`C=5/9 (F-32)`
The equation above shows how a temperature F, measured in degrees Fahrenheit, relates to a temperature C, measured in degrees Celsius. Based on the equation, which of the following must be true?
I. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of `5/9` degree Celsius.
II. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.
III. A temperature increase of `5/9` degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

Options

A. I only
B. II only
C. III only
D. I and II only

Answer

To solve the problem, we need to analyze the given equation \( C = \frac{5}{9}(F - 32) \) and determine the validity of the three statements regarding temperature changes in Fahrenheit and Celsius.

### Step 1: Analyze Statement I
**Statement I:** A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of \( \frac{5}{9} \) degree Celsius.

To verify this, we can find the change in Celsius when there is a 1 degree increase in Fahrenheit.

1. Start with the equation:
\[ C = \frac{5}{9}(F - 32) \]
2. If \( F \) increases by 1 degree, we have:
\[ C' = \frac{5}{9}((F + 1) - 32) = \frac{5}{9}(F - 32 + 1) = \frac{5}{9}(F - 32) + \frac{5}{9} \]
3. The change in Celsius, \( \Delta C \), is:
\[ \Delta C = C' - C = \left(\frac{5}{9}(F - 32) + \frac{5}{9}\right) - \frac{5}{9}(F - 32) = \frac{5}{9} \]

Thus, a 1 degree increase in Fahrenheit results in a \( \frac{5}{9} \) degree increase in Celsius. **Statement I is true.**

### Step 2: Analyze Statement II
**Statement II:** A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

To verify this, we can find the change in Fahrenheit when there is a 1 degree increase in Celsius.

1. We start with the rearranged equation:
\[ F = \frac{9}{5}C + 32 \]
2. If \( C \) increases by 1 degree, we have:
\[ F' = \frac{9}{5}(C + 1) + 32 = \frac{9}{5}C + \frac{9}{5} + 32 \]
3. The change in Fahrenheit, \( \Delta F \), is:
\[ \Delta F = F' - F = \left(\frac{9}{5}C + \frac{9}{5} + 32\right) - \left(\frac{9}{5}C + 32\right) = \frac{9}{5} \]

Now, converting \( \frac{9}{5} \) to decimal gives us \( 1.8 \). Thus, a 1 degree increase in Celsius results in a 1.8 degree increase in Fahrenheit. **Statement II is true.**

### Step 3: Analyze Statement III
**Statement III:** A temperature increase of \( \frac{5}{9} \) degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

To verify this, we can find the change in Celsius when there is a \( \frac{5}{9} \) degree increase in Fahrenheit.

1. If \( F \) increases by \( \frac{5}{9} \), we have:
\[ C' = \frac{5}{9}\left(F + \frac{5}{9} - 32\right) = \frac{5}{9}(F - 32) + \frac{5}{9}\cdot\frac{5}{9} = C + \frac{25}{81} \]
2. The change in Celsius, \( \Delta C \), is:
\[ \Delta C = C' - C = \frac{25}{81} \]

Since \( \frac{25}{81} \) is not equal to 1, a \( \frac{5}{9} \) degree increase in Fahrenheit does not correspond to a 1 degree increase in Celsius. **Statement III is false.**

### Conclusion
Based on the analysis:
- Statement I is true.
- Statement II is true.
- Statement III is false.

Thus, the correct answer is that only statements I and II are true.

### Final Answer
The correct statements are I and II.

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