Direct and Inverse Variation Worksheet for Easy Practice

Understanding Direct and Inverse Variation

Direct and inverse variation are fundamental concepts in algebra and mathematics, representing relationships between variables. Understanding these concepts is crucial for problem-solving in various fields, including physics, engineering, and economics. In this worksheet, we will delve into the world of direct and inverse variation, providing examples and exercises for easy practice.

What is Direct Variation?

Direct variation occurs when one variable increases or decreases in direct proportion to the change in another variable. This relationship can be represented by the equation:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of variation.

📝 Note: The constant of variation k can be positive or negative, but it must be non-zero.

Examples of Direct Variation

1. The cost of apples varies directly with the number of apples purchased. If 1 apple costs 1, then 2 apples will cost 2, and 3 apples will cost $3.
2. The distance traveled by a car varies directly with the time spent driving, assuming a constant speed.

What is Inverse Variation?

Inverse variation occurs when one variable increases as the other variable decreases, and vice versa. This relationship can be represented by the equation:

y = k/x

where y is the dependent variable, x is the independent variable, and k is the constant of variation.

📝 Note: The constant of variation k can be positive or negative, but it must be non-zero.

Examples of Inverse Variation

1. The pressure of a gas varies inversely with its volume, assuming a constant temperature.
2. The time it takes to complete a task varies inversely with the number of people working on it.

Practice Exercises

Now it’s time to practice! Solve the following exercises to reinforce your understanding of direct and inverse variation.

Direct Variation Exercises

1. If y varies directly with x, and y = 12 when x = 3, find the constant of variation k.
2. If y = 2x, and x = 4, find the value of y.
3. The cost of renting a car varies directly with the number of days rented. If the cost is $40 for 2 days, find the cost for 5 days.

Inverse Variation Exercises

1. If y varies inversely with x, and y = 6 when x = 2, find the constant of variation k.
2. If y = 12/x, and x = 3, find the value of y.
3. The volume of a gas varies inversely with its pressure. If the volume is 20 liters when the pressure is 2 atm, find the volume when the pressure is 4 atm.

Solutions

Direct Variation Solutions

1. k = ¹²⁄₃ = 4
2. y = 2(4) = 8
3. y = 40(⁵⁄₂) = 100

Inverse Variation Solutions

1. k = 6(2) = 12
2. y = ¹²⁄₃ = 4
3. y = 20(²⁄₄) = 10

By now, you should have a solid understanding of direct and inverse variation. Remember to practice regularly to reinforce your skills.

Some common formulas and equations related to direct and inverse variation are summarized in the table below:

In conclusion, direct and inverse variation are essential concepts in mathematics, representing fundamental relationships between variables. By understanding these concepts and practicing with exercises, you’ll become proficient in solving problems involving direct and inverse variation.