Variation

This chapter formalises the everyday idea that things change together. In direct variation one quantity rises as another rises; in inverse variation one rises as another falls; combined variation mixes several relationships.

You learn to form the equation, find the constant, and solve for an unknown.

Paper 2 gives a relationship in words (“y varies directly with the square of x”), a pair of values to find k, and then asks you to predict another value. The single most common slip is misreading the type of variation, so the first sentence deserves care.

Beyond simple direct and inverse variation, a quantity can depend on two or more others at the same time. In joint variation, y varies directly with both x and z, written y ∝ xz, so the equation is y = kxz.

In combined variation you mix the two ideas: for example y varies directly with x and inversely with z gives y ∝ x/z, so y = kx/z. The key move is to build the whole relationship into one equation before you do anything else, every quantity it varies directly with goes on top, every quantity it varies inversely with goes on the bottom, and the single constant k multiplies the lot.

You still find k the same way, by substituting one complete set of matching values, and then you can predict a new value by substituting the rest.

The most common error in this chapter is misreading whether a relationship is direct or inverse, and there are two quick tests that stop it. From the words: in direct variation the two quantities move the same way, as one rises the other rises, while in inverse variation they move in opposite ways, as one rises the other falls.

From a table of values: in direct variation the ratio y ÷ x stays the same for every pair (that constant ratio is k), while in inverse variation the product x × y stays the same for every pair (that constant product is k). So if a table shows the numbers multiply to the same value each time, it is inverse; if they divide to the same value each time, it is direct.

Checking one of these before you write the equation catches the mistake early, while it is still cheap to fix.

Variation often involves a power, and the safe rule is to put the power into the equation from the very start rather than adding it later. If y varies directly with the square of x, the relationship is y = kx², not y = kx; if y varies inversely with the square root of x, it is y = k/√x.

Find k with the power already in place, substitute the given pair, work out x² or √x first, then solve for k, and keep the power there when you predict the next value. The most common slip is to find k as if the relationship were linear and only remember the square at the very end, which throws the constant off.

Reading carefully for phrases like 'the square of', 'the cube of' or 'the square root of' tells you exactly which power to write down.

This example is a combined variation, mixing a direct part and an inverse-square part, exactly the shape a Paper 2 question tends to take.