If p and q are intergers such that `6 lt q lt 17 and p/q=3/4`, how many possible values are there for p?

Options

A. two
B. three
C. four
D. five

Answer

To solve the problem, we need to determine how many possible integer values there are for \( p \) given the conditions on \( q \) and the relationship between \( p \) and \( q \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We are given that \( \frac{p}{q} = \frac{3}{4} \). This implies that:
\[ p = \frac{3}{4} q \]

2. **Finding the Range for \( q \)**:
We know that \( q \) must be an integer such that:
\[ 6 < q < 17 \]
Therefore, the possible integer values for \( q \) are:
\[ 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 \]
This gives us a total of 10 possible values for \( q \).

3. **Determining Valid Values for \( p \)**:
Since \( p \) must also be an integer, \( \frac{3}{4} q \) must yield an integer value. This means that \( q \) must be a multiple of 4 (because 4 is in the denominator).

4. **Identifying Multiples of 4 in the Range**:
The multiples of 4 within the range of \( 7 \) to \( 16 \) are:
- \( 8 \) (since \( 4 \times 2 = 8 \))
- \( 12 \) (since \( 4 \times 3 = 12 \))
- \( 16 \) (since \( 4 \times 4 = 16 \))

5. **Calculating Corresponding Values of \( p \)**:
Now we can calculate \( p \) for each valid \( q \):
- For \( q = 8 \):
\[ p = \frac{3}{4} \times 8 = 6 \]
- For \( q = 12 \):
\[ p = \frac{3}{4} \times 12 = 9 \]
- For \( q = 16 \):
\[ p = \frac{3}{4} \times 16 = 12 \]

6. **Counting the Possible Values of \( p \)**:
The possible integer values of \( p \) are \( 6, 9, \) and \( 12 \). Thus, there are a total of 3 possible values for \( p \).

### Final Answer:
The number of possible values for \( p \) is **3**.