A linear function has a constant rate of change — its graph is a straight line. That constant rate of change is the gradient, and gradient is itself a ratio: rise over run.
$$\text{gradient} = \frac{\text{change in }y}{\text{change in }x}$$
This connects ratios directly to linear functions: if $y$ and $x$ are in a fixed ratio, the relationship is linear and passes through the origin ($y = mx$).
Using a ratio to define a linear function: if $y : x = 3 : 2$, then $y = \frac{3}{2}x$. The gradient is $\frac{3}{2}$.
Reading gradient as a ratio from a graph: pick two clear points on the line, form the fraction $\frac{\Delta y}{\Delta x}$, and simplify — this is both the gradient and the ratio between $y$ and $x$ values (for lines through the origin).
Key distinction: a ratio $y : x = k : 1$ means direct proportion — doubling $x$ doubles $y$. This is the definition of a linear function through the origin.
Common error: confusing the ratio $y : x$ with the gradient — they are equal only when the line passes through the origin.