If $a : b = 2 : 5$, then the value of $(2a + 3b) : (7a + 5b)$ is

### Concept & Ratio Substitution When given a ratio of variables like $a : b = m : n$ and asked to find the value of a homogeneous expression (where all terms have the same total degree), we can directly substitute the proportional values $a = m$ and $b = n$. ### Step-by-Step Solution - **Given:** $a : b = 2 : 5$. - **Substitute:** Let $a = 2$ and $b = 5$. - **Expression:** $(2a + 3b) : (7a + 5b)$ - **Calculate Numerator:** $2(2) + 3(5) = 4 + 15 = 19$ - **Calculate Denominator:** $7(2) + 5(5) = 14 + 25 = 39$ - **Result:** The ratio is $19 : 39$. ### Exam Strategy & Shortcut Since the degree of every term in the numerator and denominator is $1$, we can bypass using a constant $k$ (e.g., $a = 2k, b = 5k$) because $k$ will completely factor out and cancel. Direct substitution is the fastest path. ### Common Pitfall A common mistake is trying to solve for $a$ or $b$ individually, forgetting that we only need their relative sizes. Also, simple arithmetic errors during multiplication or addition can lead to choosing a distractor option. ### Final Answer Therefore, the correct answer is **$19 : 39$**.