Ratio and proportion
Direct proportion
Two quantities are in direct proportion when one increases (or decreases) and the other increases (or decreases) in the same ratio. In this unit you scale recipes, prices, models and shapes up and down.
What 'in proportion' means
When two quantities are in direct proportion, their ratio stays the same. When one increases the other increases in the same ratio; when one decreases the other decreases in the same ratio. For example, the more hours Kate works the more money she earns: 1 hour → $20, 2 hours → $40, 3 hours → $60.
Scaling up
To scale up, work out what you multiply the first amount by, then multiply the other amount by the same number. If 350 g of pasta feeds 4 people, then for 12 people you multiply by 3 (because 12 ÷ 4 = 3), giving 350 × 3 = 1050 g.
Scaling down
To scale down, you divide instead. If a recipe for 8 people uses 400 ml of cream, then for 4 people you halve it (because 4 is half of 8), giving 400 ÷ 2 = 200 ml.
Scale and enlargements
A model that is one-tenth of the real object has every length divided by 10; the real object has every length multiplied by 10. When a rectangle is enlarged, every side is multiplied by the same number, and the whole shape stays in proportion.
Worked examples
What in proportion means
- Kate earns $20 for every hour she works.
- 1 hour → $20, 2 hours → $40, 3 hours → $60.
- The ratio of hours to money stays the same, so they are in direct proportion.
Answer: The money earned is in direct proportion to the hours worked.
Scaling a quantity up (multiply)
- 350 g of pasta feeds 4 people. Find the amount for 12 people.
- 12 ÷ 4 = 3, so you multiply by 3.
- 350 × 3.
Answer: 350 × 3 = 1050 g of pasta for 12 people.
Scaling with a different multiplier
- A sauce recipe for 6 people uses 300 g of tomatoes. Find the amount for 12 people.
- 12 ÷ 6 = 2, so you multiply by 2.
- 300 × 2.
Answer: 300 × 2 = 600 g of tomatoes for 12 people.
Cost in proportion
- 3 melons cost $2. Find the cost of 15 melons.
- 15 ÷ 3 = 5, so you multiply by 5.
- $2 × 5.
Answer: 15 melons cost $2 × 5 = $10.
Scale of a model
- A model car is one-tenth the size of the real car. The model is 40 cm long.
- The real car is 10 times bigger, so multiply by 10.
- 40 × 10.
Answer: The real car is 40 × 10 = 400 cm long.
Enlarging a rectangle
- Rectangle A is 5 cm by 2 cm. Rectangle B has every side twice as long.
- So B is 10 cm by 4 cm.
- Perimeter of B = 2 × (10 + 4).
Answer: Perimeter of rectangle B = 2 × 14 = 28 cm.
Where you see this in real life
- Scaling a cookie or ice cream recipe up or down for more or fewer people.
- Working out the real size of a car from a scale model.
- Enlarging a photo so the lengths stay in proportion and the angles stay the same.
Key words
- Direct proportion
- When one quantity increases or decreases, the other changes in the same ratio.
- In proportion
- Two quantities whose ratio stays the same as they change.
- Scale
- How many times bigger the real object is than the model, such as 1:24.
- Enlargement
- A larger (or smaller) copy where every length is multiplied by the same number.