A dealer buys dry fruit at the rate of ₹ $100$, ₹ $80$ and ₹ $60$ per kg. He bought them in the ratio $12 : 15 : 20$ by weight. He in total gets $20\%$ profit by selling the first two and at last he finds he has no gain no loss in selling the whole quantity which he had. What was the percentage loss he suffered for the third quantity? (M.A.T., 2007)

### Concept & Break-Even Analysis in Mixtures "No gain no loss" on the whole transaction means the Total Selling Price (SP) exactly equals the Total Cost Price (CP). The profit made on the first two items must perfectly offset the absolute monetary loss incurred on the third item. ### Step-by-Step Solution * **Given:** - Prices: $P_1$ = ₹ $100$, $P_2$ = ₹ $80$, $P_3$ = ₹ $60$ - Weights ratio: $W_1 = 12$, $W_2 = 15$, $W_3 = 20$ - Profit on first two combined = $20\%$ - Overall profit/loss = $0\%$ (Total CP = Total SP) * **Calculation:** 1. Let the actual weights be $12$ kg, $15$ kg, and $20$ kg. 2. Calculate the CP of each variety: $CP_1 = 12 \times 100 = 1200$ $CP_2 = 15 \times 80 = 1200$ $CP_3 = 20 \times 60 = 1200$ 3. Total CP of the first two varieties = $1200 + 1200 = 2400$. 4. Total CP of all three = $1200 + 1200 + 1200 = 3600$. 5. The dealer gets a $20\%$ profit on the first two varieties. Profit amount on first two = $20\%$ of $2400 = 0.20 \times 2400 = 480$. 6. Since there is "no gain no loss" overall, the profit made on the first two ($+480$) must equal the loss on the third variety. Loss on third variety = ₹ $480$. 7. Calculate the percentage loss on the third variety based on its CP ($1200$): Loss % = $\left( \frac{\text{Loss}}{CP_3} \right) \times 100$ Loss % = $\left( \frac{480}{1200} \right) \times 100 = 40\%$ ### Exam Strategy & Shortcut Notice the CP of all three components is identical: $12 \times 100 = 1200$, $15 \times 80 = 1200$, and $20 \times 60 = 1200$. Let the CP of each part be $1$ unit. Total CP = $3$ units. Profit on the first two parts is $20\%$. Since they represent $2$ equal units of cost, the total profit value is $2 \times 20\% = +40\%$. To balance this out to $0\%$ overall, the third part (which is $1$ equal unit of cost) must suffer a loss of exactly $-40\%$. ### Common Pitfall A common error is calculating the $20\%$ profit on the selling price instead of the cost price, or taking the average of the rates instead of using the total cost amounts. ### Final Answer Therefore, the correct answer is **$40\%$**.