In life, not everything grows together. Sometimes, when one thing increases, another thing decreases. Think about speed and travel time: the faster you drive, the less time it takes to get there. Or think about sharing a pizza: the more people you invite, the fewer slices each person gets.
This "seesaw" relationship, where one quantity goes up while the other goes down, is called Inverse Proportion (or Inverse Variation). It is just as common as direct proportion but works in the opposite way.
1. What is Inverse Proportion?
Two quantities are inversely proportional if their product is constant. If you double one quantity, the other is cut in half. If you triple one, the other becomes one-third.
The Core Rule: The product of the two quantities always remains the same (constant).
x * y = k (constant)
2. Visualizing Inverse Proportion
Unlike direct proportion, which creates a straight line, inverse proportion creates a curve called a hyperbola. It never touches the axes because neither quantity can ever be zero.
[Image of inverse proportion graph curve]Think about a construction job that requires 100 hours of labor:
- 1 worker takes 100 hours.
- 2 workers take 50 hours.
- 4 workers take 25 hours.
- 10 workers take 10 hours.
As the number of workers shoots up, the time drops drastically.
3. The Constant of Proportionality (k)
In inverse proportion, the constant k represents the "Total Work" or "Total Value" available.
In our construction example, k is 100 (the total man-hours required). The formula for time is Time = 100 / Workers.
4. Solving Inverse Proportion Problems
When solving these problems, remember: Case 1 multiplied equals Case 2 multiplied.
Example Problem
Problem: If 6 pumps can fill a tank in 2 hours, how long will it take 3 pumps to fill the same tank?
Step 1: Identify the variables.
x1 = 6 pumps, y1 = 2 hours.
x2 = 3 pumps, y2 = ? hours.
Step 2: Use the formula.
6 * 2 = 3 * y2
12 = 3 * y2
Step 3: Solve.
y2 = 12 / 3
y2 = 4 hours.
Logic Check: Fewer pumps should take more time. 4 hours is more than 2 hours, so the answer makes sense.
5. Real-Life Examples
Inverse proportion governs many physical laws and daily situations.
1. Speed and Time
If you need to travel 100 miles, driving at 50 mph takes 2 hours. Driving at 100 mph takes 1 hour. Doubling your speed halves your time.
2. Sharing Resources
If you have $100 to give away as a prize, and 1 winner takes it, they get $100. If there are 10 winners, they each get $10. As the number of winners goes up, the prize per person goes down.
3. Gears and Pulleys
In a bicycle, a large gear turning slowly can drive a small gear turning quickly. The number of teeth on a gear is inversely proportional to its rotation speed. (More teeth = Slower rotation).
6. Summary Comparison
| Feature | Direct Proportion | Inverse Proportion |
|---|---|---|
| Relationship | Both go UP together | One goes UP, one goes DOWN |
| Formula | y = kx | y = k / x |
| Constant | y / x = k | x * y = k |
| Graph | Straight Line | Curved Hyperbola |
7. Conclusion
Inverse Proportion helps us understand trade-offs. It reminds us that resources are finite - whether that resource is time, money, or energy. By increasing the intensity or the workforce, we can reduce the time required, but we must respect the mathematical balance that governs the relationship.