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

Explanation

To solve the problem of finding the temperature at which the Fahrenheit scale reading is equal to twice and half of the Celsius scale, we will use the conversion formula between Celsius (C) and Fahrenheit (F):

\[ F = \frac{9}{5}C + 32 \]

### Part (a): Fahrenheit is twice Celsius

1. **Set up the equation**: According to the problem, we want to find the temperature where Fahrenheit (F) is equal to twice Celsius (C). This can be expressed as:
\[ F = 2C \]

2. **Substitute into the conversion formula**: We can substitute \( F = 2C \) into the conversion formula:
\[ 2C = \frac{9}{5}C + 32 \]

3. **Clear the equation**: To eliminate the fraction, multiply the entire equation by 5:
\[ 10C = 9C + 160 \]

4. **Rearrange the equation**: Subtract \( 9C \) from both sides:
\[ 10C - 9C = 160 \]
\[ C = 160 \]

5. **Calculate Fahrenheit**: Now substitute \( C = 160 \) back into the equation \( F = 2C \):
\[ F = 2 \times 160 = 320 \]

Thus, the temperature at which the Fahrenheit scale reading is equal to twice the Celsius is **320°F**.

### Part (b): Fahrenheit is half of Celsius

1. **Set up the equation**: Here, we want to find the temperature where Fahrenheit (F) is equal to half of Celsius (C). This can be expressed as:
\[ F = \frac{1}{2}C \]

2. **Substitute into the conversion formula**: Substitute \( F = \frac{1}{2}C \) into the conversion formula:
\[ \frac{1}{2}C = \frac{9}{5}C + 32 \]

3. **Clear the equation**: Multiply the entire equation by 10 to eliminate the fraction:
\[ 5C = 18C + 320 \]

4. **Rearrange the equation**: Subtract \( 18C \) from both sides:
\[ 5C - 18C = 320 \]
\[ -13C = 320 \]
\[ C = -\frac{320}{13} \approx -24.615 \]

5. **Calculate Fahrenheit**: Now substitute \( C \approx -24.615 \) back into \( F = \frac{1}{2}C \):
\[ F = \frac{1}{2} \times -24.615 \approx -12.3075 \]

Thus, the temperature at which the Fahrenheit scale reading is equal to half of the Celsius is approximately **-12.3°F**.

### Summary of Answers:
- (a) 320°F
- (b) -12.3°F