If `c + d = 11` and c and d postive integers , which of the following is a possible value for `5c + 8d` ?

Options

A. 55
B. 61
C. 69
D. 83

Answer

To solve the problem, we start with the equations provided:

1. **Given equations**:
\[ c + d = 11 \]
where \(c\) and \(d\) are positive integers.

2. **Express \(c\) in terms of \(d\)**:
\[ c = 11 - d \]

3. **Substitute \(c\) into the expression \(5c + 8d\)**:
\[ 5c + 8d = 5(11 - d) + 8d \]

4. **Distribute the 5**:
\[ = 55 - 5d + 8d \]

5. **Combine like terms**:
\[ = 55 + 3d \]

Now we have the expression for \(5c + 8d\):
\[ 5c + 8d = 55 + 3d \]

6. **Identify possible values**: We need to check which of the following options can be expressed as \(55 + 3d\):
- 55
- 61
- 69
- 83

7. **Check each option**:
- For **55**:
\[ 55 + 3d = 55 \implies 3d = 0 \implies d = 0 \quad (\text{not a positive integer}) \]
- For **61**:
\[ 55 + 3d = 61 \implies 3d = 6 \implies d = 2 \quad (\text{positive integer}) \]
- For **69**:
\[ 55 + 3d = 69 \implies 3d = 14 \implies d = \frac{14}{3} \quad (\text{not an integer}) \]
- For **83**:
\[ 55 + 3d = 83 \implies 3d = 28 \implies d = \frac{28}{3} \quad (\text{not an integer}) \]

8. **Conclusion**: The only option that yields a positive integer for \(d\) is **61**.

Thus, the possible value for \(5c + 8d\) is:
\[ \boxed{61} \]