Proportion

You see this whenever one thing scales with another in a clean way, like paying per litre of fuel or sharing a fixed job between workers. The fast check is this: does the relationship keep the same multiplier rule every time, or not?

When one variable is your input and the other responds to it, you can model that link with a formula.

Usually, your independent variable is the one you choose, and your dependent variable is the one that changes because of it. If you choose how many litres to buy, then litres is the input and price is the output.

A lot of proportion questions use a fixed number called k.

Think of k like the rate locked into the situation, such as cost per litre or kilograms per container.

Three ways to spot direct proportion

A relationship is direct proportion when the ratio stays unchanged. You write this as $y\propto x$, which means

$y=kx(k\ne 0)$

and also

$\frac{y}{x}=k$

That means you can recognise it in three matching ways: the ratio $y/x$ stays the same, the equation can be written as $y=kx$, and the graph is a straight line going through $(0,0)$.

The graph idea is useful because the constant k is also the gradient. In other words,

$k=\frac{\Delta y}{\Delta x}$

So k tells you how much $y$ changes when $x$ increases by 1.

Worked example

That is why direct proportion feels so predictable. If your running app says a route is twice as long and your pace stays the same, your time doubles too. Same scale factor in, same scale factor out.