Mathematics Class 8 - Direct And-Inverse-Variations Notes

Direct and Inverse Variations

Welcome to the chapter on Direct and Inverse Variations for Class 8. In this chapter, you will learn how quantities change together in direct and inverse ways. By the end of this chapter, you will be able to identify, solve, and use direct and inverse variations in real-life problems!

Introduction

Sometimes, two quantities change together. If one increases and the other also increases, it is called direct variation. If one increases and the other decreases, it is called inverse variation.

Direct Variation

In direct variation, as one quantity increases, the other also increases in the same ratio. If one decreases, the other also decreases.

• If x and y are in direct variation, then y = kx, where k is a constant.
• Example: The cost of apples is directly proportional to the number of apples bought.
• If 1 apple costs ₹10, then 5 apples cost ₹50.

Inverse Variation

In inverse variation, as one quantity increases, the other decreases in the same ratio.

• If x and y are in inverse variation, then xy = k, where k is a constant.
• Example: The time taken to finish a job is inversely proportional to the number of people working.
• If 2 people take 6 hours, 3 people will take 4 hours (since 2 × 6 = 3 × 4 = 12).

How to Identify Variation

• If both quantities increase or decrease together, it is direct variation.
• If one increases and the other decreases, it is inverse variation.

Solving Problems

• For direct variation, use y = kx to find missing values.
• For inverse variation, use xy = k to find missing values.

Example (Direct): If 4 notebooks cost ₹80, how much do 7 notebooks cost?
Find cost per notebook: ₹80 ÷ 4 = ₹20
Cost for 7 notebooks: 7 × ₹20 = ₹140

Example (Inverse): If 5 workers finish a task in 12 days, how many days will 10 workers take?
5 × 12 = 10 × ?
60 = 10 × ?
? = 60 ÷ 10 = 6 days

Fun Activity: Variation Hunt!

Look for examples of direct and inverse variation at home or school. Write them down and explain why they are direct or inverse!

Summary

• Direct variation: both quantities change together.
• Inverse variation: one increases, the other decreases.
• Use formulas to solve problems.

Practice Questions

1. If 3 pens cost ₹45, how much do 8 pens cost?
2. If 6 workers finish a job in 10 days, how many days will 12 workers take?
3. Is the relationship between speed and time direct or inverse?
4. If x and y are in direct variation and x = 5, y = 20, find k.
5. If xy = 24 and x = 8, find y.

Challenge Yourself

• Make your own word problem for direct variation.
• Make your own word problem for inverse variation.

Did You Know?

• Direct and inverse variations are used in science, economics, and daily life!
• The formula for inverse variation is also used in physics for speed and time.

Glossary

• Direct Variation: Both quantities increase or decrease together.
• Inverse Variation: One quantity increases, the other decreases.
• Constant (k): A fixed value in the relationship.

Answers to Practice Questions

1. ₹120
2. 5 days
3. Inverse
4. k = 4
5. y = 3

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Practice direct and inverse variations to solve real-life problems with ease!