Variation

Variation describes how one quantity changes in relation to another. When two or more quantities are connected by a rule, we say they vary with each other. Understanding variation allows us to express real-life relationships algebraically, find unknown quantities, and solve practical problems.

Before You Begin: Make sure you are confident with Ratios and Percentages and Rates of Change, especially direct and inverse proportion. Variation extends those ideas using index notation and the constant of variation k.

In this article

Variation
Direct Variation
Inverse Variation
Joint Variation
Combined Variation
Summary Table

Direct Variation

When we say y varies directly as x, we mean that as x increases, y increases in the same proportion, and as x decreases, y decreases in the same proportion. The ratio y : x always stays constant.

Examples of direct variation in real life:

The cost of fuel varies directly as the number of litres bought.
The distance a car travels varies directly as the time spent driving at a constant speed.
The area of a square varies directly as the square of its side length.

Notation and Equation

We write direct variation using the proportionality symbol ∝:

y ∝ xⁿ where n ∈ ℚ

This is then expressed as an equation by introducing the constant of variation k:

y = kxⁿ

Where:

k is the constant of variation (always positive for direct variation)
n is a rational number (commonly 1, 2, 3, or ½)
When n = 1: y varies directly as x → y = kx
When n = 2: y varies directly as the square of x → y = kx²
When n = ½: y varies directly as the square root of x → y = k√x

Finding k

To find k, substitute a known pair of values (x, y) into the equation y = kxⁿ and solve for k.

Example 1: Direct Variation (n = 1)

y varies directly as x. When x = 4, y = 20. Find the equation connecting y and x, then find y when x = 7.

Step 1: Write the variation statement as an equation.

y = kx

Step 2: Substitute the known values to find k.

20 = k × 4
k = 20 ÷ 4 = 5

Step 3: Write the complete equation.

y = 5x

Step 4: Substitute x = 7 to find y.

y = 5 × 7 = 35

Answer: The equation is y = 5x. When x = 7, y = 35.

Example 2: Direct Variation (n = 2)

y varies directly as the square of x. When x = 3, y = 36. Find y when x = 5.

Step 1: Write the equation.

y = kx²

Step 2: Substitute x = 3, y = 36 to find k.

36 = k × 3²
36 = 9k
k = 36 ÷ 9 = 4

Step 3: Write the complete equation.

y = 4x²

Step 4: Find y when x = 5.

y = 4 × 5² = 4 × 25 = 100

Answer: The equation is y = 4x². When x = 5, y = 100.

Example 3: Finding x given y

Using the equation y = 4x² from Example 2, find x when y = 64.

Step 1: Substitute y = 64 into y = 4x².

64 = 4x²
x² = 64 ÷ 4 = 16
x = √16 = 4

Answer: When y = 64, x = 4.

Key Rule For Direct Variation:

y ∝ xⁿ means y = kxⁿ. To find k, substitute a known pair of values. Once k is known, substitute any value of x to find y, or any value of y to find x.

Inverse Variation

When we say y varies inversely as x, we mean that as x increases, y decreases in the same proportion, and as x decreases, y increases. The product y × xⁿ always stays constant (equal to k).

Examples of inverse variation in real life:

The time taken to complete a job varies inversely as the number of workers.
The pressure of a gas varies inversely as its volume (at constant temperature).
The brightness of a light source varies inversely as the square of the distance from it.

Notation and Equation

y ∝ $\frac{1}{x^{n}}$ where n ∈ ℚ

Expressed as an equation:

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

Where:

When n = 1: y varies inversely as x → y = k/x
When n = 2: y varies inversely as the square of x → y = k/x²
When n = ½: y varies inversely as the square root of x → y = k/√x

Example 4: Inverse Variation (n = 1)

y varies inversely as x. When x = 5, y = 12. Find the equation connecting y and x, then find y when x = 20.

Step 1: Write the equation.

y = k/x

Step 2: Substitute x = 5, y = 12 to find k.

12 = k/5
k = 12 × 5 = 60

Step 3: Write the complete equation.

y = 60/x

Step 4: Find y when x = 20.

y = 60/20 = 3

Answer: The equation is y = 60/x. When x = 20, y = 3.

