Inversely Proportional: Meaning, Formula and Examples
Some quantities move in opposite directions. Add more workers and a job takes fewer days. Drive faster and the trip takes less time. When one quantity goes up by a factor and the other goes down by the same factor, they are inversely proportional. This guide explains what that means, the formula, the graph, and how to solve inverse proportion problems, with real examples from work, travel and science.
Key takeaways
- Two quantities are inversely proportional when their product stays constant: x × y = k.
- The formula is y = k/x, written with the symbol as y ∝ 1/x.
- If x doubles, y halves. If x triples, y becomes one third.
- To solve a problem, use x₁ × y₁ = x₂ × y₂, then divide.
- The graph is a curve called a hyperbola that never touches either axis.
Table of Contents
- What Does Inversely Proportional Mean?
- The Inverse Proportion Formula
- How to Find the Constant of Proportionality
- How to Solve Inverse Proportion Problems
- The Graph of an Inverse Proportion
- Directly vs Inversely Proportional
- Real World Examples of Inverse Proportion
- Inverse Proportion vs Inverse Square
- Common Mistakes
- Practice Problems With Answers
- Frequently Asked Questions
- Keep Learning
What Does Inversely Proportional Mean?
Two quantities are inversely proportional when one increases at the same rate that the other decreases, so that their product never changes. If 6 workers need 10 days to build a shed, the job takes 6 × 10 = 60 worker days. With 12 workers it takes 5 days, and with 3 workers it takes 20 days. The product is always 60.
You will also hear this called inverse proportion, inverse variation, indirect proportion or “y varies inversely with x”. They all mean the same thing.
The key test
Multiply the two quantities. If the product is the same for every pair, they are inversely proportional. If the ratio is the same instead, they are directly proportional.
The Inverse Proportion Formula
If y is inversely proportional to x, you can write the relationship in three equivalent ways:
| Form | Formula | Best for |
|---|---|---|
| Symbol form | y ∝ 1/x | Stating the relationship in words or science |
| Product form | x × y = k | Checking a table of values |
| Function form | y = k/x | Graphing and finding any y from x |
| Two pair form | x₁ × y₁ = x₂ × y₂ | Solving word problems quickly |
Here, k is the constant of proportionality, the fixed product of the two quantities. For the shed, k = 60 worker days. Notice that y ∝ 1/x means y is directly proportional to the reciprocal of x. That is where the name “inverse” comes from.
How to Find the Constant of Proportionality
Multiply any matching pair of values. If a car takes 3 hours at 60 mph, then k = 60 × 3 = 180, which is the distance in miles. Then write the equation y = 180/x, where x is the speed and y is the time. At 90 mph the trip takes 180 ÷ 90 = 2 hours, and at 45 mph it takes 4 hours.
How to Solve Inverse Proportion Problems
Most inverse proportion questions give you one complete pair and one value from a second pair. Use these four steps.
- Confirm it is inverse. Ask: if one quantity goes up, should the other go down? If yes, it is inverse.
- Multiply the complete pair to find k.
- Divide k by the known value from the second pair.
- Check that the second pair multiplies to the same k.
Example 1: Workers and days
6 workers can finish a job in 10 days. How long will 4 workers take? More workers would be faster, so fewer workers will take longer. This is inverse. k = 6 × 10 = 60, so the time is 60 ÷ 4 = 15 days. Check: 4 × 15 = 60.
Example 2: Speed and time
A trip takes 3 hours at 60 mph. How long does it take at 90 mph? k = 60 × 3 = 180 miles, so the time is 180 ÷ 90 = 2 hours. Faster speed, less time, as expected.
Example 3: Pipes filling a tank
3 identical pipes fill a tank in 8 hours. How long would 4 pipes take? k = 3 × 8 = 24 pipe hours, so 4 pipes take 24 ÷ 4 = 6 hours.
Let the calculator do the work
Our proportion calculator has an inverse mode. Enter three values, switch to Inverse, and get the answer with every step and a curved graph.
The Graph of an Inverse Proportion
When you plot y = k/x for positive values, you get a smooth curve called a hyperbola. It drops steeply at first, then flattens out. The curve gets closer and closer to both axes but never touches them, because neither x nor y can ever be zero when their product is a fixed positive number.
| x | 1 | 2 | 3 | 4 | 6 | 12 |
|---|---|---|---|---|---|---|
| y = 12/x | 12 | 6 | 4 | 3 | 2 | 1 |
| x × y | 12 | 12 | 12 | 12 | 12 | 12 |
Compare this with a direct proportion, whose graph is a straight line through the origin. Our guide to proportional relationships shows that graph side by side with a line that is not proportional.
Directly vs Inversely Proportional
| Feature | Directly proportional | Inversely proportional |
|---|---|---|
| What happens | Both go up or down together | One goes up as the other goes down |
| What stays constant | The ratio y/x | The product x × y |
| Formula | y = kx | y = k/x |
| Symbol | y ∝ x | y ∝ 1/x |
| If x doubles | y doubles | y halves |
| Graph | Straight line through the origin | Hyperbola that never touches the axes |
| Solving setup | x₁/y₁ = x₂/y₂ | x₁ × y₁ = x₂ × y₂ |
| Example | More gallons, more miles | More speed, less time |
How to tell which one you have
Ask yourself what should happen in real life if one quantity doubles. If the other should double too, the relationship is direct. If the other should halve, it is inverse. Then confirm with numbers: a constant ratio means direct, and a constant product means inverse. If neither stays constant, the relationship is not proportional at all.
