📌 What is Variation?

Variation describes a mathematical relationship between two or more quantities. When one quantity changes, the other quantity (or quantities) changes in a predictable way.

Definition:

Two quantities are in direct variation if one is a constant multiple of the other. When one increases, the other increases proportionally.

\( y \propto x \) or \( y = kx \)

where \( k \) is the constant of variation (or constant of proportionality)

Key Properties of Direct Variation:

📝 Example 1 - Finding Constant and Equation:

If \( y \) varies directly with \( x \), and \( y = 12 \) when \( x = 3 \), find the equation.

Step 1: Use formula \( y = kx \)

Substitute: \( 12 = k(3) \)
Solve: \( k = 4 \)

Step 2: Write equation

\( y = 4x \)

📝 Example 2 - Solving for Unknown:

If \( y \) varies directly with \( x \), and \( y = 20 \) when \( x = 5 \), find \( y \) when \( x = 8 \).

Method 1: Find \( k \) first

\( 20 = k(5) \) → \( k = 4 \)
Equation: \( y = 4x \)
When \( x = 8 \): \( y = 4(8) = 32 \)

Method 2: Use proportion

\( \frac{y_1}{x_1} = \frac{y_2}{x_2} \)
\( \frac{20}{5} = \frac{y}{8} \)
\( 4 = \frac{y}{8} \)
\( y = 32 \)

Definition:

Two quantities are in inverse variation if their product is constant. When one increases, the other decreases proportionally.

\( y \propto \frac{1}{x} \) or \( y = \frac{k}{x} \) or \( xy = k \)

where \( k \) is the constant of variation

Key Properties of Inverse Variation:

📝 Example 1 - Inverse Variation Equation:

If \( y \) varies inversely with \( x \), and \( y = 6 \) when \( x = 4 \), find the equation.

Step 1: Use formula \( y = \frac{k}{x} \) or \( xy = k \)

Substitute: \( (6)(4) = k \)
Solve: \( k = 24 \)

Step 2: Write equation

\( y = \frac{24}{x} \) or \( xy = 24 \)

📝 Example 2 - Solving for Unknown:

If \( y \) varies inversely with \( x \), and \( y = 10 \) when \( x = 3 \), find \( y \) when \( x = 5 \).

Step 1: Find \( k \)

\( xy = k \) → \( (10)(3) = k \) → \( k = 30 \)

Step 2: Use equation to find new value

\( y = \frac{30}{x} \)
When \( x = 5 \): \( y = \frac{30}{5} = 6 \)

Alternative: Use product rule

\( x_1 y_1 = x_2 y_2 \)
\( (3)(10) = (5)y \)
\( 30 = 5y \)
\( y = 6 \)

How to Identify Types of Variation:

📝 Example - Classify from Data:

Given table, determine if it shows direct, inverse, or neither:

Check ratios: \( \frac{12}{2} = 6, \frac{6}{4} = 1.5, \frac{4}{6} \approx 0.67 \) → Not constant (not direct)
Check products: \( (2)(12) = 24, (4)(6) = 24, (6)(4) = 24, (8)(3) = 24 \) → Constant!
Answer: Inverse Variation with \( k = 24 \)

Definition:

Joint variation occurs when one quantity varies directly with two or more other quantities.

\( z = kxy \)

"z varies jointly with x and y"

📝 Example - Joint Variation:

If \( z \) varies jointly with \( x \) and \( y \), and \( z = 24 \) when \( x = 3 \) and \( y = 2 \), find \( z \) when \( x = 5 \) and \( y = 4 \).

Step 1: Find \( k \)

\( z = kxy \)
\( 24 = k(3)(2) \)
\( 24 = 6k \)
\( k = 4 \)

Step 2: Write equation and solve

\( z = 4xy \)
\( z = 4(5)(4) = 80 \)

Answer: \( z = 80 \)

Definition:

Combined variation involves both direct and inverse variation together.

📝 Example 1 - Combined Variation:

If \( z \) varies directly with \( x \) and inversely with \( y \), and \( z = 6 \) when \( x = 12 \) and \( y = 4 \), find \( z \) when \( x = 15 \) and \( y = 5 \).

Step 1: Write equation and find \( k \)

\( z = \frac{kx}{y} \)
\( 6 = \frac{k(12)}{4} \)
\( 6 = 3k \)
\( k = 2 \)

Step 2: Use equation to find new value

\( z = \frac{2x}{y} \)
\( z = \frac{2(15)}{5} = \frac{30}{5} = 6 \)

Answer: \( z = 6 \)

📝 Example 2 - More Complex:

If \( w \) varies jointly with \( x \) and \( y \), and inversely with \( z \), and \( w = 8 \) when \( x = 4, y = 6, z = 3 \), find \( w \) when \( x = 10, y = 2, z = 5 \).

Step 1: Write equation and find \( k \)

\( w = \frac{kxy}{z} \)
\( 8 = \frac{k(4)(6)}{3} \)
\( 8 = \frac{24k}{3} \)
\( 8 = 8k \)
\( k = 1 \)

Step 2: Calculate new value

\( w = \frac{xy}{z} \)
\( w = \frac{(10)(2)}{5} = \frac{20}{5} = 4 \)

Answer: \( w = 4 \)

How to Find \( k \):

📝 Practice Examples:

1. Find \( k \) if \( y \) varies directly with \( x \), and \( y = 15 \) when \( x = 3 \).

\( k = \frac{y}{x} = \frac{15}{3} = 5 \)

2. Find \( k \) if \( y \) varies inversely with \( x \), and \( y = 8 \) when \( x = 5 \).

\( k = xy = (8)(5) = 40 \)

3. Find \( k \) if \( z \) varies jointly with \( x \) and \( y \), and \( z = 36 \) when \( x = 3 \) and \( y = 4 \).

\( z = kxy \)
\( 36 = k(3)(4) = 12k \)
\( k = 3 \)

General Strategy:

1. Identify the type of variation from the problem statement
2. Write the appropriate equation with \( k \)
3. Substitute given values to find \( k \)
4. Rewrite the equation with the value of \( k \)
5. Use the equation to find unknown values

📝 Complete Problem Example:

The time \( t \) it takes to complete a job varies inversely with the number of workers \( w \). If 5 workers can complete the job in 12 hours, how long will it take 8 workers?

Step 1: Identify variation type

Inverse variation: \( t = \frac{k}{w} \)

Step 2: Find \( k \)

Given: \( t = 12 \) when \( w = 5 \)
\( 12 = \frac{k}{5} \)
\( k = 60 \)

Step 3: Write equation

\( t = \frac{60}{w} \)

Step 4: Find answer

When \( w = 8 \):
\( t = \frac{60}{8} = 7.5 \) hours

Answer: 7.5 hours (or 7 hours 30 minutes)

⚡ Quick Summary

• Direct: Both increase/decrease together; ratio is constant
• Inverse: One increases, other decreases; product is constant
• Joint: One varies directly with multiple variables
• Combined: Mix of direct and inverse
• Always find \( k \) first, then use it to solve for unknowns

📚 Real-World Applications