Question

Find the compound interest, if $\$4000$ was invested for 1 year at 10% per annum
compounded half-yearly?

• A. $\$400$
• B. $\$820$
• C. $\$500$
• D. $\$410$

Answer 1

The correct option is D $\$410$

$\text{Amount}=P{(1+\frac{r}{100\times n})}^{n\times t}$

$\text{Where,}P\text{= Principal amount}$
$r=\text{rate of interest (\%)}$
$t=\text{time period (years)}$
$n=\text{number of times interest is compounded per year}$

$\text{Given,}P\text{= 4000}$
$r=10$
$t=1$
$n=2,\text{as interest is compounded 2 times per year}$

$\therefore \text{Amount}=4000{(1+\frac{10}{100\times 2})}^{2\times 1}$
$=4000{(1+\frac{10}{10\times 10\times 2})}^2$
$=4000{(1+\frac{1}{10\times 2})}^2$
$=4000{(1+\frac{1}{20})}^2$
$=4000{(\frac{20}{20}+\frac{1}{20})}^2$
$=4000{\left(\frac{20+1}{20}\right)}^2$
$=4000{\left(\frac{21}{20}\right)}^2$
$=4000\times \frac{(21{)}^2}{(20{)}^2}$
$=4000\times \frac{21\times 21}{20\times 20}$
$=\frac{400\times 10\times 21\times 21}{400}$
$=10\times 21\times 21$
$=441\times 10=4410$

$\because \text{Amount = Principal + Compound Interest}$
$\Rightarrow \text{Compound Interest = Amount - Principal}$
$\Rightarrow \text{Compound Interest = 4410 - 4000}$
$\Rightarrow \text{Compound Interest = 4410 - 4000}$
$\Rightarrow \text{Compound Interest = 410}$

Hence, compound interest is $\$410.$