Ashish started a business with an initial investment of ₹ 500000. In the first year, he incurred a loss of 4%. However, during the second year, he earned a profit of 5% which in the third year rose to 10%. Calculate his net profit for the entire period of three years.

### Concept & Formula This requires calculating the final amount after three successive percentage changes (one loss, two profits), and then finding the difference between the final amount and the initial investment to determine the net profit. $$ \text{Final Amount} = \text{Initial} \times \left(1 - \frac{L_1}{100}\right) \times \left(1 + \frac{P_1}{100}\right) \times \left(1 + \frac{P_2}{100}\right) $$ $$ \text{Net Profit} = \text{Final Amount} - \text{Initial Amount} $$ ### Step-by-Step Solution * **Given:** Initial investment = $₹ 500000$. Year 1: $4\%$ loss (multiplier $0.96$). Year 2: $5\%$ profit (multiplier $1.05$). Year 3: $10\%$ profit (multiplier $1.10$). * **Set up the final amount equation:** $$ \text{Final Amount} = 500000 \times \frac{96}{100} \times \frac{105}{100} \times \frac{110}{100} $$ * **Simplify by cancelling zeros:** There are $5$ zeros in $500000$ and $1$ zero in $110$. That's $6$ zeros in the numerator. The denominator has $100 \times 100 \times 100 = 1000000$ ($6$ zeros). They cancel out perfectly! * **Calculate the product:** $$ \text{Final Amount} = 5 \times 96 \times 105 \times 11 $$ $$ \text{Final Amount} = 480 \times 105 \times 11 $$ $$ \text{Final Amount} = 50400 \times 11 = 554400 $$ * **Calculate Net Profit:** $$ \text{Net Profit} = 554400 - 500000 = 54400 $$ ### Exam Strategy & Shortcut **Effective Percentage Method:** You can combine the percentage changes successively using the formula $x + y + \frac{xy}{100}$. 1. Combine Year 1 ($-4\%$) and Year 2 ($+5\%$): $$ -4 + 5 + \frac{(-4 \times 5)}{100} = 1 - 0.20 = 0.8\% $$ (Net $0.8\%$ profit after 2 years) 2. Combine this result ($+0.8\%$) with Year 3 ($+10\%$): $$ 0.8 + 10 + \frac{(0.8 \times 10)}{100} = 10.8 + 0.08 = 10.88\% $$ The total net profit is exactly $10.88\%$ of the initial investment. $$ 10.88\% \text{ of } 500000 = 10.88 \times 5000 = 54400 $$ This eliminates heavy multiplication and delivers the answer directly. ### Common Pitfall A frequent mistake is applying the $10\%$ profit in the third year to the base $5\%$ profit of the second year (meaning only a $15\%$ total profit), rather than applying a $10\%$ compound growth onto the actual final accrued capital of the second year. Always use successive multipliers. ### Final Answer **Therefore, the correct answer is ₹ 54400.**