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Neeraj lent Rs.65536 for 2 years at \[12\dfrac{1}{2}\% \] per annum, compounded annually. How much interest more could he earn if the interest were compounded half-yearly?

Answer Verified Verified
Hint: Here we have to find the final amount using the given values, the value we have found is taken annually with the help of it we will find the required interest.

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

Complete step-by-step answer:
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.

Note: Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously accumulated interest.
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