Proportion in Practice: Recipes, Rates and Best Buys
Many everyday proportion problems can be solved by finding the value for one unit, then scaling to the amount you need. This is called the unitary method.
Foundation & Higher. This lesson uses proportion tables and practical contexts before any formal proportional equation.
Scaling recipes
A recipe for 4 people uses 300 g flour. For 1 person: 300 ÷ 4 = 75 g. For 6 people: 75 × 6 = 450 g.
You can also scale directly: 6 is 1.5 times 4, so every ingredient is multiplied by 1.5.
The unitary method
If 5 notebooks cost £12, one notebook costs £12 ÷ 5 = £2.40. Eight notebooks cost 8 × £2.40 = £19.20.
Unitary thinking works for quantities, time, distance, pay and many other rates.
Worked example — pay and time
At £12 per hour, 2 hours 30 minutes is 2 + 30/60 = 2.5 hours, so pay is 12 × 2.5 = £30. It is not 2.30 hours. This assumes a constant hourly rate, with no different overtime rate or unpaid time.
Visual method — build a proportion table
If 7 cinema tickets cost £59.50, a table lets you scale through one ticket before moving to another number.
| Tickets | Cost |
|---|---|
| 7 | £59.50 |
| 1 | £8.50 |
| 12 | £102.00 |
The move from 7 tickets to 1 ticket is ÷7. The move from 1 ticket to 12 tickets is ×12. The same operations must be applied to the cost column.
Currency conversion
If £1 = €1.16, then £75 = 75 × 1.16 = €87. To convert in the opposite direction, divide by the rate if the statement is still £1 = €1.16.
For example, at that same stated rate, €58 buys 58 ÷ 1.16 = £50. These are example exchange rates, not live quotations. Unless stated otherwise, assume no fee. Keep unrounded values during a comparison and round a final money amount to the nearest penny when appropriate.
Best buys and unit prices
Compare the same unit. A 750 g pack for £3.30 costs £3.30 ÷ 750 × 100 = 44p per 100 g. A 500 g pack for £2.35 costs 47p per 100 g. The first is better value.
The cheapest unit price is not always the sensible purchase if you cannot use the quantity.
Worked example — best buy with different pack sizes
Shop A sells 6 cans for £4.50. Shop B sells 10 cans for £7.20.
Shop A: £4.50 ÷ 6 = 75p per can. Shop B: £7.20 ÷ 10 = 72p per can.
Shop B is the better unit-price buy. If you need at least 12 cans and must buy whole packs from one shop, Shop A needs 2 packs costing £9; Shop B needs 2 packs costing £14.40. Shop A costs less for that purchase despite its higher price per can. Separate the lowest unit price from the lowest total cost that meets the requirement.
Pack sizes and whole items
If one pack covers 2.4 m² and you need 10 m², 10 ÷ 2.4 ≈ 4.17. You cannot buy 4.17 packs, so you need 5 packs. Context determines whether to round up, down or normally.
Offers involving percentages
If a comparison uses percentage discounts, use the existing Percentage Increase and Decrease lesson rather than inventing a new ratio method.
Think Like a Mathematician
The unitary method and a scale factor are two descriptions of the same proportional structure. One finds the value of a single unit first; the other moves directly between equivalent quantities.
Common Mistakes
Comparing unlike units: convert before comparing prices or rates.
Using the wrong direction for currency: read what one unit of the starting currency is worth.
Ignoring whole-pack constraints: a calculated 4.2 packs means 5 packs must be bought.
Practice — practical proportion
1. A recipe for 5 people uses 400 g rice. How much for 8 people?
One person: 80 g. Eight: 640 g.
2. 6 tickets cost £27. Find the cost of 10 tickets.
One ticket = £4.50. Ten = £45.
3. £1 = $1.30. Convert £48 to dollars.
48 × 1.30 = $62.40.
4. £1 = €1.18. Convert €88.50 to pounds.
88.50 ÷ 1.18 = £75.
5. Pack A: 600 g for £2.64. Pack B: 900 g for £4.14. Which is cheaper per 100 g?
A: 44p per 100 g. B: 46p per 100 g. Pack A.
6. A worker earns £84 for 7 hours. Find the hourly rate.
84 ÷ 7 = £12 per hour.
7. 3 litres of paint cover 36 m². How much area does 5 litres cover at the same rate?
12 m² per litre, so 60 m².
8. Tiles come in packs covering 1.8 m². You need 11 m². How many packs?
11 ÷ 1.8 ≈ 6.11, so buy 7 packs.
9. A car travels 156 miles using 12 litres. Find miles per litre.
156 ÷ 12 = 13 miles per litre.
10. A 750 ml bottle costs £2.10 and a 1.2 L bottle costs £3.24. Which is better value per litre?
750 ml = 0.75 L: £2.10 ÷ 0.75 = £2.80/L. £3.24 ÷ 1.2 = £2.70/L. 1.2 L bottle.
Frequently Asked Questions
Do I always have to find one unit first?
No. A direct scale factor is equally valid when it is easier. The same multiplier must apply to both quantities.
Is the lowest unit price always the cheapest purchase?
No. Whole packs, the amount needed and any stated fees or offers can change the total cost.