Translate the following statements.
In a certain word, the number of consonants is `1//4` more than the number of vowels. Which of the following is a possibility for the number of letters in the word?

Options

A. 8
B. 9
C. 10
D. 11

Answer

To solve the problem, we need to translate the statement and find the total number of letters in a word based on the relationship between the number of consonants and vowels.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the number of vowels in the word be represented as \( V \).
- According to the problem, the number of consonants \( C \) is \( \frac{1}{4} \) more than the number of vowels.

2. **Establish the Relationship**:
- From the statement, we can express the number of consonants as:
\[ C = V + \frac{1}{4}V = \frac{5}{4}V \]

3. **Total Letters in the Word**:
- The total number of letters \( T \) in the word is the sum of the number of vowels and consonants:
\[ T = V + C = V + \frac{5}{4}V = \frac{9}{4}V \]

4. **Finding Possible Values for \( T \)**:
- Since \( T \) must be a whole number, \( \frac{9}{4}V \) must also be a whole number. This implies that \( V \) must be a multiple of 4 (to cancel out the denominator).
- Let \( V = 4x \) for some integer \( x \). Then substituting this into the equation for \( T \):
\[ T = \frac{9}{4}(4x) = 9x \]

5. **Evaluate Possible Options**:
- The options given are 8, 9, 10, and 11. We need to check which of these can be expressed as \( 9x \) for some integer \( x \).
- The only option that is a multiple of 9 is 9 (when \( x = 1 \)).
- Therefore, the total number of letters \( T \) can only be 9.

### Conclusion:
The only possibility for the number of letters in the word is **9**.

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