The numerical value of $\alpha $ expressed per $^\circ C$ is equal to _______ times the numeric value expressed per$^\circ F$.

Hint: To convert the Celsius into Fahrenheit we should get the knowledge about Celsius scale and Fahrenheit scale and use the boiling and freezing point of water to in Celsius scale and Fahrenheit scale to relate them.

Complete step by step answer
$\alpha $ is the thermal coefficient of expansion.
Thermal expansion: The tendency of matter to change its shape, area, volume, and density in response to a change in temperature.
Coefficient of thermal expansion: It shows how the shape, area, volume, or density of an object changes with a change in temperature.
The discovery of temperature scales is the most important discovery in science history.

Celsius scale: Celsius is also known as centigrade. It is the standard unit of temperature. It was invented in 1742 by the Swedish astronomer Anders Celsius and the Celsius scale was named after him. The Celsius scale is expressed in degree Celsius, represented as $^\circ C$ The Celsius scale is in general use and used in scientific work everywhere.
1. The freezing point of water in Celsius is $0^\circ C$
2. The boiling point of water in Fahrenheit is $100^\circ C$

Fahrenheit scale: The Fahrenheit scale was introduced by Daniel Fahrenheit, a German physicist and so the Fahrenheit scale was named after him. The Fahrenheit scale is expressed in degree Fahrenheit, represented as $^\circ F$
1. The freezing point of water in Fahrenheit is $32^\circ F$
2. The boiling point of water in Fahrenheit is $212^\circ F$
Freezing temperature of water in Celsius scale is $0^\circ C$ and in Fahrenheit scale is $32^\circ F$
Boiling point of water in Celsius scale is $100^\circ C$ and in Fahrenheit scale is $212^\circ F$
From the above statement we can see there is 100 division difference in the boiling and freezing point of water in Celsius scale and 180 division difference in the boiling and freezing point of water in Fahrenheit scale.

So, per 1 division rise in temperature in Celsius scale is equivalent to$\dfrac{{180}}{{100}}$ which is equal to $\dfrac{9}{5}$ division in Fahrenheit scale.

Hence we get,
$0^\circ C{\text{ = }}32^\circ F$
${\text{1 division}}$rise in Fahrenheit is equal to $\dfrac{9}{5}$ division in Celsius.

Hence, the numerical value of $\alpha $ expressed per $^\circ C$ is equal to $\dfrac{9}{5}$ times the numeric value expressed per$^\circ F$.
Note: For any temperature conversion from Fahrenheit to Celsius we have to subtract 32 from given °F temperature then multiply it with 5/9 to get the equivalent Celsius scale of temperature.