Inverse Proportion

Inverse proportion is a math concept where one quantity increases while the other decreases, but their product stays constant. For instance, if workers double in number, the time to finish a job halves, like sharing the load evenly. We use the symbol "∝̸". It shows the opposite behavior of direct proportion.

Table of Contents

• What is Inverse Proportion?
• Inverse Proportion Formula
• Inverse Proportion Graph
• Direct and Inverse Proportion
• Inverse Proportion Solved Examples
• Inverse Proportion Practice Questions

What is Inverse Proportion?

When two quantities are related in such a way that an increase in one causes a decrease in the other by the same factor, they are said to be in inverse proportion. The relationship works in both directions. If one quantity doubles, the other becomes half. If one quantity is reduced to one-third, the other becomes three times as large.

The most important thing to understand is that the product of the two quantities always stays the same. This fixed value is called the constant of proportionality, usually written as k.

Inverse Proportion Formula

The standard formula for inverse proportion is written as:

y = k / x

This can also be written as:

x × y = k

where, k is the constant of proportionality.

y increases as x decreases.

y decreases as x increases.

In this formula, x is the input value, y is the output value, and k is the constant of proportionality. The symbol k represents the fixed product that stays the same no matter what values x and y take.

Inverse Proportion Graph

When the values of x and y in an inverse proportion relationship are plotted on a graph, the result is a curve called a rectangular hyperbola. This is one of the most recognisable curves in mathematics.

The curve starts very high on the y-axis side and sweeps down smoothly toward the x-axis as x increases. It never actually touches either axis, no matter how far it is extended. The two axes act as asymptotes, meaning the curve approaches them infinitely but never reaches or crosses them.

Direct and Inverse Proportion

Direct proportion and inverse proportion are often studied together because they both describe how two quantities relate to each other. However, they behave in completely opposite ways.

Direct Proportion: When one quantity increases, the other increases by the same factor. When one decreases, the other decreases too. The formula is y = kx, and the graph is a straight line passing through the origin. The ratio of y to x stays constant, that is, y divided by x always equals k.

Inverse Proportion: When one quantity increases, the other decreases by the same factor. The formula is y = k/x, and the graph is a rectangular hyperbola. The product of x and y stays constant, that is, x multiplied by y always equals k.

Aspect	Direct Proportion	Inverse Proportion
Relationship	y increases as x increases	y decreases as x increases
Formula	y = kx	y = k/x
Constant	y/x = k	x × y = k
Graph	Straight line through origin	Rectangular hyperbola
If x doubles	y doubles	y halves

Inverse Proportion Solved Examples

Example 1: Six workers can build a wall in 10 days. How many days will it take if 12 workers are used?

Solution: Step 1: Find k. k = 6 × 10, so k = 60.

Step 2: Find the new time. Days = 60 ÷ 12, so days = 5.

12 workers will take 5 days.

Example 2: A car travelling at 60 kilometres per hour takes 4 hours to reach its destination. How long will the journey take if the car travels at 80 kilometres per hour?

Solution: Step 1: Find k. k = 60 × 4, so k = 240.

Step 2: Find the new time. Time = 240 ÷ 80, so time = 3.

The journey will take 3 hours.

Example 3: Three pipes take 8 hours to fill a tank. How long will it take if 4 pipes are used?

Solution: Step 1: Find k. k = 3 × 8, so k = 24.

Step 2: Find the new time. Time = 24 ÷ 4, so time = 6.

Four pipes will fill the tank in 6 hours.

Example 4: The table below shows values of x and y that are in inverse proportion.

x: 2, 4, 8 y: 24, 12?

Solution: Step 1: Find k using the first pair. k = 2 × 24, so k = 48.

Step 2: Find the missing y. y = 48 ÷ 8, so y = 6.

The missing value is 6.

Example 5: y is inversely proportional to x. When x = 9, y = 4. Find x when y = 6.

Solution: Step 1: Find k. k = 9 × 4, so k = 36.

Step 2: Find x. x = 36 ÷ 6, so x = 6.

x = 6.

Inverse Proportion Practice Questions

1. Five taps can fill a tank in 18 minutes. How long will it take 9 taps to fill the same tank?
2. y is inversely proportional to x. When x = 7, y = 4. Find y when x = 14.
3. 12 machines can pack boxes in 5 hours. How many machines are needed to complete the same job in 3 hours?
4. x and y are in inverse proportion. When x = 3, y = 20. Find y when x = 12.
5. A train travelling at 80 kilometres per hour takes 6 hours to complete a journey. How fast must it travel to complete the same journey in 4 hours?