Two packs of different sizes
A shop sells the same pens in two packs. One pack holds 4 pens and costs $12. The other holds 6 pens and also costs $12. Which pack is the better buy?
The prices are the same, but the packs do not hold the same number of pens, so the two prices cannot be compared as they are. What can be compared is the price of one pen in each pack.
Each bar is $12, so the two bars are the same length. The first is cut into 4 pens and the second into 6.
Find the price of one
Share each price between the pens in its pack. In the pack of 4, one pen costs 12 ÷ 4 = $3. In the pack of 6, one pen costs 12 ÷ 6 = $2. The pack of 6 is the better buy, because each pen costs less.
The price of one item is called the unit price. It is a rate, a price per pen, and it is the one number that makes two packs of different sizes comparable. The lower unit price is the better buy.
The same $12 cut into 4 pieces makes each piece bigger than when it is cut into 6: $3 a pen against $2 a pen.
When the prices differ too
Now suppose the pack of 4 costs $12 and a pack of 6 costs $15. The pack of 6 costs more money, but it also holds more pens. Find both unit prices: 12 ÷ 4 = $3 a pen, and 15 ÷ 6 = $2.50 a pen. The pack of 6 is still the better buy, by 50 cents a pen.
Sometimes the unit prices come out the same. $6 for 2 pens and $9 for 3 pens are both $3 a pen, so neither pack is better value than the other.
The price of one pen from each pack, in dollars: $3 from the pack of 4 and $2.50 from the pack of 6.
The usual mistakes
Choosing the pack with the lower price. A lower price for fewer items is not always the better buy. Compare the price of one item in each pack.
Deciding that a bigger pack must be cheaper. Many bigger packs are cheaper per item, but not all of them. Only the unit prices tell you.
Dividing the wrong way. 4 ÷ 12 is the number of pens for each dollar, not the price of a pen. To find a price per pen, divide the price by the number of pens.
Comparing by mass
Food is often sold by mass, in bags of different sizes. Then the unit price is a price per 100 g or per kilogram, and a shelf label in a supermarket often prints it under the price. In the next problem, each bag is cut into lots of 100 g, and the price of one lot decides the better buy.
Worked example: Two Bag Sizes of Coffee Beans: The Price per 100 g, and a Café's Weekly Saving
Question A shop sells the same coffee beans in two bag sizes: a 250 g bag for $6.00 and a 400 g bag for $8.80. (a) Find the price per 100 g of each bag. Which bag is the better buy? (b) A café uses exactly 2 kg of these beans each week. How much does it save each week by buying only the better bags instead of only the other bags?
1.Cut each bag into 100 g parts: the 250 g bag is 212 parts and the 400 g bag is 4 parts.
Cut each bag into 100 g parts: the 250 g bag is 212 parts and the 400 g bag is 4 parts. 2.One part of the 250 g bag costs 6.00 ÷ 2.5 = $2.40.
One part of the 250 g bag costs 6.00 ÷ 2.5 = $2.40. 3.One part of the 400 g bag costs 8.80 ÷ 4 = $2.20.
One part of the 400 g bag costs 8.80 ÷ 4 = $2.20. 4.(a) The 400 g bag is the better buy: each 100 g costs 2.40 − 2.20 = $0.20 less.
(a) The 400 g bag is the better buy, by $0.20 for each 100 g. 5.2 kg is 2000 g, which is 5 bags of 400 g: 5 × 8.80 = $44.00.
Five 400 g bags make 2000 g: 5 × 8.80 = $44.00. 6.In 250 g bags, 2000 g is 8 bags: 8 × 6.00 = $48.00.
Eight 250 g bags make 2000 g: 8 × 6.00 = $48.00. 7.(b) The café saves 48.00 − 44.00 = $4.00 a week. Check: 2000 g is 20 parts of 100 g, and 20 × 0.20 = $4.00.
(b) The café saves 48.00 − 44.00 = $4.00 a week.
Answer: (a) $2.40 and $2.20 per 100 g, so the 400 g bag is the better buy; (b) $4.00 a week
Common mistakes
- Calling the cheaper bag, $6.00 against $8.80, the better buy: the bags hold different amounts of coffee, so only the prices for the same mass, 100 g, can be compared.
- Giving $0.20 as the weekly saving: that is the saving on each 100 g, and 2 kg holds 20 lots of 100 g.