If s and r are integers, `8ltrlt40`, and `(s)/(r)=(4)/(7)`, how many possible values are there for s?

Options

A. Two
B. Three
C. Four
D. Five

Answer

To solve the problem, we need to find how many possible integer values there are for \( s \) given that \( s \) and \( r \) are integers, \( 8 < r < 40 \), and \( \frac{s}{r} = \frac{4}{7} \).

### Step-by-step Solution:

1. **Set up the equation**:
From the equation \( \frac{s}{r} = \frac{4}{7} \), we can express \( s \) in terms of \( r \):
\[ s = \frac{4}{7} r \]

2. **Determine the conditions for \( s \)**:
Since \( s \) must be an integer, \( \frac{4}{7} r \) must also be an integer. This implies that \( r \) must be a multiple of 7, because 4 is not divisible by 7.

3. **Identify the range for \( r \)**:
We know that \( r \) must satisfy \( 8 < r < 40 \). We need to find the multiples of 7 within this range.

4. **List the multiples of 7**:
The multiples of 7 are:
\[ 7, 14, 21, 28, 35, 42, \ldots \]
We only need to consider those that fall between 8 and 40.

5. **Find the valid multiples of 7**:
The multiples of 7 that are greater than 8 and less than 40 are:
- 14
- 21
- 28
- 35

6. **Count the valid values of \( r \)**:
We have found 4 valid values for \( r \): 14, 21, 28, and 35.

7. **Calculate corresponding values of \( s \)**:
For each valid \( r \), we can calculate \( s \):
- If \( r = 14 \), then \( s = \frac{4}{7} \times 14 = 8 \)
- If \( r = 21 \), then \( s = \frac{4}{7} \times 21 = 12 \)
- If \( r = 28 \), then \( s = \frac{4}{7} \times 28 = 16 \)
- If \( r = 35 \), then \( s = \frac{4}{7} \times 35 = 20 \)

All these values of \( s \) (8, 12, 16, and 20) are integers.

8. **Conclusion**:
Since there are 4 valid values of \( r \), there will be 4 corresponding values of \( s \). Thus, the total number of possible values for \( s \) is:
\[ \text{Total values of } s = 4 \]

### Final Answer:
The number of possible values for \( s \) is **4**.