A faulty thermometer reads `5^@ C` melting ice and `99^@ C` in steam. Find the correct temperature in `^@ C` when this faulty thermometer reads `52^@ C`.

Explanation

To solve the problem step by step, we need to convert the reading of the faulty thermometer to the correct temperature in degrees Celsius.

### Step-by-Step Solution:

1. **Identify the given values**:
- The faulty thermometer reads:
- \( T_f(ice) = 5^\circ C \) (melting ice)
- \( T_f(steam) = 99^\circ C \) (steam)
- We need to find the correct temperature \( T_c \) when the faulty thermometer reads \( T_f = 52^\circ C \).

2. **Set up the conversion formula**:
The relationship between the faulty thermometer reading and the correct temperature can be expressed as:
\[ \frac{T_f - T_f(ice)}{T_f(steam) - T_f(ice)} = \frac{T_c - T_c(ice)}{T_c(steam) - T_c(ice)} \]
Where:
- \( T_c(ice) = 0^\circ C \) (correct temperature for melting ice)
- \( T_c(steam) = 100^\circ C \) (correct temperature for steam)

3. **Substituting the known values**:
Substitute the known values into the formula:
\[ \frac{52 - 5}{99 - 5} = \frac{T_c - 0}{100 - 0} \]
Simplifying the left side:
\[ \frac{47}{94} = \frac{T_c}{100} \]

4. **Cross-multiply to solve for \( T_c \)**:
Cross-multiplying gives:
\[ 47 \cdot 100 = 94 \cdot T_c \]
This simplifies to:
\[ 4700 = 94 T_c \]

5. **Isolate \( T_c \)**:
Now, divide both sides by 94 to find \( T_c \):
\[ T_c = \frac{4700}{94} \]

6. **Calculate \( T_c \)**:
Performing the division:
\[ T_c \approx 50^\circ C \]

### Final Answer:
The correct temperature when the faulty thermometer reads \( 52^\circ C \) is approximately \( 50^\circ C \).

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