Ratio and proportion
Using direct proportion
One packet of rice costs $3.25, so two packets cost twice as much. When quantities keep the same ratio as they grow, they are in direct proportion.
What direct proportion means
Two quantities are in DIRECT PROPORTION when their ratio stays the same as they increase or decrease. One packet of rice costs $3.25, so two packets cost 2 × $3.25 = $6.50. Six concert tickets cost $120, so three tickets cost half as much: $120 ÷ 2 = $60. Double the amount, double the cost. Halve it, halve the cost.
The unitary method
The UNITARY METHOD means finding the value of ONE first. Three books cost $12. How much are 10 books? One book: $12 ÷ 3 = $4 Ten books: 10 × $4 = $40 Divide to get to one, then multiply up to whatever you need. It works for any number you are asked for.
The build-up method
Sometimes scaling directly is quicker. A recipe uses 2 eggs for 12 cupcakes. How many eggs for 36? 36 ÷ 12 = 3, so you need three times the recipe. Three times the eggs is 2 × 3 = 6. Use build-up when the numbers divide neatly; use unitary when they don't.
Choosing your method
A recipe uses 450 g of rice for six people. Jamila (unitary): one person needs 450 ÷ 6 = 75 g, so nine people need 9 × 75 = 675 g. Xun (build-up): three people need 450 ÷ 2 = 225 g, so nine people need 450 + 225 = 675 g. Both are right. Unitary always works; build-up is faster when the numbers are friendly.
Worked examples
Three books cost $12 — what do 10 cost?
- One book: 12 ÷ 3 = 4.
- Ten books: 10 × 4 = 40.
Answer: $40
2 eggs make 12 cupcakes — how many for 36?
- 36 ÷ 12 = 3, so triple the recipe.
- Eggs: 2 × 3 = 6.
Answer: 6 eggs
Three bananas weigh 375 g — what do eight weigh?
- One banana: 375 ÷ 3 = 125 g.
- Eight bananas: 8 × 125 = 1000 g.
Answer: 1000 g, or 1 kg
USD 100 buys ZAR 1400 — what does USD 6 buy?
- One dollar: 1400 ÷ 100 = 14 rand.
- Six dollars: 6 × 14 = 84 rand.
Answer: ZAR 84
Seven tickets cost $112 — what do three cost?
- One ticket: 112 ÷ 7 = 16.
- Three tickets: 3 × 16 = 48.
Answer: $48
450 g of rice feeds six — how much for nine?
- One person: 450 ÷ 6 = 75 g.
- Nine people: 9 × 75 = 675 g.
Answer: 675 g
Where you see this in real life
- Scaling a recipe up for more guests. Every ingredient is multiplied by the same number.
- Working out the best value at a shop by finding the price of one unit and comparing.
Key words
- Direct proportion
- two quantities whose ratio stays the same as they change
- Unitary method
- find the value of one, then multiply up
- Twice as much
- two times as much
- Half as much
- divided by two
- Scale up
- multiply every quantity by the same number