Chapter 11: Variation

Overview

Welcome to Form 5 Mathematics! In Chapter 11, you'll explore the fascinating world of variation - how quantities relate to each other in proportional ways. You'll learn about direct variation, inverse variation, and combined variation, and how to solve real-world problems involving these relationships. Variation is fundamental to understanding relationships in physics, economics, and many other fields.

What You'll Learn:

• Understand direct, inverse, and combined variation
• Convert variation relationships to equations
• Solve problems involving various types of variation
• Apply variation concepts to real-world scenarios

Learning Objectives

After completing this chapter, you will be able to:

• Explain the meaning of direct, inverse, and combined variation
• Solve problems involving direct, inverse, and combined variation

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable increases when another increases, and vice versa, at the same rate. Written as y ∝ xⁿ.

General Form:

$$y \propto x$$

Equation Form:

$$y = kx$$

Where k is the constant of variation.

Relationship Diagram:

Visual Representation:

Examples:

• Distance traveled varies directly with time at constant speed
• Area of circle varies directly with radius squared ($A = \pi r^2$)
• Cost varies directly with quantity
• Wages vary directly with hours worked

Inverse Variation

Inverse variation describes a relationship where one variable increases when another decreases, and vice versa. Written as y ∝ 1/xⁿ.

General Form:

$$y \propto \frac{1}{x}$$

Equation Form:

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

Where k is the constant of variation.

Relationship Diagram:

Visual Representation:

Mathematical Properties:

• Hyperbolic relationship: The graph forms a hyperbola
• Constant product: $x \times y = k$ (always constant)
• Asymptotes: $x = 0$ and $y = 0$ are vertical/horizontal asymptotes

Examples:

• Time taken varies inversely with speed for fixed distance ($t = \frac{d}{v}$)
• Pressure varies inversely with volume (Boyle's Law: $PV = k$)
• Number of workers varies inversely with time for fixed task
• Brightness varies inversely with square of distance from light source

Combined Variation

Combined variation is a combination of direct and inverse variation. There are different types:

Relationship Diagram:

Joint Variation (Direct with multiple variables):

$$y \propto xz$$

Equation Form:

$$y = kxz$$

Examples:

• Volume of rectangular prism varies directly with length, width, and height
• Area of rectangle varies directly with length and width
• Cost varies directly with price and quantity

Mixed Variation (Direct with some, inverse with others): Example: y varies directly with x and inversely with z

General Form:

$$y \propto \frac{x}{z}$$

Equation Form:

$$y = \frac{kx}{z}$$

Examples:

• Time varies directly with distance and inversely with speed
• Cost varies directly with quantity and inversely with efficiency
• Pressure varies directly with force and inversely with area ($P = \frac{F}{A}$)

Important Formulas and Methods

Variation Problem-Solving Steps

1. Write the relationship in variation form:

   • Direct: y ∝ x
   • Inverse: y ∝ 1/x
   • Combined: y ∝ x/z (example)

2. Convert to equation form with constant k:

   • Direct: y = kx
   • Inverse: y = k/x
   • Combined: y = kx/z

3. Substitute given values to find k:

   • Use one set of given values to solve for k

4. Write the complete equation:

   • Substitute k back into the equation

5. Use the equation to solve the problem:

   • Substitute the new values to find the unknown

Common Variation Relationships

Variation Graphs Visualization

Direct Variation Graph:

Inverse Variation Graph:

Problem-Solving Workflow:

Step-by-Step Solved Examples

Example 1: Direct Variation

Problem: If y varies directly as x, and y = 12 when x = 4, find: a) The constant of variation k b) The equation relating y and x c) The value of y when x = 7

Solution Flow:

Solution: a) Find k:

$$y = kx$$ $$12 = k \times 4$$ $$k = \frac{12}{4} = 3$$

b) Write equation:

$$y = 3x$$

c) Find y when x = 7:

$$y = 3 \times 7 = 21$$

Verification:

$$\frac{y}{x} = \frac{12}{4} = 3 \quad \text{and} \quad \frac{21}{7} = 3 \quad \text{(consistent ratio)}$$

Answer: a) k = 3, b) y = 3x, c) y = 21

Example 2: Inverse Variation

Problem: If y varies inversely as x, and y = 8 when x = 3, find: a) The constant of variation k b) The equation relating y and x c) The value of y when x = 12

Solution Flow:

Solution: a) Find k:

$$y = \frac{k}{x}$$ $$8 = \frac{k}{3}$$ $$k = 8 \times 3 = 24$$

b) Write equation:

$$y = \frac{24}{x}$$

c) Find y when x = 12:

$$y = \frac{24}{12} = 2$$

Verification:

$$x \times y = 3 \times 8 = 24 \quad \text{and} \quad 12 \times 2 = 24 \quad \text{(constant product)}$$

Answer: a) k = 24, b) y = 24/x, c) y = 2

Example 3: Combined Variation

Problem: If z varies directly as x and inversely as y, and z = 6 when x = 8 and y = 4, find: a) The constant of variation k b) The equation relating z, x, and y c) The value of z when x = 10 and y = 5

Solution Flow:

Solution: a) Find k:

$$z = \frac{kx}{y}$$ $$6 = \frac{k \times 8}{4}$$ $$6 = 2k$$ $$k = \frac{6}{2} = 3$$

b) Write equation:

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

c) Find z when x = 10 and y = 5:

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

Verification:

$$\frac{z \times y}{x} = \frac{6 \times 4}{8} = \frac{24}{8} = 3 \quad \text{and} \quad \frac{6 \times 5}{10} = \frac{30}{10} = 3 \quad \text{(consistent)}$$

Answer: a) k = 3, b) z = 3x/y, c) z = 6

Example 4: Real-world Application - Physics

Problem: The pressure P of a gas varies inversely as its volume V when temperature is constant. If P = 100 kPa when V = 2 L, find the pressure when V = 5 L.

