Question

Amrutha deposited rupees $20000$ in a bank which compounds interest half yearly. The annual rate of interest is $8\%$. How much would she get back after one year?

Answer 1

The problem can be solved easily with the concept of compound interest. Compound interest is the interest calculated on the principal and the interest of the previous period. The amount in compound interest to be cumulated depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .

Complete answer:
In the given problem,
Principal $= P = Rs\,20,000$
Rate of interest $= 8\%$
Time Duration $= 1\,year$
In the question, the period after which the compound interest is compounded or evaluated is given as half a year. Since the total time duration is one year.
So, Number of time periods$= n = 2$
Now, The amount A to be paid after a certain number of time periods n on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .
Hence, Amount $= A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Now, substituting all the values that we have with us in the formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$, we get,
$\Rightarrow A = Rs\,20,000{\left( {1 + \dfrac{{\left( {\dfrac{8}{2}} \right)}}{{100}}} \right)^2}$
Simplifying the calculations,
$\Rightarrow A = Rs\,20,000{\left( {1 + \dfrac{4}{{100}}} \right)^2}$
Taking LCM within the bracket, we get,
$\Rightarrow A = Rs\,20,000{\left( {\dfrac{{100 + 4}}{{100}}} \right)^2}$
Opening the bracket,
$\Rightarrow A = Rs\,20,000 \times \dfrac{{104}}{{100}} \times \dfrac{{104}}{{100}}$
Cancelling common factors in numerator and denominator, we get,
$\Rightarrow A = Rs\,2 \times \dfrac{{104}}{1} \times \dfrac{{104}}{1}$
$\Rightarrow A = Rs\,2 \times {104^2}$
Now, simplifying the calculations, we get,
$\Rightarrow A = Rs\,21632$
Therefore, the sum of $20000$ rupees will amount to $Rs\,21632$ in one year at $8\%$ per annum compound interest compounded half yearly.

Note:
Time duration is not always equal to the number of time periods. The equality holds only when the compound interest is compounded annually. If the compound interest is compounded half yearly, then the number of time periods doubles in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest.