Lesson 8: Direct and Inverse Proportion

§Text Passage

1Summary 重點摘要

Direct proportion: one goes up, the other goes up proportionally (e.g. more buns, more money). Inverse proportion: one goes up, the other goes down (e.g. faster speed, less time).

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Direct Proportion ExampleBuying things is a classic direct proportion example: when the unit price is fixed, buying more costs more. Example: pencils cost $2 each, so 1 pencil is $2, 2 are $4 and 3 are $6. Working out total ÷ quantity always gives $2; the quotient is fixed, so total price and quantity are in direct proportion. Daily life offers many more direct proportion examples: for the same rice, total price grows with weight; at a steady speed, distance grows with time. In every case, when one quantity changes by some factor, the other changes by the same factor.
2. Block 2: Inverse Proportion ExampleSharing a cake is a classic inverse proportion example: with the cake fixed, more people means less for each. Example: a cake cut into 12 pieces gives 4 each for 3 people, 3 each for 4 people, 2 each for 6 people. People × pieces each is always 12; the product is fixed, so it is inverse proportion. More inverse examples: covering a fixed distance, speed and time are in inverse proportion; finishing a job, the number of workers and the days needed are in inverse proportion. In every case one quantity grows as many times as the other shrinks, the product never changing.
3. Block 3: How to TellTo decide between the two, look at how the quantities change: if the quotient of the two numbers is fixed (y ÷ x = constant), it is direct proportion; if the product is fixed (x × y = constant), it is inverse proportion. Remember: fixed quotient means direct, fixed product means inverse. Decide first, then calculate. Applied examples: for a fixed journey, speed × time is fixed, so it is inverse proportion; at a fixed unit price, total ÷ quantity is fixed, so it is direct proportion. A wrong decision wastes all the working after it, so deciding always comes first.
4. Block 4: Solving Direct ProblemsExample: 3 apples cost $15; how much do 7 apples cost? This is direct proportion. First find the unit price: $15 ÷ 3 = $5 each, then multiply for 7: $5 × 7 = $35. You can also write 3:15 = 7:x and cross multiply to get 3x = 105, so x = 35; both methods agree. Check: 35 ÷ 7 = 5, agreeing with the unit price 15 ÷ 3 = 5, so the answer fits the direct proportion; whether the unit-price method or a proportion statement is used, the results must be the same.
5. Block 5: Solving Inverse ProblemsExample: 6 people build a wall in 12 days; how long would 8 people take? This is inverse proportion because the total work is fixed: 6 × 12 = 72 person-days. With 8 people the time = 72 ÷ 8 = 9 days. More people means fewer days, exactly the fixed-product pattern of inverse proportion. If the question asks about 9 people instead, the time = 72 ÷ 9 = 8 days; whatever the headcount, work out the fixed total effort first, then divide by the new number of people and the days follow at once.
6. Block 6: Common MistakeThe most common mistake is seeing the word 'proportion' and assuming it is direct without checking. Remember: walking a fixed distance, speed and time are in inverse proportion — a faster speed takes less time; shopping, quantity and total price are in direct proportion. Ask which quantity is fixed: fixed quotient is direct, fixed product is inverse. For example, 'each book costs $8, how much for n books' is direct proportion, since total ÷ number is fixed at 8; '24 sweets shared equally among n people' is inverse proportion, since people × sweets each is fixed at 24.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Direct: quotient fixed (y ÷ x = constant)
• Inverse: product fixed (x × y = constant)
• Quantity and price = direct
• People and work time = inverse
• Speed and time (fixed distance) = inverse