Question

At what temperature does the numerical reading on a Celsius thermometer equal that on a Fahrenheit thermometer?

Short answer

The numerical reading on a Celsius thermometer equals that on a Fahrenheit thermometer at -40 degrees.

Step-by-step solution

Understand Fahrenheit and Celsius conversions

The Fahrenheit (F) to Celsius (C) conversion formula is \(C = (F - 32) * \frac{5}{9}\). Conversely, the Celsius to Fahrenheit conversion can be written as \(F = C * \frac{9}{5} + 32 \).

Set Fahrenheit and Celsius equal

To figure out the temperature at which the Fahrenheit and Celsius scales are the same, we need to set the two formulas equal to each other, resulting in the following equation: \(C = F = C * \frac{9}{5} + 32\).

Solve the equation

First, subtract \(C\) from both sides of the equation, resulting in: \( 0 = F - C * \frac{9}{5} + 32\). Next, subtract \(F\) from both sides of the equation: \(-F = -C * \frac{9}{5} + 32\). Finally, Solve the equation for \(F\). This gives: \(C * \frac{9}{5} - F = 32 \).

Final computation

Substitute \(F\) with \(C\) in the equation derived from step 3 as the condition is such that \(F = C\) at the specific temperature, thus giving \(C * \frac{9}{5} - C = 32 \). Upon simplification, the equation becomes \(-\frac{4}{5} C = 32 \). Solving this equation gives \(C = -40\). Thus, the numerical readings on both Celsius and Fahrenheit thermometers would be the same at -40.