Variation in Mathematics Explained — Meaning, Types & Examples (JAMB & WAEC Maths)

Variation in mathematics describes how one quantity changes in relation to another — whether they increase together, one increases as the other decreases, or several quantities are related at once.

This topic is a regular feature in JAMB and WAEC Mathematics, usually tested through word problems that require setting up an equation of variation and solving for an unknown constant or value. This lesson explains each type of variation with worked examples.

Quick takeaways

• Variation describes how one quantity changes in relation to another, and it’s classified into direct, inverse, joint, and partial variation.
• Direct variation means two quantities increase or decrease together at a constant rate, written as y = kx.
• Inverse variation means one quantity increases as the other decreases, written as y = k/x.
• Joint variation involves one quantity depending on two or more other quantities multiplied together, such as y = kxz.
• Every variation problem requires first finding the constant of variation (k) using given values, before solving for the unknown the question asks for.

What is variation in mathematics?

Variation describes a specific type of relationship between two or more quantities, where a change in one quantity causes a predictable, proportional change in another.

Rather than being a fixed equation with specific numbers, a variation relationship contains a constant, called the constant of variation (k), which must be determined from given information before the relationship can be used to solve for other values.

What is direct variation?

Direct variation occurs when two quantities increase or decrease together at a constant rate — as one quantity increases, the other increases proportionally, and as one decreases, so does the other. It’s written as:

y = kx

Where k is the constant of variation. For example, if the cost of fuel varies directly with the number of litres purchased, buying more litres proportionally increases the total cost.

What is inverse variation?

Inverse variation occurs when one quantity increases as the other decreases, in such a way that their product remains constant. It’s written as:

y = k/x

For example, if the time taken to complete a journey varies inversely with speed, traveling faster reduces the time taken, and traveling slower increases it, assuming the distance stays the same.

What is joint variation?

Joint variation occurs when one quantity depends on two or more other quantities, typically multiplied together. It’s written as:

y = kxz

Where y varies jointly with both x and z. For example, the volume of a cylinder varies jointly with the square of its radius and its height, since both factors affect the volume together, not independently.

What is partial variation?

Partial variation occurs when a quantity is made up of two parts: one part that stays constant, and another part that varies directly with another quantity. It’s written as:

y = a + kx

Where “a” is a fixed amount and “kx” is the part that varies. For example, a taxi fare might include a fixed base charge (the constant part) plus an additional charge that varies directly with the distance traveled.

How do you solve variation word problems?

Most variation problems follow the same general approach:

1. Identify the type of variation described in the problem (direct, inverse, joint, or partial)
2. Write the general equation for that type of variation, using the constant k
3. Substitute the given values into the equation to solve for k
4. Rewrite the equation with the known value of k, then use it to solve for whatever the question actually asks

Common mistakes students make with variation

• Confusing direct variation (y = kx) with inverse variation (y = k/x), especially when a word problem doesn’t explicitly name the type
• Forgetting to find the constant of variation first before attempting to answer the actual question
• Mixing up joint variation (multiple variables multiplied together) with partial variation (a fixed amount plus a variable part)
• Misreading word problems and missing key phrases like “varies directly,” “varies inversely,” or “varies jointly,” which indicate exactly which type of variation applies

Frequently asked questions

Q: What is the difference between direct and inverse variation? In direct variation, two quantities increase or decrease together proportionally (y = kx), while in inverse variation, one quantity increases as the other decreases, keeping their product constant (y = k/x).

Q: What is joint variation? Joint variation occurs when one quantity depends on two or more other quantities multiplied together, written generally as y = kxz.

Q: What is partial variation? Partial variation occurs when a quantity consists of a fixed part plus another part that varies directly with a second quantity, written as y = a + kx.

Q: How do you find the constant of variation in a word problem? Substitute the given corresponding values for the variables into the appropriate variation equation, then solve algebraically for the constant k.

Want to strengthen your algebra fundamentals further? Read our lesson on changing the subject of a formula →

This is one topic from our full JAMB & WAEC Maths course — see everything covered →