What is the amount for Rs 10000 by compound interest at 8 rate for 2 years answer

Answer

Verified

Hint: First we recall the definition and formula of compound interest and then calculate the compound interest. The formula used to calculate the compound interest is
Compound interest = Amount – Principal
And \[\text{Amount =}P{{\left( 1+\dfrac{R}{100} \right)}^{T}}\]
Where, $P=$ Principal
\[R=\] Rate of interest
$T=$Time period

Complete step by step answer:
Now, we have given that Principal sum $=10,000$
Rate of interest $=4%$ per annum
Time period \[=2\text{ years}\]
We have given that the compound interest being compounded half yearly, so the time period will be $4\text{ years}$and the rate of interest will be half i.e. $2%$ because when interest is compounded half yearly the rate of interest will be $\dfrac{R}{2}$.
Now, we have to calculate the Amount, so we put all values in the formula
\[\text{Amount =}P{{\left( 1+\dfrac{R}{100} \right)}^{T}}\]
$\Rightarrow 10000{{\left( 1+\dfrac{2}{100} \right)}^{4}}$
$\begin{align}
  & \Rightarrow 10000{{\left( 1+\dfrac{1}{50} \right)}^{4}} \\
 & \Rightarrow 10000\times \left( \dfrac{51}{50} \right)\times \left( \dfrac{51}{50} \right)\times \left( \dfrac{51}{50} \right)\times \left( \dfrac{51}{50} \right) \\
 & \Rightarrow 10.2\times 10.2\times 10.2\times 10.2 \\
 & \Rightarrow 10824.32 \\
\end{align}$
The Amount will be Rs. $10824.32$
Now we have to calculate compound interest.
We know that Compound interest = Amount – Principal
Putting the values, Compound interest will be
 $\begin{align}
  & =10824.32-10000 \\
 & =824.32 \\
\end{align}$
So, the compound interest on Rs $10000$ in $2$ years at $4%$ per annum being compounded half yearly is $Rs.824.32$.

So, the correct answer is “Option C”.

Note: Compound interest is interest on interest; it means compound interest is additional amount of interest to the principal sum. Before calculating compound interest students have to calculate the amount by using the formula and then subtract principal from amount. Students must read questions carefully about the compounding frequency i.e. interest compounded yearly, half-yearly, quarterly, monthly or weekly. The time period will be changed accordingly.

Answer

Verified

Hint:
 In this question, we are given the principal amount, rate of interest compounded half-yearly, and time period. We will first change the rate of interest per annum and then change the time period as per the number of times the amount increases. After that, we will use the formula of the compound amount and compound interest to find out the required answer. Formula for compound amount is given by $ A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}} $ where P is the principal amount, r is the numerical value of rate of interest and n is the time period.
The formula for compound interest is given by,
Compound interest = Compounds amount - principal amount.

Complete step by step answer:
Here we are given the principal amount as Rs.10000. Therefore, P = 10000.
Now we are given the rate of interest as 8% compounded half yearly but rate of interest is in per annum. So we will change it according to the half year that is interest becomes half so that it can be compounded after every half year. Hence, the rate of interest becomes $ \dfrac{8}{2}=4\% $ . Therefore, r = 4%.
Interest is compounded half-yearly so time period should also be half years. As the number of months are 12 so it means we have 2 half years. Therefore, n = 12.
Now we know compound amount is given by $ A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}} $ where A is compound amount, P is the principal amount, r is rate of interest and n is time period. Putting in all the values we get,
\[A=10000{{\left( 1+\dfrac{4}{100} \right)}^{2}}\]
Taking LCM as 100 we get,
\[\begin{align}
  & A=10000{{\left( \dfrac{100+4}{100} \right)}^{2}} \\
 & \Rightarrow A=10000{{\left( \dfrac{104}{100} \right)}^{2}} \\
 & \Rightarrow A=10000\times \dfrac{104}{100}\times \dfrac{104}{100} \\
\end{align}\]
Cancelling $ 100\times 100 $ with 10000 we get,
\[A=104\times 104=10816\]
Hence the amount after 12 months becomes Rs.10816. Now we know that compound interest can be calculated using the formula,
Compound interest = compound amount - principal amount.
Putting in the values we get,
Compound interest = Rs.10816 - Rs.10000 = Rs.816.
Hence the required compound interest is Rs.816

Note:
 Students should not forget to convert the rate of interest and time as for half-yearly. Students should note that in the formula for calculating the compound amount, we fill the numerical value of r only. For percentage, 100 is already divided.

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