Question

What temperature is the same whether it is expressed on the Celsius or Fahrenheit scale?

Short answer

-40 degrees is the temperature that is the same on both Celsius and Fahrenheit scales.

Step-by-step solution

Write down the conversion formula

Use the formula for converting Celsius (C) to Fahrenheit (F): \( F = \frac{9}{5}C + 32 \).

Set up the equation

Since we are looking for the temperature that is the same in both scales, we set \( C = F \). Now plug \( C \) into the conversion formula: \( C = \frac{9}{5}C + 32 \).

Rearrange the equation

Subtract \( \frac{9}{5}C \) from both sides to get \( C - \frac{9}{5}C = 32 \).

Combine like terms

To combine the terms on the left, find a common denominator, which is 5: \( \frac{5}{5}C - \frac{9}{5}C = 32 \), and simplify to \( -\frac{4}{5}C = 32 \).

Solve for C

Multiply both sides of the equation by \( -\frac{5}{4} \) to solve for C: \( C = 32 \cdot -\frac{5}{4} \).

Calculate the temperature in Celsius

Calculating gives us \( C = -40 \), which means \( -40 ^\circ C = -40 ^\circ F \).

Key concepts

Celsius to Fahrenheit Conversion

Understanding how to switch between temperature scales is a fundamental skill in both daily life and scientific endeavors. The Celsius (C) and Fahrenheit (F) scales are two of the most commonly used temperature scales globally. To convert Celsius to Fahrenheit, we use the following formula:
\[ F = \frac{9}{5}C + 32 \].
For example, to convert 25 degrees Celsius to Fahrenheit, you would complete the following calculation: \( F = \frac{9}{5} \times 25 + 32 = 77 \) degrees Fahrenheit. This conversion is essential since weather forecasts, oven temperatures, and many scientific experiments often require an understanding of both scales.

Equation Solving

The process of solving equations is one of the foundational tools in mathematics and science. Solving equations involves finding the value(s) of the variable(s) that make the equation true. When working with temperature conversions, equations allow us to find unknown values based on known relationships. Here's a logic thread to follow when solving equations:

• Write down the equation and identify the variables.
• Rearrange the equation to isolate the variable you're solving for by performing opposite operations on both sides.
• Combine like terms to simplify the equation.
• Perform the necessary calculations to find the variable's value.

The correct approach and a good understanding of algebra will help unravel even the most complex equations.

Temperature Scales

The concept of temperature scales is crucial as it allows us to measure and communicate temperatures in a standardized way. There are several temperature scales used across the world, but the Celsius and Fahrenheit scales are the most common in everyday contexts. The Celsius scale, used primarily outside the United States, is based on 0 degrees for the freezing point of water and 100 degrees for the boiling point, under normal atmospheric conditions. On the other hand, the Fahrenheit scale, used mainly in the United States, assigns 32 degrees to the freezing point of water and 212 degrees to the boiling point. Understanding these scales and their origins helps us communicate and understand temperature-related information more effectively.

Scientific Problem-Solving

Scientific problem-solving is a systematic approach to understanding and solving issues using scientific methods. It's a step-by-step process that involves observation, hypothesis formulation, testing, and conclusions. When applied to temperature conversion, the scientific method can help explain why and how the values are calculated. In the context of our temperature exercise, scientific problem-solving involves:

• Understanding the temperature conversion formula.
• Setting up an equation based on the conditions presented.
• Rearranging and solving the equation.
• Interpreting the results in the context of the problem.

By employing this methodical approach, students can deepen their comprehension and tackle complex scientific problems with greater confidence.