Real World Examples of Inverse Proportion
| Situation | Constant product (k) | Worked example |
|---|---|---|
| Workers and days to finish a job | Total worker days | 6 workers × 10 days = 60, so 4 workers need 15 days |
| Speed and travel time | Distance | 60 mph × 3 h = 180 miles, so 90 mph takes 2 h |
| People sharing a bill | Total bill | $48 split 4 ways is $12, split 6 ways is $8 |
| Gear teeth and rotation speed | Teeth × rpm | 40 teeth at 300 rpm drives a 20 tooth gear at 600 rpm |
| Pressure and volume of a gas (Boyle’s law) | P × V at constant temperature | 2.0 L at 1.0 atm becomes 0.5 L at 4.0 atm |
| Current and resistance at a fixed voltage | Voltage (V = I × R) | 12 V across 4 Ω gives 3 A, across 6 Ω gives 2 A |
| Sound frequency and wavelength | Speed of sound | At 343 m/s, 343 Hz has a 1 m wavelength and 686 Hz has 0.5 m |
Inverse proportion in science: Boyle’s law
Boyle’s law is a classic inverse proportion. For a fixed amount of gas at constant temperature, pressure and volume are inversely proportional, so P₁ × V₁ = P₂ × V₂. Squeeze a gas to a quarter of its volume and its pressure becomes four times higher. The free OpenStax Chemistry chapter on the gas laws explains Boyle’s law with more examples and data.
Inverse proportion in electronics: Ohm’s law
Ohm’s law says V = I × R. When the voltage V stays fixed, current I and resistance R are inversely proportional. Doubling the resistance halves the current. This is why a dimmer switch that adds resistance makes a bulb glow less brightly.
Inverse Proportion vs Inverse Square
Be careful with laws that say “inversely proportional to the square”. Gravity and the brightness of light follow an inverse square law: y = k/x². If the distance doubles, the strength drops to one quarter, not one half. That is a different relationship from a simple inverse proportion, and the product x × y does not stay constant. Instead, x² × y stays constant.
Common Mistakes
- Using the direct method. Writing 6/10 = 4/x for the workers problem gives about 6.7 days, which is wrong, because fewer workers should take longer. Multiply instead.
- Assuming any decrease is inverse. A candle that burns down 1 inch per hour is decreasing, but height × time is not constant. It is a linear decrease, not an inverse proportion.
- Mixing up inverse and inverse square. Light and gravity weaken with the square of the distance.
- Forgetting the conditions. Workers and days are only inversely proportional if every worker works at the same pace on a job that can be shared.
- Allowing zero. In an inverse proportion, neither quantity can be zero, because k divided by 0 has no value.
Practice Problems With Answers
1. y is inversely proportional to x. When x = 5, y = 8. Find y when x = 10.
Show answer
k = 5 × 8 = 40, so y = 40 ÷ 10 = 4.
2. 8 people can paint a house in 6 days. How many days would 12 people take?
Show answer
k = 8 × 6 = 48 person days, so 48 ÷ 12 = 4 days.
3. A trip takes 4 hours at 55 mph. How fast must you drive to make it in 5 hours?
Show answer
k = 55 × 4 = 220 miles, so the speed is 220 ÷ 5 = 44 mph.
4. Is this table inversely proportional? x: 2, 4, 5 and y: 30, 15, 12.
Show answer
Yes. 2 × 30 = 60, 4 × 15 = 60 and 5 × 12 = 60, so y = 60/x.
5. A gas fills 3 L at 2 atm. What is its volume at 6 atm, at the same temperature?
Show answer
By Boyle’s law, 2 × 3 = 6 × V, so V = 1 L.
6. A 30 tooth gear turns at 200 rpm and drives a 50 tooth gear. How fast does the second gear turn?
Show answer
30 × 200 = 6,000, so the second gear turns at 6,000 ÷ 50 = 120 rpm.
Frequently Asked Questions
What does inversely proportional mean?
It means that as one quantity increases, the other decreases by the same factor, so their product stays the same. If one doubles, the other halves.
What is the formula for inversely proportional?
The formula is y = k/x, where k is a constant. You can also write it as x × y = k, or as x₁ × y₁ = x₂ × y₂ when you compare two pairs of values.
What is the symbol for inversely proportional?
Use the proportionality symbol ∝ with a reciprocal: y ∝ 1/x. It reads “y is inversely proportional to x”.
What is an example of inversely proportional?
Speed and travel time over a fixed distance. A 120 mile trip takes 2 hours at 60 mph and 3 hours at 40 mph, because speed × time always equals 120.
What does an inversely proportional graph look like?
It is a curve called a hyperbola. It falls steeply, then levels off, and it approaches both axes without ever touching them.
What is the difference between directly and inversely proportional?
In a direct proportion, both quantities rise together and their ratio is constant. In an inverse proportion, one rises as the other falls and their product is constant.
How do you find k in an inverse proportion?
Multiply any matching pair of values. If 5 workers take 12 days, k = 5 × 12 = 60 worker days.
How do you solve an inverse proportion problem?
Multiply the complete pair to find k, then divide k by the known value of the second pair. If 3 pipes fill a tank in 8 hours, 4 pipes take 24 ÷ 4 = 6 hours.
Is inversely proportional the same as indirectly proportional?
Yes. Inversely proportional, indirectly proportional, inverse proportion and inverse variation all describe the same relationship, y = k/x.
Can a value be zero in an inverse proportion?
No. Since x × y equals a fixed nonzero number, neither x nor y can be zero. That is why the graph never touches the axes.
Keep Learning
- Review the basics of what a proportion is, including its types and properties.
- See the opposite pattern in our guide to proportional relationships.
- Solve inverse problems instantly with the inverse proportion calculator.
- Simplify, scale or split ratios with the ratio calculator.