Inverse Variation – Definition, Formula, and Real-Life Examples

In Mathematics, inverse variation is a type of relationship between two variables where their product remains constant. When one variable increases, the other decreases proportionally. Represented by the equation xy = k, where k is a non-zero constant, inverse variation is the opposite of direct variation. This concept appears frequently in algebra, physics, and real-life situations such as speed and time, or pressure and volume. Understanding inverse variation helps in solving problems efficiently using logical reasoning and algebraic methods. 

1.0What is Inverse Variation?

Inverse variation describes a relationship between two variables such that when one increases, the other decreases proportionally, and their product remains constant.

Definition of Inverse Variation

Two quantities are said to be in inverse variation if their product is constant.
Mathematically, if x and y vary inversely, then:

$x\times y=k\text{(where k is a constant)}$

This is also referred to as the equation for inverse variation.

Formula of Inverse Variation

The standard formula of inverse variation is:

$y=\frac{k}{x}$

Where:

• y varies inversely as xx
• k is the constant of variation
• $x\ne 0$

You can rearrange it to verify:

$x.y=k$

2.0Direct and Inverse Variation

Let’s understand the difference clearly:

3.0Solved Example of Inverse Variation

$$\begin{gathered} \text{Example 1:}Ifx=4,y=6,\text{and x and y vary inversely, find the value of y when x = 8.} \\ \text{Solution:} \\ \text{From inverse variation:} \\ x\cdot y=k\Rightarrow 4\cdot 6=24 \\ \text{Now use k }=24: \\ 8\cdot y=24\Rightarrow y=\frac{24}{8}=3 \\ \text{Answer:}y=3 \end{gathered}$$

Example 2: If the speed of a car increases, the time taken to cover a fixed distance decreases.

Solution:

Let speed s and time t vary inversely:

$$\begin{gathered} s.t=k \\ \Rightarrow Speed\times Time=Distance \end{gathered}$$

This is an inverse variation in real life.

Example 3: If x = 6 and y = 4, and x and y vary inversely, find the value of y when x = 8.

Solution:

$$\begin{gathered} Sincex\cdot y=k: \\ 6\cdot 4=24 \\ \Rightarrow k=24 \\ Now,whenx=8: \\ 8\cdot y=24 \\ \Rightarrow y=\frac{24}{8}=3 \end{gathered}$$

Example 4: In an inverse variation, y = 10 when x = 5. What is the constant of variation?

Solution:

Use the formula:

$$\begin{gathered} x\cdot y=k \\ \Rightarrow 5\cdot 10=50 \end{gathered}$$

Example 5: A car travels a fixed distance in 4 hours at 60 km/h. How long will it take if the speed is increased to 80 km/h?

Solution:

$$\begin{gathered} \text{Speed and time are inversely related:} \\ {s}_1{t}_1=s_2{t}_2 \\ 60\cdot 4=80\cdot t \\ 240=80t \\ t=\frac{240}{80}=3\text{hours} \end{gathered}$$

Example 6: If y varies inversely as xx, and y = 9 when x = 2, find x when y = 6.

Solution:

$$\begin{gathered} x\cdot y=2\cdot 9=18 \\ x=\frac{18}{6}=3 \end{gathered}$$

Example 7: The variables x and y vary inversely. If x = 1, y = 6, plot the values of y for x = 2, 3, 4, 6.

Solution:

Since xy = 6,

$$\begin{gathered} x=2\Rightarrow y=3 \\ x=3\Rightarrow y=2 \\ x=4\Rightarrow y=1.5 \\ x=6\Rightarrow y=1 \end{gathered}$$

You’ll get a hyperbola when you plot these on a graph.

Example 8: 8 workers can complete a job in 12 days. How many workers are needed to complete it in 6 days?

