If $5a + 3b : 2a - 3b = 23 : 5$, then the value of $a : b$ is

### Concept & Cross-Multiplication To find the base ratio of two variables from a fractional equation, express the ratio as a fraction and use cross-multiplication to isolate the variables on opposite sides of the equation. ### Step-by-Step Solution - **Given:** $\frac{5a + 3b}{2a - 3b} = \frac{23}{5}$ - **Cross-multiply:** $5(5a + 3b) = 23(2a - 3b)$ - **Expand both sides:** $25a + 15b = 46a - 69b$ - **Group like terms:** - Move terms with $b$ to the left: $15b + 69b = 84b$ - Move terms with $a$ to the right: $46a - 25a = 21a$ - **Equation:** $84b = 21a$ - **Solve for $a/b$:** $\frac{a}{b} = \frac{84}{21}$ - **Simplify:** $\frac{a}{b} = \frac{4}{1} = 4 : 1$. ### Exam Strategy & Shortcut You can quickly group terms mentally: $(23 \cdot 2 - 5 \cdot 5)a = (23 \cdot 3 + 5 \cdot 3)b \implies 21a = 84b \implies a : b = 84 : 21 = 4 : 1$. This saves writing down the intermediate expansion steps. ### Common Pitfall A common pitfall is incorrectly managing the negative signs during expansion, such as writing $-69b$ as $+69b$, which drastically changes the resulting ratio. ### Final Answer Therefore, the correct answer is **$4 : 1$**.