Step 1: Understanding the Concept:
This is a problem of ratios and proportions. We should express all variables (\(a, b, c\)) in terms of a single variable to simplify the target expression. Step 2: Detailed Explanation:
Given:
\[ \frac{a}{b} = \frac{1}{3} \implies b = 3a \]
\[ \frac{b}{c} = \frac{3}{4} \implies c = \frac{4b}{3} \]
Substitute \(b = 3a\) into the equation for \(c\):
\[ c = \frac{4(3a)}{3} = 4a \]
Now, substitute \(b = 3a\) and \(c = 4a\) into the target expression:
\[ \frac{a + 2b}{b + 2c} = \frac{a + 2(3a)}{3a + 2(4a)} \]
\[ = \frac{a + 6a}{3a + 8a} = \frac{7a}{11a} = \frac{7}{11} \] Step 3: Final Answer:
The value is \(7/11\).