Question

Find the compound interest on Rs. $32000$ at $20\% $ per annum for $1$ year, compounded half yearly.
(a) Rs. $6320$ (b) Rs. $6720$
(c) Rs. $6400$ (d) Rs. $6500$

Answer 1

Hint:In this example, given that the rate of interest is annual and the interest is compounded half yearly (that is, six months). That means the interest paid at the end of every six months is one-half of the rate of interest per annum. So, the rate of annual interest is $\dfrac{R}{2}\% $ and the number of years is doubled (that is, $2T$). First we will find the amount $A$ for $1$ year by using the formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$. After that we will find compound interest by subtracting the principal amount $P$ from the amount $A$.

Complete step-by-step solution:
Here given that the principal amount $P = $ Rs. $32,000$, rate of interest $R = 20\% $ per annum and time $T = 1$ year. Also given that the interest is compounded half yearly (that is, six months). So, the interest paid at the end of every six months is one-half of the rate of interest per annum. So, the rate of annual interest is $R = \dfrac{{20}}{2}\% = 10\% $ and the number of years is doubled. That is, $T = 2$ half years.
Now we are going to find the amount for $1$ year by using the formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ where $P$ is principal amount, $R$ is rate of annual interest and $T$ is time in half years.
Now we are going to substitute the values of $P$, $R$ and $T$ in the formula of amount for $1$ year.
Therefore, $A = 32000{\left( {1 + \dfrac{{10}}{{100}}} \right)^2}$
$ \Rightarrow $ $A = 32000{\left( {\dfrac{{100 + 10}}{{100}}} \right)^2}$
$ \Rightarrow $$A = 32000{\left( {\dfrac{{110}}{{100}}} \right)^2} = 32000{\left( {\dfrac{{11}}{{10}}} \right)^2} = 32000\left( {\dfrac{{121}}{{100}}} \right)$
$ \Rightarrow $ $A = 320 \times 121$ $ = $ Rs. $38720$
Now we will find compound interest by subtracting the principal amount $P$ from the amount $A$.
Therefore, compound interest (CI) $ = $ $A - P$
$ \Rightarrow $ Compound interest (CI) $ = $ $38720 - 32000$ $ = $ Rs. $6720$.

Therefore, the compound interest is Rs. $6720$ for $T = 1$ year.

Note:Simple interest is calculated only on the principal amount but compound interest is calculated on principal amount as well as previous year’s interest. If interest is paid only for $T = 1$ year then there is no distinction between simple interest and compound interest.