A merchant invests a certain sum and his annual gain percent is 25. If at the end of the third year, his capital is ₹ 10000, then the amount invested by him is equal to

### Concept & Formula This problem follows the principle of compound interest. A constant annual percentage gain applied to the total capital (principal + previous gains) is mathematically identical to successive percentage growth. We use the reverse compound formula to find the initial principal: $$ A = P \left(1 + \frac{R}{100}\right)^n $$ Where $A$ is the final amount, $P$ is the initial investment, $R$ is the rate, and $n$ is time. ### Step-by-Step Solution * **Given:** Final Amount ($A$) = $₹10000$. Rate ($R$) = $25\%$. Time ($n$) = $3$ years. * **Let the initial investment be $P$.** * **Determine Fractional Multiplier:** A $25\%$ gain corresponds to a multiplier of $125\%$, which simplifies to the fraction $\frac{5}{4}$. * **Set up the equation for 3 years:** $$ P \times \left(\frac{5}{4}\right)^3 = 10000 $$ $$ P \times \frac{125}{64} = 10000 $$ * **Isolate $P$:** $$ P = 10000 \times \frac{64}{125} $$ * **Simplify:** We know that $1000 \div 125 = 8$, so $10000 \div 125 = 80$. $$ P = 80 \times 64 $$ $$ P = 5120 $$ ### Exam Strategy & Shortcut **Fractional Ratios:** Memorize standard fraction equivalents. $25\% = \frac{1}{4}$. This means an investment of $4$ becomes $5$ in one year. Over 3 years, the principal to amount ratio is $4^3 : 5^3 = 64 : 125$. The final amount ($125$ units) equals $10000$. $1\text{ unit} = \frac{10000}{125} = 80$. The initial investment is $64\text{ units}$. $64 \times 80 = 5120$. This mental mapping bypasses writing out the full algebraic equation. ### Common Pitfall Students often confuse "annual gain percent" with Simple Interest. If calculated as Simple Interest, the total gain over 3 years would be $75\%$, making the final amount $175\%$ of the principal (which leads to the incorrect option d). However, in business reinvestment (capital growth), gains compound annually unless explicitly stated otherwise. ### Final Answer **Therefore, the correct answer is ₹ 5120.**