Proportionality Constant
The proportionality constant (k) is the fixed ratio between two directly proportional variables, appearing as the slope in the direct variation equation y = kx.
Formula
k = \frac{y}{x}
Definition
The proportionality constant, usually called $k$, is the fixed number connecting two variables in a direct variation, $k = y/x$, staying the same no matter what values $x$ and $y$ take; it represents the unit rate, how much $y$ changes for every one unit of $x$, and graphically it is the slope of the line through the origin, found by dividing any (non-zero) y-value by its corresponding x-value. Formally, $k$ is the unique element of $F$ such that the linear map $f(x) = kx$ satisfies $f(x)/x = k$ for all $x \neq 0$; in dimensional analysis $k$ carries the units of $y/x$, and for power laws $y = kx^n$, $k$ remains the proportionality constant but is no longer the slope of the function except when $n = 1$ (the Buckingham Pi theorem systematizes identifying such constants in physics).
Example
If $2$ pounds of apples cost $\$4$, then $k = 4/2 = 2$, so cost $= 2 \cdot \text{pounds}$: five pounds cost $\$10$, ten pounds cost $\$20$. A car traveling $180$ miles in $3$ hours has $k = 60$ mph, so distance $= 60 \cdot \text{time}$, and after $4$ hours the distance is $240$ miles. In Hooke's Law $F = -kx$, the spring constant $k$ (units N/m) is the proportionality constant between displacement $x$ and restoring force $F$, determined experimentally for each spring.
Key Insight
The proportionality constant is the "rate" in a direct variation, miles per gallon, dollars per hour, heartbeats per minute, staying constant as the variables themselves change, and because every direct-variation line passes through the origin, $k$ is uniquely determined by any single non-origin point on that line. Proportionality constants like Planck's constant $h$, the speed of light $c$, and the gravitational constant $G$ are fundamental parameters in physical laws, encoding the intrinsic scale of each relationship and required for dimensional consistency.