Problem 37 - Prealgebra

Question

The formula \(F=\frac{9}{5} C+32\) is used to find the Fahrenheit temperature when a Celsius temperature is known. For what value are the Celsius and Fahrenheit temperatures the same? A. \(-72^{\circ}\) B. \(-40^{\circ}\) C. \(0^{\circ}\) D. \(32^{\circ}\)

Step-by-Step Solution

Answer

The temperatures are the same at \(-40^{\circ}\).

Step 1: Set the Variables Equal

To find the temperature where Celsius and Fahrenheit are the same, we set \( F = C \).

Step 2: Substitute the Variables into the Formula

Replace \( F \) with \( C \) in the formula \( F = \frac{9}{5}C + 32 \), resulting in the equation \( C = \frac{9}{5}C + 32 \).

Step 3: Eliminate the Fraction

Multiply both sides by 5 to get rid of the fraction: \( 5C = 9C + 160 \).

Step 4: Isolate the Variable

Subtract \( 9C \) from both sides: \( 5C - 9C = 160 \). This simplifies to \( -4C = 160 \).

Step 5: Solve for C

Divide both sides by \(-4\) to solve for \( C \): \( C = \frac{160}{-4} \). This simplifies to \( C = -40 \).

Step 6: Conclusion

Therefore, the Celsius and Fahrenheit temperatures are the same at \( -40^{\circ} \).

Key Concepts

Celsius and Fahrenheit

When discussing temperature, two of the most common scales are Celsius and Fahrenheit. Celsius, also known as Centigrade, is frequently used in most parts of the world and is based on the properties of water. Water freezes at 0°C and boils at 100°C under normal atmospheric conditions. On the other hand, Fahrenheit is primarily used in the United States. It sets water's freezing point at 32°F and boiling at 212°F.

Understanding these scales can be challenging due to their different starting points and increments. Understanding how to convert between these two scales can help in interpreting weather data or scientific information from various countries. For example, to determine equivalent temperatures in both scales, we use specific formulas that help bridge the gap between them.

temperature formula

The temperature conversion formula between Celsius and Fahrenheit is crucial for transitioning between the two scales. The formula provided is:

• \( F = \frac{9}{5}C + 32 \)

This formula allows us to convert a temperature in Celsius (\( C \)) to its equivalent in Fahrenheit (\( F \)). Here's how it works:

• The fraction \( \frac{9}{5} \) is used to scale the Celsius temperature to match the Fahrenheit scale increments.
• The +32 accounts for the starting offset between the two scales, as 0°C is equivalent to 32°F.

By understanding and using this formula, one can easily switch between these two temperature measurements, facilitating better communication and understanding, especially in diverse scientific and international contexts.

fraction elimination

In mathematical equations, dealing with fractions can often complicate problem-solving, making "fraction elimination" a helpful strategy. In our exercise, we have the formula:

• \( C = \frac{9}{5}C + 32 \)

To simplify the equation, we want to get rid of the fraction \( \frac{9}{5} \). This is achieved by multiplying every term on both sides of the equation by the denominator, which is 5:

• \( 5C = 9C + 160 \)

This step removes the fraction and makes future calculations easier to follow, paving the way to clearer and less error-prone arithmetic processes on both sides of the equation.

variable isolation

"Variable isolation" is a step in solving algebraic equations that involves simplifying the formula to solve for a specific variable. In this case, we need to find when Celsius (\( C \)) equals Fahrenheit (\( F \)), which results in setting \( F = C \).After eliminating the fraction, we simplify the equation \( 5C = 9C + 160 \) to find the value for \( C \):

• Subtract \( 9C \) from both sides to gather like terms: \( 5C - 9C = 160 \).
• Combine the \( C \) terms, yielding \( -4C = 160 \).
• Divide both sides by \(-4\) to isolate \( C \): \( C = \frac{160}{-4} \).

Eventually, this gives the solution \( C = -40 \). This process of isolating the variable helps find the precise point where both Celsius and Fahrenheit scales indicate the same temperature, which is \(-40\) degrees.