Ratio, proportion & rates of change
Direct & Inverse Proportion: Unitary Method & Best Buys
Year 7 · Year 8 · Year 9
- ✓By the end of this lesson students will be able to identify and solve problems involving direct proportion.
- ✓By the end of this lesson students will be able to use the unitary method to find unknown quantities.
- ✓By the end of this lesson students will be able to apply proportional reasoning to solve best-buy problems.
- ✓By the end of this lesson students will be able to identify and solve simple problems involving inverse proportion.
Key concepts
Two quantities are in direct proportion if, as one quantity increases, the other quantity increases at the same rate, or as one quantity decreases, the other quantity decreases at the same rate. This means their ratio remains constant. For example, if you buy more items, the total cost increases proportionally. We can write this as y = kx, where k is the constant of proportionality.
The unitary method is a technique used to solve proportion problems by first finding the value of a single unit. Once the value of one unit is known, it can be used to find the value of any number of units. This method is particularly useful for direct proportion problems.
A 'best buy' problem involves comparing the prices of different quantities of the same item to determine which option offers the most value for money. This is typically done by calculating the price per unit (e.g., price per 100g, price per item) for each option and then choosing the one with the lowest unit price.
Two quantities are in inverse proportion if, as one quantity increases, the other quantity decreases proportionally, and vice versa. This means their product remains constant. For example, if more workers are assigned to a task, the time taken to complete the task decreases. We can write this as y = k/x, where k is the constant of proportionality.
Key facts to remember
- 1Direct proportion means that as one quantity increases, the other increases by the same factor (or decreases by the same factor).
- 2In direct proportion, the ratio of the two quantities is constant (y/x = k).
- 3Inverse proportion means that as one quantity increases, the other decreases by the same factor.
- 4In inverse proportion, the product of the two quantities is constant (xy = k).
- 5The unitary method involves finding the value of one unit first.
- 6To find the best buy, calculate the price per unit (e.g., per kg, per litre, per item) for each option and choose the lowest.
- 7Always check if a problem involves direct or inverse proportion before solving.
Worked examples
Example 1
If 7 apples cost £2.45, how much would 12 apples cost?
Answer
12 apples would cost £4.20.
This is a direct proportion problem because as the number of apples increases, the total cost increases.
Example 2
A supermarket offers two deals for orange juice: Deal A is 1.5 litres for £1.80. Deal B is 2 litres for £2.20. Which deal is the better buy?
Answer
Deal B is the better buy as it costs £1.10 per litre, compared to £1.20 per litre for Deal A.
Always calculate the price for a common unit (e.g., 1 litre, 1 kg, 1 item) to make a fair comparison.
Example 3
It takes 5 painters 8 hours to paint a room. How long would it take 4 painters to paint the same room, assuming they work at the same rate?
Answer
It would take 4 painters 10 hours to paint the same room.
For inverse proportion, the product of the two quantities remains constant.
Common mistakes
- ✗Confusing direct and inverse proportion, leading to incorrect calculations.
- ✗Not using the unitary method correctly, for example, multiplying when division is needed.
- ✗Failing to find a common unit when comparing items for 'best buy' problems.
- ✗Making calculation errors, especially with decimals or fractions.
- ✗Not showing full working out, which can lead to loss of marks in exams.
Exam tips
- ★Read the question carefully to determine if it is a direct or inverse proportion problem.
- ★Always show your working step-by-step, especially when using the unitary method, as this can earn method marks.
- ★Pay attention to units and ensure consistency throughout your calculations (e.g., convert everything to grams or kilograms).
- ★After solving, consider if your answer seems reasonable in the context of the problem.
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