A shopkeeper gives 3 consecutive discounts of 10%, 15% and 15% after which he sells his goods at a percentage profit of 30.05 percent on the cost price. Find the value of the percentage profit that the shopkeeper would have earned if he had given discounts of 10% and 15% only. (M.B.A., 2008)

### Concept & Linking Marked Price, Selling Price, and Cost Price The relationship between Cost Price ($CP$), Marked Price ($MP$), Profit Percentage ($P\%$), and successive discounts ($d_1, d_2, d_3$) is foundational. The Selling Price ($SP$) links them: $$SP = CP \times \left(1 + \frac{P}{100}\right) = MP \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \times \left(1 - \frac{d_3}{100}\right)$$ ### Step-by-Step Solution * **Set up a baseline for Marked Price:** Let the Marked Price ($MP$) of the goods be $100$. * **Calculate actual Selling Price with 3 discounts:** * Discounts are $10\%$, $15\%$, and $15\%$. * $SP = 100 \times 0.90 \times 0.85 \times 0.85$ * $SP = 90 \times 0.7225 = 65.025$ * **Find the Cost Price (CP):** * The profit is given as $30.05\%$, meaning $SP = 1.3005 \times CP$. * $65.025 = 1.3005 \times CP$ * $CP = \frac{65.025}{1.3005} = 50$ * **Calculate Hypothetical Selling Price with 2 discounts:** * If he gives only two discounts ($10\%$ and $15\%$): * $New SP = 100 \times 0.90 \times 0.85 = 76.5$ * **Calculate the New Profit Percentage:** * New Profit = $New SP - CP = 76.5 - 50 = 26.5$ * $New Profit \% = \left(\frac{26.5}{50}\right) \times 100 = 26.5 \times 2 = 53\%$ ### Exam Strategy & Shortcut Instead of finding exact absolute values, work entirely in ratios. Let $SP_1$ be the price after 3 discounts and $SP_2$ be the price after 2 discounts. Notice that $SP_1$ is just $SP_2$ with an additional $15\%$ discount applied. So, $SP_1 = SP_2 \times 0.85$. We know $SP_1 = CP \times 1.3005$. Substitute: $SP_2 \times 0.85 = CP \times 1.3005 \rightarrow SP_2 = CP \times \frac{1.3005}{0.85} = CP \times 1.53$. Since $SP_2 = 1.53 \times CP$, the profit is directly $53\%$. This bypasses calculating MP completely! ### Common Pitfall A common pitfall is assuming you need the absolute values of the Cost Price and Marked Price in rupees to solve the problem, which leads to algebraic gridlock. Assume an easy index base like $100$ or use ratio properties. ### Final Answer Therefore, the correct answer is **53%**.