Question

Calculate the compound interest of \[$2000\] for a period of 1 year, compounded semi-annually at the rate 0.2 percent per six months.

Answer 1

Hint: Now we are given with the principal amount which is \[$ 2000\]. Now rate of interest is 0.2 percent per six months. And the time duration is 1 year. Hence we can calculate the final amount by formula $P{{\left( 1+\dfrac{r}{n\times 100} \right)}^{nt}}$ where n is the number of time Interest is compounded in a year, r is rate of interest per annum. P is principal amount and t is time in years. Now total interest if Final amount – principal amount.

Complete step by step answer:
Now we are given that \[$2000\] is compounded semi-annually for a period of 1 year at the rate 0.2 percent per six months.
Now we know that principal amount is \[$ 2000.\]
Now rate of interest is 0.2 percent per six months.
Hence the rate of interest is 0.2 × 2 = 0.4 percent per annum.
Now time is given as 1 year.
And since the interest is compounded semi-annually we have n = 2.
Now we will use a formula of compound interest which is $P{{\left( 1+\dfrac{r}{n\times 100} \right)}^{nt}}$ to find the final amount.
Hence we have $P{{\left( 1+\dfrac{r}{n\times 100} \right)}^{nt}}$
$\begin{align}
  & A=2000{{\left( 1+\dfrac{0.4}{2\times 100} \right)}^{2\times 1}} \\
 & \Rightarrow A=2000{{\left( 1+0.002 \right)}^{2}} \\
 & \Rightarrow A=2000{{\left( 1.002 \right)}^{2}} \\
 & \Rightarrow A=2000\left( 1.004 \right) \\
 & \therefore A=2008 \\
\end{align}$
Now we have Interest is given by the formula Total amount – principal amount.
Hence we get Interest = 2008 – 2000 = \[$ 8.\]

Hence the interest is \[$ 8.\]

Note: Now note that while substituting the rate in the formula it must be per annum and the time should be in years. If not we convert them accordingly. Also n in the formula is the number of times interest is compounded in a year. Hence we have if the interest is compounded semi-annually that is every six months in a year it will be compounded twice.