Ram bought 1600 eggs at ₹ 3.75 a dozen. He sold 900 of them at 2 for ₹ 1 and the remaining at 5 for ₹ 2. His percent gain or loss is (R.R.B., 2006)

### Concept & Multi-stage Selling To calculate the overall profit or loss percentage, you must determine the absolute total Cost Price (CP) for the entire inventory and sum up the Selling Price (SP) fragments from all different sales groups. $$ \text{Total Gain} = (\text{SP}_1 + \text{SP}_2) - \text{Total CP} $$ ### Step-by-Step Solution 1. **Total Cost Price (CP):** Ram bought 1600 eggs. Total dozens = $1600 / 12 = 400 / 3$ dozens. Cost rate = ₹ 3.75 per dozen = $15 / 4$ per dozen. $$ \text{Total CP} = \left(\frac{400}{3}\right) \times \left(\frac{15}{4}\right) $$ $$ \text{Total CP} = 100 \times 5 = \text{₹ } 500 $$ 2. **Selling Price Part 1 (SP1):** He sells 900 eggs at 2 for ₹ 1. Rate per egg = $1 / 2$ = ₹ 0.50. $$ \text{SP1} = 900 \times 0.50 = \text{₹ } 450 $$ 3. **Selling Price Part 2 (SP2):** Remaining eggs = $1600 - 900 = 700$ eggs. He sells these 700 eggs at 5 for ₹ 2. Rate per egg = $2 / 5$ = ₹ 0.40. $$ \text{SP2} = 700 \times 0.40 = \text{₹ } 280 $$ 4. **Total Selling Price and Gain:** $$ \text{Total SP} = \text{SP1} + \text{SP2} = 450 + 280 = \text{₹ } 730 $$ $$ \text{Gain} = \text{Total SP} - \text{Total CP} = 730 - 500 = \text{₹ } 230 $$ 5. **Gain Percentage:** $$ \text{Gain \%} = \left(\frac{\text{Gain}}{\text{Total CP}}\right) \times 100 $$ $$ \text{Gain \%} = \left(\frac{230}{500}\right) \times 100 = \frac{230}{5} = 46\% $$ ### Exam Strategy & Shortcut Break the fractions down smartly to save time. CP = $(1600/12) \times 3.75 = (400/3) \times (15/4) = 500$. SP of 900 = $900 \times (1/2) = 450$. SP of 700 = $700 \times (2/5) = 280$. Total SP = $730$. Gain = $230$. Gain% = $(230/500)\times 100 = 46\%$. ### Common Pitfall Converting $400/3$ to a decimal early on (133.33) and multiplying by 3.75 leads to severe rounding errors. Always keep intermediate steps in fractional form and simplify cross-multiplications. ### Final Answer Therefore, the correct answer is **46%**.