At what temperature is the Fahrenheit scale reading equal to
(a) twice (b) half of Celsius ?

Explanation

To solve the problem, we need to use the relationship between the Celsius (Tc) and Fahrenheit (Tf) temperature scales. The formula to convert Celsius to Fahrenheit is:

\[ Tf = \frac{9}{5} Tc + 32 \]

### Part (a): When is Fahrenheit equal to twice Celsius?

1. **Set up the equation**: We want to find the temperature where Fahrenheit is twice Celsius. So, we can write:
\[ Tf = 2 Tc \]

2. **Substitute the conversion formula**: Substitute the expression for Tf from the conversion formula into the equation:
\[ \frac{9}{5} Tc + 32 = 2 Tc \]

3. **Rearrange the equation**: To isolate Tc, we can rearrange the equation:
\[ \frac{9}{5} Tc + 32 = 2 Tc \]
\[ 32 = 2 Tc - \frac{9}{5} Tc \]

4. **Combine like terms**: Convert 2 Tc into a fraction with a common denominator:
\[ 2 Tc = \frac{10}{5} Tc \]
\[ 32 = \frac{10}{5} Tc - \frac{9}{5} Tc \]
\[ 32 = \frac{1}{5} Tc \]

5. **Solve for Tc**: Multiply both sides by 5 to solve for Tc:
\[ Tc = 32 \times 5 \]
\[ Tc = 160 \text{ degrees Celsius} \]

### Part (b): When is Fahrenheit equal to half of Celsius?

1. **Set up the equation**: Now we want to find the temperature where Fahrenheit is half of Celsius:
\[ Tf = \frac{1}{2} Tc \]

2. **Substitute the conversion formula**: Substitute Tf in the equation:
\[ \frac{9}{5} Tc + 32 = \frac{1}{2} Tc \]

3. **Rearrange the equation**: Isolate Tc:
\[ \frac{9}{5} Tc + 32 = \frac{1}{2} Tc \]
\[ 32 = \frac{1}{2} Tc - \frac{9}{5} Tc \]

4. **Combine like terms**: Convert \(\frac{1}{2} Tc\) into a fraction with a common denominator:
\[ \frac{1}{2} Tc = \frac{5}{10} Tc \]
\[ \frac{9}{5} Tc = \frac{18}{10} Tc \]
\[ 32 = \frac{5}{10} Tc - \frac{18}{10} Tc \]
\[ 32 = -\frac{13}{10} Tc \]

5. **Solve for Tc**: Multiply both sides by -10/13:
\[ Tc = -32 \times \frac{10}{13} \]
\[ Tc \approx -24.6 \text{ degrees Celsius} \]

### Final Answers:
- (a) The temperature at which the Fahrenheit scale reading is twice that of Celsius is **160 degrees Celsius**.
- (b) The temperature at which the Fahrenheit scale reading is half that of Celsius is approximately **-24.6 degrees Celsius**.