Proportional Reasoning-2 · Lesson 6 of 7
Understanding Inverse Proportion
“Learn how inverse proportion differs from direct proportion and how a constant product reveals relationships where one quantity rises as the other falls.”
- Recall the difference between direct and inverse proportional relationships.
- Recognise inverse proportion from how two quantities change.
- Use the constant-product relationship xy = k.
- Use x1y1 = x2y2 to find missing values.
- Test tables of values to decide whether they show inverse proportion.
Direct Proportion as a Starting Point
In a direct proportion, the quantities change in the same direction by the same factor. If one quantity becomes four times as large, the other also becomes four times as large, provided the conditions stay the same.
Problem
Five workers can move 4500 bricks in one day. At the same working rate, how many workers are needed to move 18,000 bricks in one day?
- 1.The amount of work increases from 4500 to 18,000 bricks, which is a factor of 4.
- 2.For the same one-day time limit and same worker rate, the number of workers must also be multiplied by 4.
- 3.Workers needed = 5 × 4 = 20.
- 4.Equivalently, 4500 : 18,000 :: 5 : 20.
When One Quantity Increases and the Other Decreases
Some relationships behave in the opposite way. For a fixed journey, travelling faster means taking less time. If the speed doubles, the time is halved. The quantities are still linked by a consistent factor, but they change in opposite directions.
| Mode | Speed (km/h) | Time (hours) | Speed × time |
|---|---|---|---|
| Walk | 5 | 18 | 90 |
| Bicycle | 15 | 6 | 90 |
| Motorcycle | 30 | 3 | 90 |
| Car | 60 | 1.5 | 90 |
Notice that every product is 90. Here that constant is the fixed distance of the journey. When speed is multiplied by 3, time is divided by 3; when speed is doubled, time is halved. This opposite-factor behaviour is the key feature of inverse proportion.
Two quantities are inversely proportional when multiplying one by a factor causes the other to be divided by the same factor, so their product remains constant.
Problem
A journey takes 3 hours at 30 km/h. How long would the same journey take at 60 km/h?
- 1.The distance is fixed, so speed and time are inversely proportional.
- 2.Use speed × time = constant: 30 × 3 = 60 × t.
- 3.90 = 60t.
- 4.t = 90 ÷ 60 = 1.5 hours.
- 5.The speed doubled, so the time halved, which matches the inverse relationship.
Testing a Table for Inverse Proportion
A quick way to test a table is to multiply each x-value by its matching y-value. If every product is the same, the data show inverse proportion. A pattern in which y merely decreases while x increases is not enough; the product must stay constant.
Problem
Test these three sets: (i) x = 40, 80, 25, 16 and y = 20, 10, 32, 50; (ii) x = 40, 80, 25, 16 and y = 20, 10, 12.5, 8; (iii) x = 30, 90, 150, 10 and y = 15, 5, 3, 45.
- 1.For (i), the products are 800, 800, 800 and 800, so it is inverse proportion.
- 2.For (ii), the products are 800, 800, 312.5 and 128, so the product is not constant and the table is not inverse proportion.
- 3.For (iii), the products are 450, 450, 450 and 450, so it is inverse proportion.
Problem
x and y are inversely proportional. One pair is x = 16, y = 9. Find y when x = 12, find x when y = 48, and find y when x = 36.
- 1.Find the constant: k = 16 × 9 = 144.
- 2.When x = 12, y = 144 ÷ 12 = 12.
- 3.When y = 48, x = 144 ÷ 48 = 3.
- 4.When x = 36, y = 144 ÷ 36 = 4.
- 5.Each completed pair has product 144.
Direct or Inverse?
The direction of change helps us decide which model to use, but we must also think about what is being held fixed. More pencils cost more money at a fixed price per pencil, so that is direct proportion. More equally efficient workers need fewer days for the same fixed amount of work, so that is inverse proportion.
| Situation | What stays fixed? | Relationship |
|---|---|---|
| Number of pencils and total price | Price per pencil | Direct |
| Speed and travel time | Journey distance | Inverse |
| Workers and days | Total work and worker rate | Inverse |
| Distance travelled and fuel used | Fuel efficiency | Direct |
Do not decide that a relationship is inverse just because one value goes up while another goes down. Check that the product remains constant and that the situation's assumptions justify that relationship.
Quiz
Which equation describes inverse proportion?
If x and y are inversely proportional and x doubles, what happens to y?
Which pair of values has the same product as 12 × 8?
For a fixed distance, speed changes from 20 km/h to 50 km/h. Which change in time is consistent with inverse proportion?
Which situation is most naturally a direct proportion under the stated condition?
Practice Problems
- Test whether x = 40, 80, 25, 16 and y = 20, 10, 32, 50 are inversely proportional.
- Test whether x = 40, 80, 25, 16 and y = 20, 10, 12.5, 8 are inversely proportional.
- Test whether x = 30, 90, 150, 10 and y = 15, 5, 3, 45 are inversely proportional.
- If x and y are inversely proportional and x = 16 when y = 9, complete these values: y when x = 12; x when y = 48; y when x = 36.
- A fixed journey takes 5 hours at 48 km/h. How long will it take at 60 km/h, assuming the speed is constant throughout?
- Explain why the number of workers and the time needed for a fixed job are only inversely proportional if the workers have comparable productivity and work for the same number of hours each day.
Key Takeaways
Direct proportion makes two quantities change by the same factor in the same direction. Inverse proportion makes one quantity change by a factor while the other changes by its reciprocal. For inverse proportion, xy = k and the product stays constant. Two inverse pairs satisfy x₁y₁ = x₂y₂. A constant product is a reliable test for inverse proportion. The real-world assumptions of a situation matter when deciding whether a proportional model is appropriate.