg, per L or per 100 g) and convert every amount into it.
- Divide each cost by its amount to get the unit price.
- Compare: the smallest unit price is the best buy.
- Saving: multiply the difference in unit prices by the amount bought.
Divide the cost by the number of kilograms.
| \(\text{unit price}\) | \(=\) | \(\dfrac{7.80}{4}\) |
| \(\) | \(=\) | \(1.95\) |
| \(\text{check: } 4 \times 1.95\) | \(=\) | \(7.80\) |
The potatoes cost \(\$1.95\) per kilogram.
Change millilitres to litres, then divide.
| \(600\text{ mL}\) | \(=\) | \(0.6\text{ L}\) |
| \(\text{price per litre}\) | \(=\) | \(\dfrac{2.70}{0.6}\) |
| \(\) | \(=\) | \(4.50\) |
The iced coffee costs \(\$4.50\) per litre.
| Size | Price |
|---|---|
| \(170\text{ g}\) | \(\$1.53\) |
| \(500\text{ g}\) | \(\$4.25\) |
| \(1\text{ kg}\) | \(\$8.90\) |
Find the price per \(100\text{ g}\) of each tub (\(170\text{ g}\) is \(1.7\) lots of \(100\text{ g}\)).
| \(170\text{ g}:\ \dfrac{1.53}{1.7}\) | \(=\) | \(\$0.90\) |
| \(500\text{ g}:\ \dfrac{4.25}{5}\) | \(=\) | \(\$0.85\) |
| \(1\text{ kg}:\ \dfrac{8.90}{10}\) | \(=\) | \(\$0.89\) |
The \(500\text{ g}\) tub is the best buy at \(\$0.85\) per \(100\text{ g}\), not the biggest tub.
Find A's price per litre, compare, then multiply the difference.
| \(\text{A}\) | \(=\) | \(\dfrac{66.50}{35} = \$1.90\text{/L}\) |
| \(\text{B}\) | \(=\) | \(\$1.84\text{/L} \lt \$1.90\text{/L}\) |
| \(\text{saving}\) | \(=\) | \((1.90 - 1.84) \times 50\) |
| \(\therefore\ \text{saving}\) | \(=\) | \(0.06 \times 50 = \$3.00\) |
(i) Station B is cheaper; (ii) it saves \(\$3.00\).
Common pitfalls
Frequently asked questions
How do you work out the best buy?
Find the unit price of each option, which is the cost of the same amount such as 1 kg or 100 g. The option with the smallest unit price is the best buy.
How do you calculate a unit price?
Divide the total cost by the amount, with every amount in the same unit. A 4 kg bag of potatoes costing $7.80 is 7.80 divided by 4, which is $1.95 per kilogram.
How do you find the price per 100 g?
Work out how many lots of 100 g are in the pack, then divide the cost by that number. A 500 g tub costing $4.25 is 5 lots of 100 g, so 4.25 divided by 5 is $0.85 per 100 g.
Is the bigger pack always the better buy?
No. Bigger packs are often cheaper per unit, but not always. Always compare the unit prices; for example a 500 g tub of yoghurt can be better value than a 1 kg tub.
How do you work out how much you save at the cheaper store?
Find the difference between the two unit prices and multiply it by the amount you buy. If petrol is $1.90 per litre at one station and $1.84 at another, the saving on 50 litres is 0.06 times 50, which is $3.00.