A grocer purchases three qualities of lemons at different rates. The first quality was purchased at 2 for ₹ 1, the second at 3 for ₹ 2 and the third at 4 for ₹ 3. He sold all the lemons at 5 for ₹ 4. If the ratio of the number of lemons of the three qualities is 1 : 2 : 3, then what is the approximate gain or loss percentage incurred by the grocer?

### Concept & Formula When dealing with mixed batches of items bought at different rates and specific ratios, calculate the exact Total Cost Price (CP) and Total Selling Price (SP) using an assumed quantity based on the Lowest Common Multiple (LCM) to avoid fractions. $$Gain \% = \left( \frac{\text{Total SP} - \text{Total CP}}{\text{Total CP}} \right) \times 100$$ ### Step-by-Step Solution * **Given:** Buying rates are 2 for ₹ 1, 3 for ₹ 2, 4 for ₹ 3. Selling rate is 5 for ₹ 4. Quantity ratio = 1:2:3. * **Calculation:** 1. Let the quantities be $x$, $2x$, and $3x$. To avoid fractions with the buying rates (divisors 2, 3, 4), let $x$ be a multiple of their LCM (which is 12). 2. Assume he buys 12, 24, and 36 lemons respectively. Total lemons = $12 + 24 + 36 = 72$. 3. Total CP = CP1 + CP2 + CP3 = $\left(\frac{1}{2} \times 12\right) + \left(\frac{2}{3} \times 24\right) + \left(\frac{3}{4} \times 36\right)$ 4. Total CP = $6 + 16 + 27 = \text{₹ } 49$. 5. Total SP of 72 lemons = $\frac{4}{5} \times 72 = \frac{288}{5} = \text{₹ } 57.60$. 6. Since SP > CP, it is a gain. Gain = $57.60 - 49 = 8.60$. 7. Gain Percentage = $\left( \frac{8.60}{49} \right) \times 100 \approx 17.551\%$. 8. Rounding to two decimal places gives 17.56% gain. ### Exam Strategy & Shortcut Always choose an integer for $x$ in the ratio $x : 2x : 3x$ that is divisible by the quantity bases (2, 3, 4) in the rates. This keeps all CP calculations as whole numbers, drastically improving speed and reducing arithmetic errors under pressure. ### Common Pitfall A common mistake is attempting to find an "average cost price" by simply averaging the rates. This fails because the items are weighted differently (1:2:3 ratio). You must calculate total values. ### Final Answer Therefore, the correct answer is **17.56% gain**.