Using direct proportion
🎯 In this topic you will
- Use the unitary method to solve problems involving ratio and direct proportion
🧠 Key Words
- direct proportion
- half as much
- twice as much
- unitary method
Show Definitions
- direct proportion: A relationship where one quantity increases or decreases at the same rate as another.
- half as much: An amount that is one-half of another amount.
- twice as much: An amount that is double another amount.
- unitary method: A technique where you first find the value of a single unit, then multiply to find the value of multiple units.
Two quantities are in direct proportion when their ratio stays the same as the quantities increase or decrease.
One packet of rice costs $3.25$, so two packets of rice cost twice as much.
Two packets of rice cost $2 \times 3.25 = 6.50$.
Six tickets to a concert cost $120$, so three tickets cost half as much.
Three tickets cost $120 \div 2 = 60$.
The method used in part a of Worked example 12.3 is called the unitary method because you find the cost of one book then use this to find the cost of 10 books.
📘 Exercise 12.3 — Answers
1. Three bananas weigh 375 g. Find the mass of one banana and eight bananas.
🔎 Reasoning Tip
You can assume that all of the bananas have the same mass.
👀 Show Answer
Eight bananas = 8 × 125 = 1000 g.
2. Tom buys two bags of chips for $2.40.
- a) Cost of one bag
- b) Cost of five bags
👀 Show Answer
b) Five bags = 5 × $1.20 = $6.00.
3. Joy exchanges USD 100 for ZAR 1400.
- a) ZAR for 1 USD
- b) ZAR for 6 USD
🔎 Reasoning Tip
Currency information:
- USD = United States dollars
- ZAR = South African rand
👀 Show Answer
a) 14 ZAR b) 6 × 14 = 84 ZAR.
4. Ivan buys four shirts for $37.80. Work out the cost of nine shirts.
👀 Show Answer
Nine shirts = 9 × 9.45 = $85.05.
5. Dakri buys seven tickets for $112. How much do three tickets cost?
👀 Show Answer
Three tickets = 3 × 16 = $48.
6. A recipe uses 360 g of flour for four people. How much flour is needed for three people?
👀 Show Answer
7. A recipe uses $450$ grams of rice for six people. How much rice is needed for:
A. $24$ people?
B. nine people?
Jamila and Xun answer this question in different ways. Jamila uses the unitary method.
Xun uses a build-up method.
a. Look through both solutions and make sure that you understand the methods.
b. Critique the methods, writing down the advantages and disadvantages of each method.
c. If you were using a calculator, which method would you choose? Explain why.
d. If you didn’t have a calculator to use, which method would you choose? Explain why.
e. Are your answers to parts c and d the same or different? Explain why.
👀 Show answer
7A. Six people need $450$ g. One person needs $450 \div 6 = 75$ g. For $24$ people: $24 \times 75 = 1800$ g $= 1.8$ kg.
7B. Nine people: $9 \times 75 = 675$ g.
a. Both methods are correct. Jamila uses the unitary approach, while Xun uses scaling (build-up).
b. Unitary method is systematic and works for any number but may require division. Build-up is faster for multiples but less flexible.
c. With a calculator, the unitary method is quicker because you can calculate directly by division and multiplication.
d. Without a calculator, the build-up method may be easier since it uses simple multiples.
e. The answers differ: with calculator → unitary; without calculator → build-up. Choice depends on ease of arithmetic.
🧠 Think like a Mathematician
Task: Compare two methods for solving ratio problems: the unitary method and the build-up method. Decide when each is most effective.
Questions:
👀 show answer
- Unitary method: - Best when you need an exact calculation. - Steps: find the value of 1 part, then scale up. - Advantages: systematic, always works
❓ EXERCISES
9. A recipe for four people uses $800$ grams of potato.
Copy and complete the workings, using the build-up method, to find the mass of potato needed for:
a. $20$ people
b. $6$ people
a Mass of potato for four people is $800$ grams.
The connection between $4$ and $20$ is: $20 \div 4 = \Box$
Mass of potato for $20$ people: $800$ grams $\times \Box = \Box$ grams
b Mass of potato for four people is $800$ grams.
Mass of potato for two people: $800$ grams $\div 2 = \Box$ grams
Mass of potato for six people: $800$ grams $+ \Box$ grams $= \Box$ grams
👀 Show answer
9a. $20 \div 4 = 5$, so mass of potato $= 800 \times 5 = 4000$ g $= 4$ kg.
9b. Two people: $800 \div 2 = 400$ g. Six people: $800 + 400 = 1200$ g $= 1.2$ kg.
10. This is part of Irene’s homework.
Explain Irene’s mistake and write out the correct solution.
👀 Show answer
Irene’s mistake: She added $6+9=15$ instead of scaling proportionally. She treated $15$ people as $6+9$ instead of a multiple of $6$.
Correct solution: For $6$ people $\to 300$ g. For $15$ people: $300 \div 6 \times 15 = 750$ g.
11. A teacher buys homework books for her class of $30$ students. She paid a total of $\$105$ for the books. Two more students join her class, so she then buys two extra books. The teacher works out that the total cost of the books is now $\$121$. Is she correct? Explain your answer.
If the teacher is incorrect, what mistake do you think she has made?
👀 Show answer
Cost per book $= 105 \div 30 = \$3.50$.
For $32$ students: $32 \times 3.50 = \$112$.
The teacher’s answer of $\$121$ is incorrect. She may have mistakenly added $\$16$ (the cost of $2$ books at $\$8$ each) instead of using the unit price of $\$3.50$ per book.
⚠️ Be careful!
- Don’t divide by the biggest ratio number — always divide by the sum of the parts.
- Keep the order given. “Sally : Bob = 2:1” means Sally’s share comes first.
- Simplify ratios first if possible (e.g., 80:40 → 2:1) to make arithmetic easier.
- Convert units before using them in a ratio (kg ↔ g, hours ↔ minutes).
- Check your total. Add the shares to ensure they equal the original amount.