Example 5: Inverse Variation (n = 2)

y varies inversely as the square of x. When x = 2, y = 25. Find y when x = 5.

Step 1: Write the equation.

y = k/x²

Step 2: Substitute x = 2, y = 25 to find k.

25 = k/2²
25 = k/4
k = 25 × 4 = 100

Step 3: Write the complete equation.

y = 100/x²

Step 4: Find y when x = 5.

y = 100/5² = 100/25 = 4

Answer: The equation is y = 100/x². When x = 5, y = 4.

Example 6: Finding x given y (Inverse Variation)

Using y = 60/x from Example 4, find x when y = 4.

Step 1: Substitute y = 4.

4 = 60/x
x = 60/4 = 15

Answer: When y = 4, x = 15.

Key Rule For Inverse Variation:

y ∝ 1/xⁿ means y = k/xⁿ, which can be rewritten as yxⁿ = k. As one variable increases, the other decreases. The product yxⁿ is always equal to k.

Joint Variation

Joint variation occurs when one quantity varies directly as two or more other quantities at the same time. All of the related quantities appear in the numerator of the equation.

Examples of joint variation in real life:

The volume of a cylinder varies jointly as its height and the square of its radius: V = πr²h.
The earnings of a worker vary jointly as the number of hours worked and the hourly rate.
The distance a vehicle travels varies jointly as its speed and the time spent travelling.

Notation and Equation

If y varies jointly as x^m and zⁿ, we write:

y ∝ x^mzⁿ where m, n ∈ ℚ

Expressed as an equation:

y = kx^mzⁿ

Where k is the constant of variation, found by substituting a known set of values.

Example 7: Joint Variation (m = 1, n = 1)

y varies jointly as x and z. When x = 2 and z = 3, y = 24. Find y when x = 5 and z = 4.

Step 1: Write the equation.

y = kxz

Step 2: Substitute x = 2, z = 3, y = 24 to find k.

24 = k × 2 × 3
24 = 6k
k = 24 ÷ 6 = 4

Step 3: Write the complete equation.

y = 4xz

Step 4: Find y when x = 5, z = 4.

y = 4 × 5 × 4 = 80

Answer: The equation is y = 4xz. When x = 5 and z = 4, y = 80.

Example 8: Joint Variation with Powers

y varies jointly as the square of x and the cube of z. When x = 2 and z = 1, y = 16. Find y when x = 3 and z = 2.

Step 1: Write the equation.

y = kx²z³

Step 2: Substitute x = 2, z = 1, y = 16 to find k.

16 = k × 2² × 1³
16 = k × 4 × 1
k = 16 ÷ 4 = 4

Step 3: Write the complete equation.

y = 4x²z³

Step 4: Find y when x = 3, z = 2.

y = 4 × 3² × 2³ = 4 × 9 × 8 = 288

Answer: The equation is y = 4x²z³. When x = 3 and z = 2, y = 288.

Example 9: Finding a Variable (Joint Variation)

Given y = 4xz from Example 7, find z when y = 60 and x = 3.

Step 1: Substitute y = 60 and x = 3.

60 = 4 × 3 × z
60 = 12z
z = 60 ÷ 12 = 5

Answer: When y = 60 and x = 3, z = 5.

Key Rule For Joint Variation:

y ∝ x^mzⁿ means y = kx^mzⁿ. All varying quantities appear in the numerator. Substitute a complete known set of values to find k first, then solve for any unknown.

Combined Variation

Combined variation (also called partial variation by some texts) occurs when one quantity varies directly as some variables and inversely as others at the same time. The direct variables appear in the numerator and the inverse variables appear in the denominator.

Examples of combined variation in real life:

The time to complete a task varies directly as the amount of work and inversely as the number of workers.
The gravitational force between two objects varies directly as their masses and inversely as the square of the distance between them.
The resistance of a wire varies directly as its length and inversely as the square of its radius.

Notation and Equation

If y varies directly as x^m and inversely as zⁿ, we write:

y ∝ $\frac{x^{m}}{z^{n}}$ where m, n ∈ ℚ

Expressed as an equation:

y = k $\frac{x^{m}}{z^{n}}$

Notice the structure: direct variables on top, inverse variables on the bottom. k is found by substituting a complete known set of values.

