An entrepreneur deposited her startup capital into a financial scheme offering 6% p.a. simple interest for a period of two years. If she had instead chosen a plan yielding 10% p.a. compound interest compounded annually for the same two years, the interest earned would have been Rs. 180 higher. Find the startup capital of the entrepreneur.

The correct answer is Rs. 2000.

Step-by-step Explanation:

Let the startup capital deposited by the entrepreneur be $P$ rupees.
The time period for both financial schemes is $t=2$ years.

Step 1: Calculate the interest earned under Simple Interest (SI)

For the first scheme:
Rate of simple interest, $R_1=6\%$ per annum.

The formula for Simple Interest is:

$\mathrm{SI}=\frac{P\times R_1\times t}{100}$

Substituting the given values into the formula:

$\mathrm{SI}=\frac{P\times 6\times 2}{100}=\frac{12P}{100}=0.12P$

Step 2: Calculate the interest earned under Compound Interest (CI)

For the second scheme:
Rate of compound interest, $R_2=10\%$ per annum compounded annually.

The formula for the total amount ($A$) under compound interest is:

$A=P{\left(1+\frac{R_2}{100}\right)}^t$

Substituting the given values:

$A=P{\left(1+\frac{10}{100}\right)}^2=P{\left(1.1\right)}^2=1.21P$

The Compound Interest ($\mathrm{CI}$) is the total amount minus the principal:

$\mathrm{CI}=A-P=1.21P-P=0.21P$

Step 3: Set up the equation using the given difference in interest

We are given that the compound interest earned is Rs. 180 higher than the simple interest earned:

$\mathrm{CI}-\mathrm{SI}=180$

Substitute the values of $\mathrm{CI}$ and $\mathrm{SI}$ obtained above:

$0.21P-0.12P=180$

$0.09P=180$

Solving for $P$:

$P=\frac{180}{0.09}=2000$

Therefore, the startup capital of the entrepreneur is Rs. 2000.