A start-up company raised Rs.8000 from an angel investor at an annual rate of 20% compound interest, compounded annually for two years. The startup then lent this entire sum to a consulting firm at a certain rate of annual simple interest for two years. If the difference between the compound interest payable and the simple interest receivable is Rs.2040, calculate the annual rate of interest at which the startup lent the money to the consulting firm.
(Note: The compound interest incurred by the start-up is greater than the simple interest earned by it)

The correct option is 9.25%.

Step 1: Calculate the Compound Interest (CI) payable by the start-up
The start-up borrows a principal amount ($P$) of Rs. 8000 at an annual compound interest rate ($r_1$) of 20% for a period ($t$) of 2 years.
The formula to calculate the Compound Interest is:

$\mathrm{CI}=P{(1+\frac{r_1}{100})}^t-P$

Substitute the given values into the formula:

$\mathrm{CI}=8000\times {(1+\frac{20}{100})}^2-8000$

$\mathrm{CI}=8000\times {\left(1.2\right)}^2-8000$

$\mathrm{CI}=8000\times 1.44-8000$

$\mathrm{CI}=11520-8000=\text{Rs. 3520}$

Step 2: Express the Simple Interest (SI) receivable from the consulting firm
The start-up lends the entire sum ($P=\text{Rs. 8000}$) for $t=2$ years at an unknown annual simple interest rate ($r_2$).
The formula for Simple Interest is:

$\mathrm{SI}=\frac{P\times r_2\times t}{100}$

Substitute the known values:

$\mathrm{SI}=\frac{8000\times r_2\times 2}{100}$

$\mathrm{SI}=160\times r_2$

Step 3: Calculate the rate of simple interest ($r_2$)
We are given that the difference between the compound interest payable and simple interest receivable is Rs. 2040:

$\mathrm{CI}-\mathrm{SI}=2040$

Substitute the expressions for $\mathrm{CI}$ and $\mathrm{SI}$:

$3520-160\times r_2=2040$

$160\times r_2=3520-2040$

$160\times r_2=1480$

$r_2=\frac{1480}{160}=9.25\%$

Thus, the startup lent the money to the consulting firm at an annual simple interest rate of 9.25%.