Example 10: Combined Variation (Direct and Inverse, n = m = 1)

y varies directly as x and inversely as z. When x = 6 and z = 2, y = 9. Find y when x = 10 and z = 5.

Step 1: Write the equation.

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

Step 2: Substitute x = 6, z = 2, y = 9 to find k.

9 = k × $\frac{6}{2}$
9 = 3k
k = 9 ÷ 3 = 3

Step 3: Write the complete equation.

y = $\frac{3 x}{z}$

Step 4: Find y when x = 10, z = 5.

y = $\frac{3 \times 10}{5}$ = $\frac{30}{5}$ = 6

Answer: The equation is y = $\frac{3 x}{z}$ . When x = 10 and z = 5, y = 6.

Example 11: Combined Variation with Powers

y varies directly as the square of x and inversely as z. When x = 3 and z = 3, y = 12. Find y when x = 6 and z = 4.

Step 1: Write the equation.

y = k $\frac{x^{2}}{z}$

Step 2: Substitute x = 3, z = 3, y = 12 to find k.

12 = k × $\frac{3^{2}}{3}$
12 = k × $\frac{9}{3}$
12 = 3k
k = 12 ÷ 3 = 4

Step 3: Write the complete equation.

y = $\frac{4 x^{2}}{z}$

Step 4: Find y when x = 6, z = 4.

y = 4 × $\frac{6^{2}}{4}$ = 4 × $\frac{36}{4}$ = $\frac{144}{4}$ = 36

Answer: The equation is y = $\frac{4 x^{2}}{z}$ . When x = 6 and z = 4, y = 36.

Example 12: Finding a Variable (Combined Variation)

Given y = $\frac{3 x}{z}$ from Example 10, find z when y = 5 and x = 15.

Step 1: Substitute y = 5 and x = 15.

5 = $\frac{3 \times 15}{z}$
5 = $\frac{45}{z}$
z = $\frac{45}{5}$ = 9

Answer: When y = 5 and x = 15, z = 9.

Example 13: Real-Life Combined Variation

The time T (in hours) taken to build a wall varies directly as the area A (in m²) of the wall and inversely as the number of workers W. When A = 30 m² and W = 5, T = 6 hours.

Find T when A = 48 m² and W = 8.

Step 1: Write the equation.

T = k $\frac{A}{W}$

Step 2: Substitute A = 30, W = 5, T = 6 to find k.

6 = k × $\frac{30}{5}$
6 = 6k
k = 1

Step 3: Write the complete equation.

T = $\frac{A}{W}$

Step 4: Find T when A = 48, W = 8.

T = $\frac{48}{8}$ = 6 hours

Answer: It will take 6 hours to build the wall with 8 workers covering 48 m².

Key Rule For Combined Variation:

y ∝ x^m/zⁿ means y = kx^m/zⁿ. Direct variables go in the numerator; inverse variables go in the denominator. Always find k first using a complete set of known values, then substitute to find the unknown.

Summary Table: Types of Variation

Type	Proportionality Statement	Equation	What stays constant?
Direct	y ∝ xⁿ	y = kxⁿ	y/xⁿ = k
Inverse	y ∝ 1/xⁿ	y = k/xⁿ	yxⁿ = k
Joint	y ∝ x^mzⁿ	y = kx^mzⁿ	y/(x^mzⁿ) = k
Combined	y ∝ x^m/zⁿ	y = kx^m/zⁿ	yzⁿ/x^m = k

Key Takeaways:

• All variation problems follow the same three steps: write the equation, find k using known values, then solve for the unknown.

• The symbol ∝ means "is proportional to." Replace it with "= k ×" to form the equation.

• Direct variation: y increases as x increases. Equation: y = kxⁿ.

• Inverse variation: y decreases as x increases. Equation: y = $\frac{k}{x^{n}}$ .

• Joint variation: y varies directly as two or more variables. Equation: y = kx^mzⁿ.

• Combined variation: y varies directly as some variables and inversely as others. Equation: y = $\frac{k x^{m}}{z^{n}}$ .

• Always find k first before solving for any other unknown.

NB: Variation appears throughout mathematics and science. Recognising the type of variation from a word problem, and knowing which equation to write, is the most important skill. Once the equation is set up and k is found, the rest is substitution and algebra.