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Find the compound interest on Rs. $6000$ at $10%$ per annum for one year, compounded half-yearly?
  & A.\text{ Rs}\text{. 1260} \\
 & \text{B}\text{. Rs}\text{. 630} \\
 & C.\ \text{Rs}\text{. 615} \\
 & D.\ \text{Rs}\text{. 600} \\

Last updated date: 20th Jun 2024
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Hint: The compound interest is the interest which is calculated by multiplying the initial amount with one plus the annual interest rate per year, power to the number of years or the term.
Use formula $A=P{{(1+\dfrac{i}{n})}^{n}}$ and compound interest is equal to the compounded amount minus the principal amount. Where P is the principal amount, i is the rate of interest and n is the number of times the interest is compounded per year.

Complete step-by-step answer:
Given that:
Principal, $P=6000\text{ Rs}\text{.}$
Interest rate, $i=10%$
Number of times interest compounded per year, $n=2$
Now, use the formula
$\Rightarrow A=P{{(1+\dfrac{i}{n})}^{n}}$
Place the known values
$\Rightarrow A=6000{{[1+\dfrac{10}{100\times 2}]}^{2}}$
Simplify the above equation –
\Rightarrow & A=6000{{[1+\dfrac{1}{10\times 2}]}^{2}} \\
\Rightarrow & A=6000{{[1+\dfrac{1}{20}]}^{2}} \\
Take LCM on the right hand side of the equation –
$\Rightarrow A=6000{{\left( \dfrac{21}{20} \right)}^{2}}$
$\Rightarrow A=6615\ \text{Rs}\text{.}$ ………………..(a)
Now, the compound interest is the amount left after subtracting the principal amount from the compounded amount
Therefore, using the value of equation (a)
$\Rightarrow$ Compounded Interest $=A-P$
$\Rightarrow$ Compounded Interest $=6615-6000$
$\Rightarrow$ Compounded Interest \[=615\ \text{Rs}\text{.}\]
Therefore, the required answer - the compound interest on Rs. $6000$ at $10%$ per annum for one year, compounded half-yearly is $615\text{ Rs}\text{.}$

So, the correct answer is “Option C”.

Note: Compound Interest is the process of compounding and it is also referred to as the term “interest on interest”. Always check the frequency of the compounding which perhaps may be yearly, half-yearly, quarterly, monthly and daily. Use standard formula, $A=P{{(1+\dfrac{R}{100})}^{n}}$ for the interest compounded yearly, and \[A=P{{(1+\dfrac{R/2}{100})}^{2n}}\]where the interest compounded is half-yearly. Where P is Principal, R is rate and time is n years.