Ratio of earnings of $A$ and $B$ is 8 : 9 respectively. If the earnings of $A$ increase by 50% and the earnings of $B$ decrease by 25%, the new ratio of their earnings becomes 16 : 9 respectively. What are $A$'s earnings? (Bank P.O., 2006)

### Concept & Sufficiency of Data Ratios and percentages represent relative values, not absolute quantities. To find an exact numerical value (like earnings in Rupees), the problem must provide at least one absolute numerical value linking the ratio to real-world quantities. ### Step-by-Step Solution * **Analyze the Given Data**: * Initial ratio of earnings $A:B = 8:9$. Let their earnings be $8x$ and $9x$. * $A$'s new earnings = $8x + (50\% \text{ of } 8x) = 8x + 4x = 12x$. * $B$'s new earnings = $9x - (25\% \text{ of } 9x) = 9x - 2.25x = 6.75x$. * **Formulate the Equation**: The problem states the new ratio is $16:9$. $$\frac{12x}{6.75x} = \frac{16}{9}$$ * **Evaluate the Equation**: The variable $x$ cancels out from the numerator and denominator on the left side: $$\frac{12}{6.75} = \frac{16}{9}$$ Multiplying numerator and denominator of the left side by $4$ gives $\frac{48}{27}$, which simplifies to $\frac{16}{9}$. So, $\frac{16}{9} = \frac{16}{9}$. This is a true statement, but it tells us nothing about the value of $x$. * **Conclusion**: Since $x$ cancels out, it can be any number. We have no absolute value (like a total sum or difference in earnings) to pin down a specific value for $x$. ### Exam Strategy & Shortcut Scan the question for absolute values (like ₹500, or a difference of 10 units). If the question only provides ratios and percentages (relative data) and asks for an exact numerical amount, the answer is almost always "Cannot be determined". ### Common Pitfall Attempting to arbitrarily assign a value to the ratio variables (like assuming $x = 1000$) and arriving at a phantom numerical answer without realizing the logic is unsupported. ### Final Answer Therefore, the correct answer is **Cannot be determined**.