Solution: Given: P ∝ 1/V (inverse variation) Equation: P = k/V

Find k: 100 = k/2 k = 100 × 2 = 200

Complete equation: P = 200/V

Find P when V = 5: P = 200/5 = 40 kPa

Answer: Pressure is 40 kPa

Example 5: Real-world Application - Business

Problem: A company finds that its revenue R varies directly with the number of units sold n, and the cost C varies directly with n. If R = RM10,000 when n = 100 and C = RM6,000 when n = 100, find: a) The equations for R and C b) The profit when n = 200 units are sold

Solution: a) Find equations:

• Revenue: R = $k_1$n, 10,000 = $k_1$ × 100, $k_1$ = 100
• Cost: C = $k_2$n, 6,000 = $k_2$ × 100, $k_2$ = 60
• Equations: R = 100n, C = 60n

b) Find profit when n = 200:

• Revenue: R = 100 × 200 = RM20,000
• Cost: C = 60 × 200 = RM12,000
• Profit: P = R - C = 20,000 - 12,000 = RM8,000

Answer: a) R = 100n, C = 60n; b) Profit = RM8,000

Example 6: Complex Variation Problem

Problem: The time T to complete a task varies directly as the amount of work W and inversely as the number of workers N. If 6 workers can complete 3 units of work in 4 hours, how long will it take 8 workers to complete 6 units of work?

Solution: Given relationship: T ∝ W/N (combined variation) Equation: T = kW/N

Find k using given values: 4 = k × 3/6 4 = k/2 k = 8

Complete equation: T = 8W/N

Find T for new conditions: W = 6, N = 8 T = 8 × 6/8 = 48/8 = 6 hours

Answer: It will take 6 hours

Advanced Variation Patterns

Power Variation

Power variation involves exponents other than 1:

Direct Power Variation:

$$y \propto x^n \quad \Rightarrow \quad y = kx^n$$

Inverse Power Variation:

$$y \propto \frac{1}{x^n} \quad \Rightarrow \quad y = \frac{k}{x^n}$$

Examples:

• Area of circle: $A = \pi r^2$ (direct power variation, n = 2)
• Volume of sphere: $V = \frac{4}{3}\pi r^3$ (direct power variation, n = 3)
• Gravitational force: $F = \frac{Gm_1m_2}{r^2}$ (inverse square variation, n = 2)

Power Variation Comparison:

Multiple Variable Variation

When more than two variables are involved:

Three Variables:

• $z \propto xy$ (joint variation)
• $z \propto \frac{x}{y}$ (combined variation)
• $z \propto \frac{xy}{w}$ (multiple combined variation)

Example: A company's profit P varies directly with sales S and inversely with costs C, and directly with advertising budget A:

$$P \propto \frac{SA}{C} \quad \Rightarrow \quad P = k\frac{SA}{C}$$

Variation Problem Types

Real-world Applications

1. Physics and Engineering

• Direct: Force ∝ acceleration (F = ma)
• Inverse: Pressure ∝ 1/volume (Boyle's Law)
• Joint: Volume ∝ length × width × height

2. Economics and Business

• Direct: Cost ∝ quantity, Revenue ∝ sales
• Inverse: Time ∝ 1/productivity
• Combined: Profit ∝ revenue - costs

3. Biology and Medicine

• Direct: Drug dosage ∝ body weight
• Inverse: Reaction time ∝ 1/concentration
• Joint: Area of skin ∝ height × width

4. Geography and Agriculture

• Direct: Crop yield ∝ rainfall, Population ∝ area
• Inverse: Population density ∝ 1/area
• Combined: Agricultural output ∝ rainfall × soil quality

Important Terms

Summary Points

• Direct Variation: y ∝ x → y = kx (same direction change)
• Inverse Variation: y ∝ 1/x → y = k/x (opposite direction change)
• Joint Variation: y ∝ xz → y = kxz (direct with multiple variables)
• Combined Variation: y ∝ x/z → y = kx/z (mix of direct/inverse)
• Problem Steps: Write relationship → Convert to equation → Find k → Solve problem
• Applications: Physics, business, biology, everyday life

Practice Tips for SPM Students

1. Master the Concepts

• Understand the difference between direct and inverse variation
• Learn to identify variation types from word problems
• Practice converting between proportionality and equation forms

2. Problem-Solving Strategies

• Follow the 5-step method consistently
• Always find the constant k first
• Write complete equations before solving for unknowns
• Check your answers for reasonableness

3. Real-world Connections

• Relate variation to everyday situations
• Practice physics and business applications
• Understand the practical importance of variation relationships

4. Common Mistakes to Avoid

• Confusing direct and inverse variation
• Forgetting to find the constant k
• Misidentifying variables in combined variation
• Calculation errors in solving for k or unknown values

SPM Exam Tips

Paper 1 (Multiple Choice)

• Look for key words indicating variation types
• Remember the standard forms for each variation type
• Practice quick k calculations
• Use elimination method for difficult questions

Paper 2 (Structured)

• Show all variation relationship steps
• Demonstrate the method of finding k
• Write complete equations before solving
• Explain the real-world context when appropriate

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Did You Know? The concept of variation has been used since ancient times to describe relationships in astronomy, physics, and mathematics. Newton's Law of Universal Gravitation (force varies inversely as the square of distance) is one of the most famous examples of inverse variation!

Next Chapter: In Chapter 2, you'll explore matrices and learn to perform matrix operations, find inverses, and solve systems of equations using matrices.

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• Chapter 10: Financial Management (Form 4)
• Chapter 2: Matrices