If $a : b = 2 : 3$ and $b : c = 4 : 5$, then $(a + b) : (b + c)$ is equal to

### Concept & Combining Ratios To evaluate an expression involving variables from two different ratios, you must first merge them into a single continuous ratio ($a : b : c$). This is done by equating the common variable (in this case, $b$) by finding the Least Common Multiple (LCM) of its corresponding values in both ratios. ### Step-by-Step Solution 1. **Given:** $a : b = 2 : 3$ $b : c = 4 : 5$ 2. The common term is $b$. In the first ratio, $b = 3$. In the second, $b = 4$. 3. Find the LCM of $3$ and $4$, which is $12$. 4. Multiply both ratios to make $b = 12$: First ratio: Multiply by $4 \implies a : b = (2 \cdot 4) : (3 \cdot 4) = 8 : 12$ Second ratio: Multiply by $3 \implies b : c = (4 \cdot 3) : (5 \cdot 3) = 12 : 15$ 5. Now, combine them into a single ratio: $$a : b : c = 8 : 12 : 15$$ 6. We can now assign proportional values: $a = 8$, $b = 12$, and $c = 15$. 7. Substitute these into the requested expression $(a + b) : (b + c)$: $$(8 + 12) : (12 + 15)$$ $$20 : 27$$ ### Exam Strategy & Shortcut Use the "Reverse N" or block multiplication method to combine $a:b$ and $b:c$ instantly. Write them stacked: $2 : 3$ $4 : 5$ Multiply down the left column ($2 \cdot 4 = 8$), diagonally up ($4 \cdot 3 = 12$), and down the right column ($3 \cdot 5 = 15$) to get $8 : 12 : 15$. Then plug the numbers in directly. ### Common Pitfall A frequent error is adding the values from the disjointed ratios directly, such as taking $a=2, b=3$ for the first part and $b=4, c=5$ for the second part, resulting in mismatched scaling. Always unify the ratios first. ### Final Answer Therefore, the correct answer is **$20 : 27$**.