If `a:b = 3:8,` find the value of `(5a - 3b) //(2a +b).` (a) 9/14 (b) 14/9 (c) - 9/14 (d) -14/9

Options

A. `9//14`
B. `14//9`
C. `-9//14`
D. `-14//9`

Answer

To solve the problem, we start with the given ratio \( a:b = 3:8 \). This means we can express \( a \) and \( b \) in terms of a common variable \( t \):

1. **Express \( a \) and \( b \) in terms of \( t \)**:
\[ a = 3t \quad \text{and} \quad b = 8t \]

2. **Substitute \( a \) and \( b \) into the expression \( \frac{5a - 3b}{2a + b} \)**:
\[ \frac{5a - 3b}{2a + b} = \frac{5(3t) - 3(8t)}{2(3t) + 8t} \]

3. **Calculate the numerator**:
\[ 5(3t) - 3(8t) = 15t - 24t = -9t \]

4. **Calculate the denominator**:
\[ 2(3t) + 8t = 6t + 8t = 14t \]

5. **Combine the results**:
\[ \frac{-9t}{14t} \]

6. **Simplify the expression**:
\[ \frac{-9t}{14t} = \frac{-9}{14} \quad (\text{since } t \neq 0) \]

Thus, the final answer is:
\[ \frac{-9}{14} \]

### Final Answer:
(c) \(-\frac{9}{14}\)

---