Variation Problems on Direct, Indirect, Joint and Partial - JSS3 Mathematics Lesson Note
Variation problems in mathematics deal with how one quantity changes in relation to another quantity. There are different types of variations:
Direct Variation
Direct variation occurs when one variable increases or decreases in proportion to another variable. Mathematically, if π¦ varies directly as π₯, we can write this relationship as:
π¦=ππ₯
where
π is called the constant of variation. This means as
π₯ increases (or decreases),
π¦ also increases (or decreases) proportionally.
Examples include:
Example 1: If the cost of apples varies directly with the number bought at a constant rate of $2 per apple, then the cost πΆ in dollars for π apples can be expressed as
πΆ=2π.
Example 2: The distance traveled by a car at a constant speed varies directly with the time traveled. If a car travels at 60 miles per hour, the distance π covered in π‘ hours is
π=60π‘.
Inverse (Indirect) Variation
Inverse variation occurs when one variable increases while the other decreases proportionally. Mathematically, if π¦ varies inversely as π₯, the relationship can be expressed as:
π¦=π/π₯
where π is the constant of variation. This means as π₯ increases, π¦ decreases, and vice versa.
Examples include:
Example 1: If the time taken for a task varies inversely with the number of workers, and it takes 5 hours with 2 workers, then with
π workers, the time π can be expressed as
π=10/π.
Example 2: The force of gravity between two masses varies inversely as the square of the distance between them. If the force πΉ is 20 N when the distance π is 2 meters, then for a distance π,
πΉ=400π/2
Joint Variation
Joint variation involves two or more variables, where one variable varies directly with two or more variables and inversely with another. Mathematically, if π§ varies jointly as π₯ and π¦, the relationship can be expressed as:
π§=ππ₯π¦
where
π is the constant of variation.
Examples include:
Example 1: If the amount of work done varies jointly with the number of workers and the time they work, then if 4 workers can complete a task in 5 hours,
π=4β 5=20 units of work. Thus,
π=πβ πβ π‘