Direct proportion and its constant

\(y\) is directly proportional to \(x\) when \(y=kx\) for a fixed constant \(k\). For nonzero \(x\), the ratio \(\dfrac yx\) is constant. Multiplying \(x\) by a factor multiplies \(y\) by the same factor. The graph is a straight line through the origin.

Example

Identical notebooks cost \(\$15\) for \(6\), with no fixed fee. Cost \(C=kn\), so \(k=\dfrac{15}{6}=2.5\) dollars per notebook. For \(10\) notebooks, \(C=2.5(10)=\$25\).

A quantity increasing as another increases is not enough to prove direct proportion. A fixed delivery charge would prevent the cost graph from passing through the origin.