A sum of money invested at 5% per annum compound interest for 2 years and 146 days amounts to Rs. 9964 more than the simple interest on Rs. 80000 at 10% per annum for 2 years. Find the sum of money.

Here's a short analysis of the solution:

• Convert time for Compound Interest (CI): The time period is $2 \text{ years and } 146 \text{ days}$.

  $$146 \text{ days} = \frac{146}{365} \text{ years} = \frac{2}{5} \text{ years} = 0.4 \text{ years}$$

  Total time for CI is $2.4 \text{ years}$.

• Calculate Principal ($x$) from CI: The CI formula for a fractional period is $A = P \left(1 + \frac{R}{100}\right)^{\text{whole years}} \left(1 + \frac{R \times \text{fractional part}}{100}\right)$. Given $P=x$, $R=5\%$, $CI = \text{Rs. } 9964$.

  $$A = x \left(1 + \frac{5}{100}\right)^2 \left(1 + \frac{5}{100} \times \frac{2}{5}\right)$$ $$A = x \left(\frac{21}{20}\right)^2 \left(1 + \frac{2}{100}\right) = x \left(\frac{441}{400}\right) \left(\frac{51}{50}\right) = x \frac{22491}{20000}$$

  The interest is $CI = A - x = x \left(\frac{22491}{20000} - 1\right) = x \left(\frac{2491}{20000}\right)$. Setting $CI = 9964$:

  $$x \left(\frac{2491}{20000}\right) = 9964 \implies x = \frac{9964 \times 20000}{2491} = 4 \times 20000 = \text{Rs. } 80000$$

• Calculate Simple Interest (SI): Now, calculate SI for $P = \text{Rs. } 80000$, $R = 10\%$, $T = 2 \text{ years}$.

  $$SI = \frac{P \times R \times T}{100}$$ $$SI = \frac{80000 \times 10 \times 2}{100} = 800 \times 20 = \text{Rs. } 16000$$

The final answer is $\text{16,000}$.