Variation · Form 5
Combined variation, explained
Combined variation is when one quantity depends on two or more others at the same time, joining direct and inverse relationships under a single constant. For example y ∝ xz becomes y = kxz, while y ∝ x/z becomes y = kx/z.
What combined variation joins together
Combined variation stitches the earlier ideas together: a quantity can vary directly with one variable and, at the same time, inversely with another. If y varies directly as x and inversely as z, we write y ∝ x/z, giving y = kx/z with a single constant k for the whole relationship.
Each variable can also carry its own power, such as y ∝ x²/√z.
Why it matters and how to recognise it
Many real formulae are combined variations, which is why the idea matters: the area of a triangle depends on both base and height, and pressure depends on force and area together. To recognise it, notice that the answer is affected by more than one quantity in the same sentence, and read each one separately 'more of this raises y' points to direct, 'more of this lowers y' points to inverse.
Holding all but one variable fixed turns combined variation back into the simple direct or inverse case you already know.
The usual misconception
The most common slip is giving each variable its own separate constant instead of using one shared k for the whole relationship, y = kx/z has a single k, not a k for x and another for z. Another trap is assuming every extra variable is direct: always check the wording, because a quantity in the denominator (like z in x/z) is an inverse part, not a direct one.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I set up the equation for a combined variation statement?
Translate the statement piece by piece: 'directly as x' contributes x, 'inversely as z' contributes 1/z, then join them under one constant k. For example, 'y varies directly as x and inversely as z' becomes y = kx/z.
Always check whether any quantity is squared or cubed in the wording.
How do exam questions usually test combined variation?
You're usually given one set of values to find k, then asked to find an unknown when the other quantities change. Substitute the known values into your combined equation, solve for k, rewrite the equation, then substitute the new values to find the answer.
Keep units and powers consistent throughout.
What's a common mistake with combined variation questions?
Students often mix up which quantity is direct and which is inverse, placing a variable on the wrong side of the fraction. Another slip is forgetting a squared or cubed term stated in the question, such as y ∝ x/z² instead of y ∝ x/z.
Re-read the wording before writing the equation.