Question

Find the amount and compound interest on Rs. 10000 at 8% per annum and in 1 year; interest being compounded half-yearly.
(a) Rs. 816
(b) Rs. 214
(c) Rs. 719
(d) None of these.

Answer 1

Hint: Use compound interest formula for the calculation of amount $A$, given by:

\[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]. From this, calculate amount $A$ after $t=1$ year.

Then subtract the principal amount $P$ from the amount $A$ to get the interest.

Complete step-by-step answer:
Compound interest is the addition of interest to the principal sum of a loan or deposit. It is the result of reinvesting interest, rather than paying it out, so the interest in the next period is then earned on the principal sum plus previously accumulated interest.

The total accumulated amount $A$ , on the principal sum \[P\] plus compound interest $I$ is given by the formula \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\].

 Here, \[A\] is the amount obtained, $t$ is the number of years, \[r\] is the rate, $P$ is the principal and \[n\] is the number of times the interest is given in a year.

The total compound interest generated is given by: $I=A-P$.

Now, we have been given that:

$P=\text{Rs}\text{. }10000\text{ , }r=8%=\dfrac{8}{100}\text{ , }t=1\text{ year and }n=2$

Therefore, amount credited in 1 year,

$\begin{align}
  & A=10000\times {{\left( 1+\dfrac{8}{2\times 100} \right)}^{2\times 1}} \\
 & =10000\times {{\left( 1+\dfrac{4}{100} \right)}^{2}} \\
 & =10000\times {{\left( \dfrac{104}{100} \right)}^{2}} \\
 & ={{\left( 104 \right)}^{2}} \\
 & =10816 \\
\end{align}$

 Also, compound interest will be
$\begin{align}
  & I=A-P \\
 & =10816-10000 \\
 & =816 \\
\end{align}$

Hence, the amount and interest in 1 year will be Rs. 10816 and Rs. 816 respectively.

Therefore, option (a) is the correct answer.

Note: We have used the value of $n$ equal to 2 because $n$ represents the number of times the interest is compounded in a year. In the question it was given that interest is compounded half-yearly, that means one time in half a year. Therefore, in one year it will be two times. Hence, $n=2$ is used.