A shopkeeper expects a gain of $22 \frac{1}{2}\%$ on his cost price. If in a week, his sale was of ₹ 392, what was his profit?

### Concept & Profit Calculation via Fractions When dealing with mixed percentage figures, converting the profit percentage to a ratio or fraction simplifies finding the exact profit amount without first calculating the absolute Cost Price. $$ \text{Profit Fraction} = \frac{\text{Gain \%}}{100} $$ ### Step-by-Step Solution * **Step 1:** Convert the gain percentage to a manageable fraction. * $\text{Gain \%} = 22 \frac{1}{2}\% = \frac{45}{2}\%$ * Fractional value = $\frac{45}{2 \times 100} = \frac{45}{200} = \frac{9}{40}$ * **Step 2:** Interpret the fraction in terms of ratios. * A gain of $\frac{9}{40}$ means that if the Cost Price (CP) is 40 units, the Profit is 9 units. * Therefore, the Selling Price (SP) = $\text{CP} + \text{Profit} = 40 + 9 = 49$ units. * **Step 3:** Relate the ratio to the given real-world value. * We are given that the total sale (Selling Price) = ₹ 392. * So, 49 units = ₹ 392 * 1 unit = $\frac{392}{49} = ₹ 8$ * **Step 4:** Calculate the profit. * The profit is 9 units. * $\text{Profit} = 9 \times 8 = ₹ 72$ ### Exam Strategy & Shortcut By immediately recognizing $22.5\%$ as $\frac{9}{40}$, you establish the CP : SP ratio as $40 : 49$. The question asks for profit (the 9 parts). You just need the multiplier for the SP: $392 / 49 = 8$. Profit is $9 \times 8 = 72$. This avoids the messy algebra of $x + 0.225x = 392$. ### Common Pitfall Attempting to find the Cost Price by dividing 392 by 1.225 can lead to arithmetic errors due to the complex decimal. Even if successful (finding CP = 320), it requires a second step ($392 - 320 = 72$), slowing you down. ### Final Answer Therefore, the correct answer is **₹ 72**.