At which temperature Celsius and Fahrenheit are equal?

Question

Prepp Team · Accepted Answer

Understanding Temperature Scales: Celsius and Fahrenheit

Temperature can be measured using different scales, with the most common being Celsius and Fahrenheit. These scales use different reference points for freezing and boiling water, which leads to different numerical values for the same physical temperature.

The question asks for the specific temperature at which the numerical values on the Celsius scale and the Fahrenheit scale are equal. To find this temperature, we need to use the conversion formula that relates the two scales.

Conversion Formulas Between Celsius and Fahrenheit

The formulas used to convert temperature between Celsius (\(C\)) and Fahrenheit (\(F\)) are:

From Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)
From Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \)

Finding the Temperature Where Celsius Equals Fahrenheit

We want to find the temperature where the reading on the Celsius scale is the same as the reading on the Fahrenheit scale. Let this temperature be \(x\). So, we are looking for the point where \(C = F = x\).

We can use either conversion formula. Let's use the formula \( F = \frac{9}{5}C + 32 \). We substitute \(C\) with \(x\) and \(F\) with \(x\):

\( x = \frac{9}{5}x + 32 \)

Now, we need to solve this algebraic equation for \(x\). To eliminate the fraction, we can multiply every term in the equation by 5:

\( 5 \times x = 5 \times \frac{9}{5}x + 5 \times 32 \)

\( 5x = 9x + 160 \)

Next, we want to get all terms with \(x\) on one side of the equation. Subtract \(9x\) from both sides:

\( 5x - 9x = 160 \)

\( -4x = 160 \)

Finally, to isolate \(x\), divide both sides by -4:

\( x = \frac{160}{-4} \)

\( x = -40 \)

So, the temperature at which Celsius and Fahrenheit scales are equal is -40 degrees.

This means that -40 degrees Celsius is the same temperature as -40 degrees Fahrenheit.

Scale	Freezing Point of Water	Boiling Point of Water	Equal Point (C=F)
Celsius (\(^{\circ}\text{C}\))	\(0^{\circ}\text{C}\)	\(100^{\circ}\text{C}\)	\(-40^{\circ}\text{C}\)
Fahrenheit (\(^{\circ}\text{F}\))	\(32^{\circ}\text{F}\)	\(212^{\circ}\text{F}\)	\(-40^{\circ}\text{F}\)

Conclusion on the Equal Temperature Point

Based on the calculation, the temperature where the numerical readings on the Celsius and Fahrenheit scales are identical is -40 degrees. This is a key point to remember when working with temperature conversions.

Revision Table: Temperature Conversion Formulas

Convert From	Convert To	Formula
Celsius (\(C\))	Fahrenheit (\(F\))	\( F = \frac{9}{5}C + 32 \)
Fahrenheit (\(F\))	Celsius (\(C\))	\( C = \frac{5}{9}(F - 32) \)

Additional Information: Other Temperature Scales

Besides Celsius and Fahrenheit, other temperature scales are used, particularly in science and engineering:

Kelvin (K): This is the SI base unit of thermodynamic temperature. It is an absolute scale, meaning that 0 K (absolute zero) is the point where particle motion theoretically stops. The Kelvin scale increment is the same as the Celsius scale increment (\(1^{\circ}\text{C} = 1\text{ K}\)). The conversion is \( \text{K} = ^{\circ}\text{C} + 273.15 \).
Rankine (\(^{\circ}\text{R}\) or \(^{\circ}\text{Ra}\)): This is another absolute scale. It is related to the Fahrenheit scale in the same way that the Kelvin scale is related to the Celsius scale. The scale increment is the same as the Fahrenheit scale increment (\(1^{\circ}\text{F} = 1^{\circ}\text{R}\)). The conversion is \( ^{\circ}\text{R} = ^{\circ}\text{F} + 459.67 \).

Understanding these different scales and their relationships is important in various scientific and practical applications involving temperature measurement.