Direct Proportion

 
What is Direct Proportion?

Direct proportion describes a consistent relationship between two variables. This means that as one variable increases, the other increases by the same factor. Similarly, if one decreases, the other decreases proportionally.

Key Characteristics of Direct Proportion:

• The ratio between the two quantities remains constant.
• There is a constant of proportionality (denoted as $k$).
• The equation follows the form: $$y = kx$$
• The graph of a directly proportional relationship is a straight line through the origin with gradient $k$.

 

Example:

If a worker earns $\pounds 15$ per hour, then:

• 2 hours = £30
• 5 hours = £75
• The ratio of hours:earnings always remains the same.

 

Using Direct Proportion with Powers and Roots

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

Common Proportional Relationships:

• $y$ is directly proportional to $x^2$: $y = kx^2$
• $y$ is directly proportional to $\sqrt{x}$: $y = k\sqrt{x}$
• $y$ is directly proportional to $x^3$: $y = kx^3$
• $y$ is directly proportional to $\sqrt[3]{x}$: $y = k\sqrt[3]{x}$

 

Example:

If the area of a circle is directly proportional to the square of its radius, then:

$$A = kr^2$$

If $A = 50$ when $r = 5$, then:

$$50 = k(5^2) \Rightarrow 50 = 25k \Rightarrow k = 2$$

So the equation is $A = 2r^2$.

 
Finding the Equation Between Two Directly Proportional Variables

To find the equation for a direct proportion problem, follow these steps:

Step 1: Write the General Formula

Identify the relationship and set up an equation with $k$.

If $y$ is proportional to $x$: $y = kx$
If $y$ is proportional to $x^2$: $y = kx^2$

Step 2: Find $k$

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

 

Step 3: Rewrite the Equation with $k$

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 directly proportional to $x^2$.

When $x = 3$, $y = 18$.
Find $y$ when $x = 4$.

Step 1: Set Up the Equation

$$y = kx^2$$

Step 2: Find $k$

$$18 = k(3^2)$$

$$18 = 9k \Rightarrow k = 2$$

Step 3: Rewrite the Equation

$$y = 2x^2$$

Step 4: Find $y$ when $x = 4$

$$y = 2(4^2)$$

$$y = 2(16) = 32$$

Final Answer: When $x = 4$, then $y = 32$.

 
Graphing Direct Proportions

The graph of $y = kx$ is a straight line through the origin.

The gradient = $k$.

For relationships involving powers, the graph will have a curved shape.

Example: If $y = 2x^2$, the graph will be a parabola passing through the origin.

 

 

 

 

 

 

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.