Lesson 5: Variation – Direct, Inverse, Joint, and Partial

👨🏽‍🏫 Introduction

Why do more workers complete a job faster? Why does distance increase with speed and time? These questions relate to variation — how one quantity changes in relation to another.

In this lesson, we’ll explore different types of variation — direct, inverse, joint, and partial — and how to solve problems involving them in everyday and exam contexts.

By the end of this lesson, you will be able to:

• Identify different types of variation
• Form equations to model variation
• Solve problems involving direct, inverse, joint, and partial variation

🧠 Key Concepts and Explanations

🔁 1. Direct Variation

Two quantities vary directly if increasing one causes the other to increase proportionally.

We write: “y varies directly as x” as:

Where k is the constant of variation.

📉 2. Inverse Variation

Two quantities vary inversely if increasing one causes the other to decrease.

We write: “y varies inversely as x” as:

🔗 3. Joint Variation

When a quantity varies directly as two or more others multiplied together, we have joint variation.

Example: “y varies jointly as x and z” means:

🌓 4. Partial (Combined) Variation

A quantity varies partly directly and partly inversely when it is affected by both relationships.

Example: “y varies partly as x and partly inversely as x” means:

Where k and m are constants.

🧮 Sample Problem Walkthroughs

1. y varies directly as x. If y = 10 when x = 2, find y when x = 7.
   First, find k:
   
   Now,
   
2. y varies inversely as x. If y = 8 when x = 3, find y when x = 6.
   
   Then,
   
3. y varies jointly as x and z. If y = 48 when x = 4 and z = 3, find y when x = 6 and z = 2.
   First, find k:
   
   Now,
   
4. y varies partly as x and partly inversely as x. If y = 10 when x = 2 and y = 13 when x = 1, find the expression for y in terms of x.
   Use:
   
   First equation:
   
   Second equation:
   
   Solve equations simultaneously to get k = 3 and m = 10.
   Final formula:
   

✍🏽 Practice Exercises

🅰️ A. Direct and Inverse Variation

1. y varies directly as x. If y = 18 when x = 6, find y when x = 10.
2. y varies inversely as x. If y = 12 when x = 5, find x when y = 20.

🅱️ B. Joint Variation

1. If y varies jointly as x and z, and y = 36 when x = 3 and z = 4, find y when x = 6 and z = 2.

🅾️ C. Partial Variation

1. If y = 12 when x = 2 and y = 10 when x = 4, and y varies partly as x and partly inversely as x, find the relationship between y and x.

✅ Answers and Explanations

🅰️ A. Direct and Inverse Variation

• A1:
  
• A2:
  

🅱️ B. Joint Variation

• B1:
  

🅾️ C. Partial Variation

• C1:
  Use
  
  For x = 2, y = 12:
  
  For x = 4, y = 10:
  
  Solving gives k = 2 and m = 8, so
  

📌 Recap

• ✅ Direct variation:
  
• ✅ Inverse variation:
  
• ✅ Joint variation:
  
• ✅ Partial variation:
  
• ✅ These equations are used in science, economics, engineering, and finance

💭 Reflection Prompt

Think of a real-life situation where increasing one thing causes another to decrease (or increase). How would you model that using variation? Try writing an equation!