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 Principal and Given Data

Principal amount raised by the startup, P = Rs. 8000
Rate of compound interest payable by the startup, r₁ = 20% per annum
Time period, t = 2 years
Compound Interest is compounded annually.

Step 2: Calculate the Compound Interest (CI) payable by the startup

The total amount A after 2 years under compound interest is given by the formula:

$A=P{(1+\frac{r_1}{100})}^t$

Substituting the values:

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

$A=8000\times {\left(1.2\right)}^2=8000\times 1.44=\text{Rs. 11520}$

Now, calculate the Compound Interest (CI):

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

Step 3: Set up the equation using Simple Interest (SI)

Let the annual rate of simple interest at which the startup lent the sum be R% per annum.
The Simple Interest receivable by the startup in 2 years is:

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

$\mathrm{SI}=\frac{8000\times R\times 2}{100}=160R$

Step 4: Solve for the rate of simple interest (R)

We are given that the difference between the compound interest payable and the simple interest receivable is Rs. 2040 (CI > SI):

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

Substitute the values of CI and SI into the equation:

$3520-160R=2040$

$160R=3520-2040$

$160R=1480$

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

Thus, the annual rate of simple interest at which the startup lent the money is 9.25%.