The cash difference between the selling prices of an article at a profit of $4\%$ and $6\%$ is ₹ 3. The ratio of the two selling prices is (C.B.I., 2003)

### Concept & Ratio of Selling Prices When asked for the ratio of two selling prices derived from the same cost price, the absolute cash difference is irrelevant. The ratio depends solely on the profit percentages. $$ \text{Ratio} = \frac{100 + P_1\%}{100 + P_2\%} $$ ### Step-by-Step Solution * Let the Cost Price (CP) of the article be $x$. * The first selling price ($SP_1$) is at a $4\%$ profit: $SP_1 = x \times \left(1 + \frac{4}{100}\right) = 1.04x$ * The second selling price ($SP_2$) is at a $6\%$ profit: $SP_2 = x \times \left(1 + \frac{6}{100}\right) = 1.06x$ * We need to find the ratio $SP_1 : SP_2$. * Ratio = $\frac{1.04x}{1.06x} = \frac{1.04}{1.06}$ * Multiply numerator and denominator by 100 to remove decimals: $\frac{104}{106}$ * Simplify the fraction by dividing both by 2: $\frac{52}{53}$ * The ratio is $52 : 53$. ### Exam Strategy & Shortcut Ignore the ₹ 3 completely; it's a distractor intended to make you calculate the actual CP. Immediately write down the ratio of the percentages representing the selling prices: $104 : 106$. Simplify it to $52 : 53$. ### Common Pitfall Wasting time calculating the actual Cost Price using the ₹ 3 difference ($2\%$ of CP = 3 $\rightarrow$ CP = 150) before calculating the selling prices and their ratio. ### Final Answer Therefore, the correct answer is **$52 : 53$**.