# Calculate compound interest for Rs 10,000 for 1 year at 8% compounded semi – annually.

(a) 817

(b) 816

(c) 815

(d) None of these

Answer

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Hint: The amount for a principal amount of Rs P for n years and at r% compound interest is given by the formula A = $P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$. The compound interest can be found by subtracting the principal amount from the total amount. If it is required to find the amount for a principal amount of Rs P for n years and at r% per annum compounded semi – annually, then we have to convert the rate r and t in terms of half years and then only we can substitute them in the above formula.

“Complete step-by-step answer:”

Before proceeding with the question, we must know the formula that will be required to solve this question.

In compound interest, the amount for a principal amount of Rs P for n years and at r% compound interest is given by the formula A = $P{{\left( 1+\dfrac{r}{100} \right)}^{n}}.................\left( 1 \right)$.

To find the compound interest, we will subtract the principal amount from this amount. $.................\left( 2 \right)$

In this question, we are required to calculate the compound interest for Rs 10,000 for 1 year at 8% compounded semi - annually. Since the rate is given 8% per annum, we have to divide it by 2 to convert it for half a year. So, the rate will be 4% per half year.

Also, the time is given as 1 year which can be written as 2 half years.

Substituting P = 10000, r = 4, n = 2 in the formula $\left( 1 \right)$, we get,

$\begin{align}

& A=10000{{\left( 1+\dfrac{4}{100} \right)}^{2}} \\

& \Rightarrow A=10000{{\left( \dfrac{100+4}{100} \right)}^{2}} \\

& \Rightarrow A=10000{{\left( \dfrac{104}{100} \right)}^{2}} \\

& \Rightarrow A=10000\left( \dfrac{10816}{10000} \right) \\

& \Rightarrow A=10816 \\

\end{align}$

From $\left( 2 \right)$, the compound interest = Amount – Principal amount

$\Rightarrow $ compound interest = 10816 – 10000

$\Rightarrow $ compound interest = 816

Hence, the answer is option (b).

Note: There is a possibility that one may commit a mistake while finding the compound interest. It is a very common mistake that one does not subtract the principal amount from the amount to find the compound interest. So, instead of compound interest, he/she will give the amount as his/her answer instead of the compound interest.

“Complete step-by-step answer:”

Before proceeding with the question, we must know the formula that will be required to solve this question.

In compound interest, the amount for a principal amount of Rs P for n years and at r% compound interest is given by the formula A = $P{{\left( 1+\dfrac{r}{100} \right)}^{n}}.................\left( 1 \right)$.

To find the compound interest, we will subtract the principal amount from this amount. $.................\left( 2 \right)$

In this question, we are required to calculate the compound interest for Rs 10,000 for 1 year at 8% compounded semi - annually. Since the rate is given 8% per annum, we have to divide it by 2 to convert it for half a year. So, the rate will be 4% per half year.

Also, the time is given as 1 year which can be written as 2 half years.

Substituting P = 10000, r = 4, n = 2 in the formula $\left( 1 \right)$, we get,

$\begin{align}

& A=10000{{\left( 1+\dfrac{4}{100} \right)}^{2}} \\

& \Rightarrow A=10000{{\left( \dfrac{100+4}{100} \right)}^{2}} \\

& \Rightarrow A=10000{{\left( \dfrac{104}{100} \right)}^{2}} \\

& \Rightarrow A=10000\left( \dfrac{10816}{10000} \right) \\

& \Rightarrow A=10816 \\

\end{align}$

From $\left( 2 \right)$, the compound interest = Amount – Principal amount

$\Rightarrow $ compound interest = 10816 – 10000

$\Rightarrow $ compound interest = 816

Hence, the answer is option (b).

Note: There is a possibility that one may commit a mistake while finding the compound interest. It is a very common mistake that one does not subtract the principal amount from the amount to find the compound interest. So, instead of compound interest, he/she will give the amount as his/her answer instead of the compound interest.

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