Inverse Proportion

 
What is Inverse Proportion?

Inverse proportion describes a relationship where as one variable increases, the other decreases by the same factor. Similarly, if one decreases, the other increases proportionally.

Key Characteristics of Inverse Proportion:

• The product of the two quantities remains constant.
• There is a constant of proportionalityy (denoted as $k$).
• The equation follows the form:  $$y = \frac{k}{x}$$
• The graph of an inverse proportion relationship is a curve that never touches the axes.

 

Example:

If a journey takes 4 hours at a speed of $60 \text{km/h}$, then:

• Doubling the speed to $120 \text{km/h}$ halves the time to 2 hours.
• Halving the speed to $30 \text{km/h}$ doubles the time to 8 hours.

 

Using Inverse Proportion with Powers and Roots

Sometimes, inverse proportion problems involve powers or roots of a variable. In these cases, the relationship follows different equations.

Common Inverse Proportional Relationships:

• $y$ is inversely proportional to $x^2$ : $y = \frac{k}{x^2}$ 
• $y$ is inversely proportional to $\sqrt{x}$ : $y = \frac{k}{\sqrt{x}}$
• $y$ is inversely proportional to $x^3$ : $y = \frac{k}{x^3}$
• $y$ is inversely proportional to $\sqrt[3]{x}$ : \(y = \frac{k}{\sqrt[3]{x}\)

 

Example:

If the intensity of light is inversely proportional to the square of the distance from the source, then:

$$I = \frac{k}{d^2}$$

If $I = 20$ when $d = 2$, then:

$$20 = \frac{k}{2^2} \quad \Rightarrow 20 = \frac{k}{4} \quad \Rightarrow k = 80$$

 
So the equation is $I = \frac{80}{d^2}$.

 

Finding the Equation Between Two Inversely Proportional Variables

To find the equation for an inverse proportion problem, follow these steps:

Step 1: Write the General Formula

Identify the relationship and set up an equation with .

• If $y$ is inversely proportional to $x$: $y = \frac{k}{x}$
• If $y$ is inversely proportional to $x^2$: $y = \frac{k}{x^2}$

 

Step 2: Sub in values

Substitute given values into the equation and solve for $k$.

 

Step 3: Substitute $k$ back into original equation

Once $k$ is found, substitute it back into the equation.

 

Step 4: Use the Equation to Find Other Values

Use the final equation to calculate other values as needed.

 

Example:

It is known that $y$  is inversely proportional to $x$ .

When $x = 6$, $y = 5$.

Find $y$ when $x = 2$.

Step 1: Set Up the Equation

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

 

Step 2: Find $k$

$$5 = \frac{k}{6}$$

 $$k = 5 \times 6 = 30$$

 

Step 3: Rewrite the Equation

$$y = \frac{30}{x}$$

 

Step 4: Find $y$ when $x = 2$ 

$$y = \frac{30}{2} = 15$$

Final Answer: When $x = 2$, then $y = 15$.

 

 

 

 

Graphing Inverse Proportions

• The graph of $y = \frac{k}{x}$ is a curved hyperbola.
• The graph never touches the axes because  never reaches zero.
• For relationships involving powers, the graph will have a different curved shape.

Example: If $y = \frac{50}{x^2}$, the graph will be a steep curve approaching the axes.