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 𝑦 varies directly as 𝑥, we can write this relationship as:

𝑦=𝑘𝑥

where 

𝑘 is called the constant of variation. This means as 

𝑥 increases (or decreases), 

𝑦 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 𝐶 in dollars for 𝑛 apples can be expressed as 

𝐶=2𝑛.

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 𝑑 covered in 𝑡 hours is 

𝑑=60𝑡.

Inverse (Indirect) Variation

Inverse variation occurs when one variable increases while the other decreases proportionally. Mathematically, if 𝑦 varies inversely as 𝑥, the relationship can be expressed as:

𝑦=𝑘/𝑥

where 𝑘 is the constant of variation. This means as 𝑥 increases, 𝑦 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 

𝑛 workers, the time 𝑇 can be expressed as 

𝑇=10/𝑛.

Example 2: The force of gravity between two masses varies inversely as the square of the distance between them. If the force 𝐹 is 20 N when the distance 𝑑 is 2 meters, then for a distance 𝑑, 

𝐹=400𝑑/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 𝑧 varies jointly as 𝑥 and 𝑦, the relationship can be expressed as:

𝑧=𝑘𝑥𝑦

where 

𝑘 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, 

𝑊=4⋅5=20 units of work. Thus, 

𝑊=𝑘⋅𝑛⋅𝑡