Solution:

$$\begin{gathered} \text{Work and number of days are inversely related (more workers, less time):} \\ 8\cdot 12=x\cdot 6 \\ x=\frac{96}{6}=16\text{workers} \end{gathered}$$

Example 9: Let f(x) and g(x) be two functions such that $f(x)\cdot g(x)=k$, where k is a non-zero constant. If f(x) = 2x + 1, find g(x) and determine the value of x when g(x) = 3.

Solution:

$$\begin{gathered} Given: \\ f(x)\cdot g(x)=k \\ (2x+1)g(x)=k \\ g(x)=\frac{k}{2x+1} \\ Ifg(x)=3,then: \\ 3=\frac{k}{2x+1} \\ 2x+1=\frac{k}{3} \\ 2x=\frac{k}{3}-1 \\ x=\frac{k-3}{6}\text{So the value of xx depends on the constant kk, but the inverse relation is clearly maintained.} \end{gathered}$$

Example 10: A variable y is inversely proportional to x, and when x = a, y = b. Express y in terms of x, a, b and evaluate the limit:$\underset{x\to \infty }{\lim }y.$

Solution:

$$\begin{gathered} \text{Since xy = ab}, \\ y=\frac{ab}{x} \\ Now,{\lim }_{x\to \infty }y={\lim }_{x\to \infty }\frac{ab}{x}=0 \\ \text{This shows that as x becomes very large,}y\to 0,\text{which is a fundamental trait of inverse variation.} \end{gathered}$$

Example 11: If x and y vary inversely and x = 3, y = 8, prove that : $\frac{1}{x^2}\propto y^2$

Solution:

$$\begin{gathered} x\cdot y=k\Rightarrow y=\frac{k}{x} \\ Since: \\ {y}^2={\left(\frac{k}{x}\right)}^2=\frac{k^2}{x^2} \\ \Rightarrow y^2\propto \frac{1}{x^2} \\ \Rightarrow \frac{1}{x^2}\propto y^2 \\ \text{Proved.} \end{gathered}$$

Example 12: In physics, the intensity II of light from a point source varies inversely as the square of the distance r from the source. If the intensity at 2 meters is 100 units, what is the intensity at 5 meters?

Solution:

$$\begin{gathered} I\propto \frac{1}{r^2} \\ \Rightarrow I=\frac{k}{r^2}Atr=2: \\ 100=\frac{k}{4} \\ \Rightarrow k=400 \\ Atr=5: \\ I=\frac{400}{25}=16\text{units} \end{gathered}$$

Example 13: The time t taken to fill a tank varies inversely with the number of pipes n used. With 6 pipes, the tank fills in 4 hours. How many more pipes are needed to fill the tank in 2 hours?

Solution:

Inverse relation:

$$\begin{gathered} t\cdot n=\text{constant} \\ \Rightarrow 6\cdot 4=24 \\ \Rightarrow 2\cdot n=24 \\ \Rightarrow n=12 \end{gathered}$$

So, additional pipes = 12 - 6 = 6

4.0Inverse Variation Examples in Real Life

• Physics: Pressure and volume of a gas (Boyle’s Law)
• Work problems: More workers reduce the time to complete a job
• Economics: Demand and price (as price increases, demand often decreases)
• Travel: Speed and time (higher speed = less time)

5.0Practice Questions on Inverse Variation 

1. If x = 5 and y = 10, what is y when x = 2.5 under inverse variation?
2. A quantity y varies inversely as x, and y = 4 when x = 3. Find y when x = 6.
3. A car covers a fixed distance in 4 hours at 60 km/h. How long will it take at 80 km/h?
4. $\text{If}y\propto \frac{1}{x},\text{and y = 10 when x = 2, find y when x = 5.}$
5. The resistance R of a wire is inversely proportional to the square of its thickness d. If R = 4 ohms when d = 2 mm, find R when d = 4 mm.
6. $\text{In a triangle, if the length of a side is inversely proportional to the sine of the opposite angle, and}\sin A=\frac{1}{2},\text{then by what factor does the side change if}\sin A\text{doubles}?$