Variation O Level Maths: Direct & Inverse

Variation in O Level Maths covers direct variation (y = kx), inverse variation (y = k/x), and joint variation, where two quantities change together according to a constant k. You will be asked to find k from given values, then use the equation to find an unknown quantity. This article explains the types, the formula, worked examples, and common pitfalls so you can handle any variation question confidently.

Formula

For direct variation, the general equation is y = kx, where k is the constant of variation. For inverse variation, it is y = k/x. In both cases, k is found using one known pair of (x, y) values, and then the equation is used to find any other value. The units of k depend on the units of y and x (e.g., if y is in kg and x is in m, then k is in kg/m).

Worked Example

Using the default values from the calculator: x = 6, y = 18, and x2 = 10. First, find k for direct variation: k = y / x = 18 / 6 = 3. So the equation is y = 3x. For x2 = 10, y2 = 3 * 10 = 30. The calculator shows k = 3.00 and y2 = 30.00, which matches.

Types of Variation in O Level Maths

Variation problems in O Level Maths usually appear in one of three forms:

• Direct variation (y ∝ x): as x increases, y increases proportionally. The graph is a straight line through the origin.
• Inverse variation (y ∝ 1/x): as x increases, y decreases. The graph is a hyperbola.
• Joint variation (y ∝ xz): y varies directly with the product of two or more variables. For example, y = kxz.

Sometimes a variation statement includes powers, like y ∝ x² or y ∝ 1/√x. The method is the same: write the equation with a constant k, substitute the given values to find k, then use the equation to find the unknown.

How to Solve Variation Problems Step by Step

Follow these steps for any variation question in the O Level exam:

1. Write the variation statement as an equation with k (e.g., y = kx or y = k/x²).
2. Substitute the given pair of values (x, y) into the equation.
3. Solve for k. Keep k as a fraction if it is not a whole number.
4. Rewrite the equation with the value of k.
5. Substitute the new x (or y) value to find the unknown quantity.

For example, if y varies inversely as x and y = 4 when x = 9, then 4 = k/9, so k = 36. The equation is y = 36/x. If x becomes 12, then y = 36/12 = 3.

Common Mistakes and How to Avoid Them

Students often mix up direct and inverse variation. Remember: in direct variation, both quantities go up or down together; in inverse, one goes up while the other goes down. Another common error is forgetting to square or root the variable when the statement says “varies as the square” or “varies as the square root.” Always read the wording carefully. Also, when finding k, do not round it prematurely; keep the exact value (as a fraction) until the final step to avoid rounding errors.

Real-World Applications of Variation

Variation is not just an exam topic; it appears in many real-world contexts. For example, the distance a car travels at constant speed varies directly with time (d = vt). The time taken for a journey varies inversely with speed (t = d/v). In physics, the force of gravity between two objects varies inversely with the square of the distance between them. In geometry, the area of a circle varies directly with the square of its radius (A = πr²). Understanding variation helps you interpret these relationships and make predictions.

Comparing Direct and Inverse Variation

The table below summarises the key differences:

Feature	Direct	Inverse
Equation	y = kx	y = k/x
Graph	Straight line through origin	Hyperbola
Effect of x doubling	y doubles	y halves
Example	Cost ∝ number of items	Speed ∝ 1/time

When solving, always check which type is stated. Some problems combine both, like “y varies directly as x and inversely as z,” which gives y = kx/z.

Frequently Asked Questions

What does ‘∝’ mean in variation?

The symbol ∝ means “is proportional to.” For example, y ∝ x means y is directly proportional to x, which translates to the equation y = kx, where k is the constant of proportionality.

How do you find the constant k in variation?

You need one known pair of values (x, y). Substitute them into the general equation (e.g., y = kx or y = k/x) and solve for k. For example, if y = 10 when x = 2 and it is direct variation, then k = 10/2 = 5.

What is joint variation in O Level Maths?

Joint variation means a variable depends on the product of two or more other variables. For instance, y varies jointly as x and z means y = kxz. To solve, you need values for x, y, and z to find k, then use the equation to find an unknown.

What is the difference between direct and inverse variation?

In direct variation, y = kx, so as x increases, y increases proportionally. In inverse variation, y = k/x, so as x increases, y decreases. The graph of direct variation is a straight line through the origin; inverse variation graphs are curves that approach the axes.

Can variation problems involve squares or square roots?

Yes. For example, y varies directly as the square of x means y = kx². If y varies inversely as the square root of x, then y = k/√x. The method is the same: substitute the given values to find k, then use the equation.

How do you solve a variation problem with three variables?

Write the equation with all variables, e.g., y = kx/z. Use the given values of x, y, and z to find k. Then substitute the new values of x and z to find the unknown y (or solve for one variable if y is given).

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in x | x (first quantity) | - | 6 | 0
in y | y (first quantity) | - | 18 | 0
in x2 | x (second quantity) | - | 10 | 0
out k | Constant k (y/x) | - | y / x | 2
out y2 | y for x2 (direct) | - | k * x2 | 2