Question

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

Short answer

Celsius equals Fahrenheit at -40 degrees.

Step-by-step solution

Understand the Relationship between Celsius and Fahrenheit

The relationship between Celsius (°C) and Fahrenheit (°F) is given by the formula: \[ °F = \frac{9}{5}°C + 32 \] We need to find the temperature where the Celsius value equals the Fahrenheit value.

Set the Equation where Celsius Equals Fahrenheit

We set \( °C = °F = x \) since we are looking for the temperature where both readings are equal. Substitute \( x \) into the Fahrenheit equation:\[ x = \frac{9}{5}x + 32 \]

Solve for x

Start solving the equation from Step 2:1. Subtract \( \frac{9}{5}x \) from both sides: \[ x - \frac{9}{5}x = 32 \]2. Combine like terms: \[ \left(1 - \frac{9}{5}\right)x = 32 \]3. Calculate the value of \( 1 - \frac{9}{5} = -\frac{4}{5} \)4. This gives: \[ -\frac{4}{5}x = 32 \]5. Solve for \( x \) by multiplying both sides by \( -\frac{5}{4}\): \[ x = 32 \times \left(-\frac{5}{4}\right) = -40 \]

Verify the Solution

Verify that when \( x = -40 \), both the Celsius and Fahrenheit values are equal:1. Substitute \( x = -40 \) back into the formula: \[ °F = \frac{9}{5}(-40) + 32 \]2. Calculate expressive terms: \[ °F = -72 + 32 = -40 \]This confirms the calculated value matches both Celsius and Fahrenheit.

Key concepts

Celsius to Fahrenheit Formula

Temperature conversion between Celsius and Fahrenheit is essential to understand, especially since these are the most commonly used temperature scales. The formula to convert a temperature from Celsius (\( °C \)) to Fahrenheit (\( °F \)) is: \[ °F = \frac{9}{5} °C + 32 \] This formula tells us how to change one temperature measurement system into another. The multiplication factor, \( \frac{9}{5} \), accounts for the different scaling between Celsius and Fahrenheit.

• A change of 1 degree Celsius is equivalent to a change of \( \frac{9}{5} \) degrees Fahrenheit.
• Additionally, the 32 in the formula translates the freezing point of water from 0°C to 32°F.

When using the formula, keep each calculation step clear in your mind; this will help avoid errors in temperature conversion.

Numerical Temperature Equality

Finding the temperature at which the readings on a Celsius and a Fahrenheit thermometer are equal is an intriguing exercise in understanding these temperature scales. To find this unique point, you equate the Celsius temperature to its Fahrenheit conversion and solve for the temperature value that satisfies this condition. The equation for this scenario uses a variable, say \(x\), where \(x = °C = °F\). Thus, it becomes: \[ x = \frac{9}{5}x + 32 \] To find \(x\), we solve this equation:

• Subtract \( \frac{9}{5}x \) from both sides to combine the terms: \( x - \frac{9}{5}x = 32 \).
• Calculate \( 1 - \frac{9}{5} \), yielding \( -\frac{4}{5} \).
• Solve for \(x\) by isolating it on one side: \( x = 32 \times -\frac{5}{4} = -40 \).

Thus, \(-40\) is the temperature where both Celsius and Fahrenheit readings are exactly the same.

Temperature Measurement Systems

The Celsius and Fahrenheit systems are widely used worldwide, each with its history and application.

• Celsius: This metric system scale is widely used in scientific contexts and by most countries. It is based on the properties of water, with the freezing point at 0°C and the boiling point at 100°C.
• Fahrenheit: Primarily used in the United States, this system places water's freezing and boiling points at 32°F and 212°F, respectively. It was developed to provide finer granularity in temperature measurement.

These systems emerged from different historical contexts and continue to be used due to their established conventions. Understanding how to convert between Celsius and Fahrenheit helps in interpreting weather, scientific data, and daily temperature-related activities. Recognizing their distinct scales and the rationale behind their specific zero points can deepen one's appreciation of global temperature reporting.