One variety of sugar is sold for ₹ 3.20 per kg at a loss of 20% and another variety is sold for ₹ 6 per kg at a gain of 20%. If equal quantities of the two are mixed together and the mixture is sold at ₹ 5.40 per kg, what is the loss or gain percentage?

### Concept & Inverse Profit Formula When selling prices and profit/loss percentages are given, you must first calculate the original Cost Price (CP) for each item before determining the profit or loss of the mixture. $$\text{CP} = \left( \frac{100}{100 \pm \text{Profit/Loss \%}} \right) \times \text{SP}$$ ### Step-by-Step Solution * **Find CP of First Variety (20% loss):** * $SP_1 = ₹ 3.20$ * $CP_1 = \left( \frac{100}{100 - 20} \right) \times 3.20 = \left( \frac{100}{80} \right) \times 3.20 = \frac{5}{4} \times 3.20 = ₹ 4.00$ per kg * **Find CP of Second Variety (20% gain):** * $SP_2 = ₹ 6.00$ * $CP_2 = \left( \frac{100}{100 + 20} \right) \times 6.00 = \left( \frac{100}{120} \right) \times 6.00 = \frac{5}{6} \times 6.00 = ₹ 5.00$ per kg * **Calculate Mixture Cost & Revenue (assume 1 kg of each):** * Total CP for 2 kg = $4.00 + 5.00 = ₹ 9.00$ * Selling price of mixture = ₹ 5.40 per kg * Total SP for 2 kg = $2 \times 5.40 = ₹ 10.80$ * **Determine Overall Profit/Loss %:** * Gain = $10.80 - 9.00 = ₹ 1.80$ * Gain \% = $\left( \frac{1.80}{9.00} \right) \times 100 = \frac{180}{9} = 20\%$ ### Exam Strategy & Shortcut Find the average Cost Price (CP) per kg first. $CP_{\text{avg}} = (4 + 5)/2 = 4.5$. The Selling Price (SP) of the mixture is $5.40$. The profit per kg is $5.40 - 4.50 = 0.90$. The gain percentage is $(0.90 / 4.50) \times 100 = (1/5) \times 100 = 20\%$. ### Common Pitfall A common pitfall is to calculate the average of the initial selling prices ($(3.20 + 6.00) / 2 = 4.60$) instead of reversing them to find the true underlying cost prices first. ### Final Answer Therefore, the correct answer is **Gain 20%**.