If the ratio of $$a : b = 2 : 3$$ and $$b : c = 4 : 5$$, find the ratio $$(a + b) : (b + c)$$.

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Accepted Answer

To solve this problem, first combine the given ratios to find the overall ratio of $$a : b : c$$.

1. **Combine Ratios**:  
   From $$a : b = 2 : 3$$ and $$b : c = 4 : 5$$, express both ratios in terms of a common variable for $$b$$.  
   Multiply the first ratio by 4 and the second ratio by 3:  
   $$
   a : b = 8 : 12, \quad b : c = 12 : 15
   $$  
   Therefore, $$a : b : c = 8 : 12 : 15$$.

2. **Express Values**:  
   Let $$a = 8k$$, $$b = 12k$$, and $$c = 15k$$.

3. **Calculate $$(a + b) : (b + c)$$**:  
   Add the respective terms:  
   $$
   a + b = 8k + 12k = 20k, \quad b + c = 12k + 15k = 27k
   $$  
   Hence, the ratio is:  
   $$
   (a + b) : (b + c) = \frac{20k}{27k} = \frac{20}{27}
   $$

Thus, the final answer is $$20 : 27$$.

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