Ranbir borrows Rs.20,000 at 12% per annum compound interest. If he repays Rs.8400 at the end of the first year and Rs.9680 at the end of the second year, find the amount of loan outstanding at the beginning of the third year.

Hint: To calculate annual compound interest, multiply the original amount of your investment or loan, or principal, by the annual interest rate. Add the amount to the principle, then multiply by interest rate again to get second year’s compounding interest. So, use this concept to reach the solution of the given problem.

Complete Step-by-Step solution:
Given that Ranbir borrows Rs.20,000 at 12% per annum compound interest.
We know that the compound interest is given by the formula \[A = P{\left( {1 + \dfrac{r}{{n \times 100}}} \right)^{nt}}\] where \[A\] is the final amount, \[P\] is the principle amount, \[t\] is the number of times interest applied per time period, \[n\] is the number of times interest applied per time period and \[R\] is the rate on interest per annum.
For the first year the given principal amount is Rs.20,000, time period is one year, number of times interest applied per time period is 1 and rate of interest is 12%.
So, final amount for first year is given by
\[
  A = 20000{\left( {1 + \dfrac{{12}}{{1 \times 100}}} \right)^{1 \times 1}} \\
  A = 20000\left( {1 + \dfrac{{12}}{{100}}} \right) \\
  A = 20000\left( {\dfrac{{112}}{{100}}} \right) \\
  \therefore A = Rs.22,440 \\
\]
Given that Ranbir repays Rs.8,400 at the end of first year.
So, principle amount for the second year \[ = Rs.22,400 - Rs.8,400 = Rs.14,000\]
And the time period is one year, the number of times interest applied per time period is 1 and rate of interest is 12%.
So, final amount for the second year is given by
\[
  A = 14000{\left( {1 + \dfrac{{12}}{{1 \times 100}}} \right)^{1 \times 1}} \\
  A = 14000\left( {1 + \dfrac{{12}}{{100}}} \right) \\
  A = 14000\left( {\dfrac{{112}}{{100}}} \right) \\
  \therefore A = Rs.15,680 \\
\]
Also, Ranbir repays Rs.9,680 at the end of second year.
Thus, the outstanding loan amount at the beginning of third year \[ = Rs.15,680 - Rs.9,680 = Rs.6,000\]
Therefore, the amount of loan outstanding at the beginning of the third year is Rs.6000.

Note: The compound interest is given by the formula \[A = P{\left( {1 + \dfrac{r}{{n \times 100}}} \right)^{nt}}\] where \[A\] is the final amount, \[P\] is the principle amount, \[t\] is the number of times interest applied per time period, \[n\] is the number of times interest applied per time period and \[R\] is the rate on interest per annum.