At what temperature do the Celsius and Fahrenheit readings have the same numerical value ?

Answer

To find the temperature at which the Celsius and Fahrenheit readings have the same numerical value, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between Celsius and Fahrenheit**:
The formula that relates Celsius (Tc) and Fahrenheit (Tf) is:
\[ Tc = \frac{5}{9}(Tf - 32) \]

2. **Set the two temperatures equal**:
Since we want to find the temperature where the readings are the same, we can set:
\[ Tc = Tf \]
Let's denote this common temperature as \( T \). Therefore, we can write:
\[ T = \frac{5}{9}(T - 32) \]

3. **Eliminate the fraction**:
To eliminate the fraction, multiply both sides of the equation by 9:
\[ 9T = 5(T - 32) \]

4. **Distribute on the right side**:
Now, distribute the 5 on the right side:
\[ 9T = 5T - 160 \]

5. **Rearrange the equation**:
Move all terms involving \( T \) to one side:
\[ 9T - 5T = -160 \]
This simplifies to:
\[ 4T = -160 \]

6. **Solve for T**:
Divide both sides by 4 to find \( T \):
\[ T = \frac{-160}{4} = -40 \]

7. **Conclusion**:
Therefore, the temperature at which the Celsius and Fahrenheit readings are the same is:
\[ T = -40^\circ \]

### Final Answer:
The temperature at which the Celsius and Fahrenheit readings have the same numerical value is **-40 degrees**.

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