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Where A= Final amount

P = Initial principal balance

r = Rate of interest

n= number of times interest applied per time period

t = number of time periods elapsed

It is given that Neeraj lent Rs.65536 for 2 years at \[12\dfrac{1}{2}\% = \dfrac{{25}}{2}\% \] per annum, compounded annually.

So the total amount Neeraj get after 2 years compounded annually is given by the formula given in the hint

Here \[\;P = 65536,{\text{ }}n = 1,{\text{ }}r = \dfrac{{25}}{2}\% ,t = 2\]

By substituting the values we get, \[A = 65536 \times {\left( {1 + \dfrac{{\dfrac{{25}}{2}}}{{\dfrac{{100}}{1}}}} \right)^{1 \times 2}}\]

On further solving we get,

\[A = 65536 \times {\left( {1 + \dfrac{{25}}{{200}}} \right)^2}\]

Which in turn imply that,

\[A = 65536 \times {\left( {1 + \dfrac{1}{8}} \right)^2}\]

That is

\[A = 65536 \times \dfrac{9}{8} \times \dfrac{9}{8}\]

\[A = 82944\]

The interest amount is found by subtracting the principal amount from the total amount.

Thus the interest amount is Rs. (82944 – 65536) = Rs.17408.

Now the total amount Neeraj get after 2 years compounded half-yearly is found,

Here\[\;P = 65536,{\text{ }}n = 2,{\text{ }}r = \dfrac{{25}}{2}\% ,t = 2\]

Since it is compounded half – yearly here we take \[n = 2\]

\[A = 65536 \times {\left( {1 + \dfrac{{\dfrac{{25}}{2}}}{{\dfrac{{100}}{1}}}} \right)^{2 \times 2}}\]

On solving the above equation,

\[A = 65536 \times {(1 + \dfrac{{25}}{{400}})^4}\]

\[A = 65536 \times {(1 + \dfrac{1}{{16}})^4}\]

Let us solve further we get,

\[A = 65536 \times \dfrac{{17}}{{16}} \times \dfrac{{17}}{{16}} \times \dfrac{{17}}{{16}} \times \dfrac{{17}}{{16}}\]

\[A = 83521\]

The interest amount is found by subtracting the principal amount from the total amount.

Thus the interest amount is Rs. (83521 – 65536) = Rs.17985.

Hence Neeraj could earn Rs. (17985 – 17408)=Rs. 577 if the interest were compounded half-yearly instead of annually.

The compound interest amount including the Principal sum P and the interest I is given by the formula,

\[A = P{(1 + \dfrac{r}{n})^{nt}}\]

Where A= Final amount

P = Initial principal balance

r = Rate of interest

n= number of times interest applied per time period

t = number of time periods elapsed