If $C$ is the temperature on a Centigrade thermometer and $F$ is the temperature on a Fahrenheit thermometer, and they are related by $\frac{C}{5}=\frac{F-32}{9}$, find the numerical temperature whose value is exactly identical on both thermometers.

We need to determine the temperature where the value on both measurement scales matches, i.e., $C=F=x$.
Substitute $x$ in place of $C$ and $F$ inside the scale conversion framework equation:

$$\frac{x}{5}=\frac{x-32}{9}$$

Cross-multiply both sides to isolate the variable $x$:

$$9x=5(x-32)$$

$$9x=5x-160$$

$$9x-5x=-160$$

$$4x=-160$$

$$x=-\frac{160}{4}=-40$$

Therefore, the temperature reading is $-40^{\circ }$, which is identical on both the Centigrade and Fahrenheit scales.