What will be the compound interest on a sum of rupees 12000 for 2 years at the rate of 20% per annum when the interest is compounded yearly?

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Find Compound interest on Rs. 12000 at 5 % p.a. for 3 years compounded annually.(a) Rs.1891.50(b) Rs2891.50(c) Rs.3891.50(d) Rs.4891.50

Answer

Verified

Hint: We will use the formula to find compound interest here. The formula is given by $\text{C}\text{.I}\text{.=Amount - Principal}$ where amount can be found out by the formula $A=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}$. Here P is called the principal, R is the rate, and T is called the time for the compound interest. With these formulas, we will solve the question directly.Complete step-by-step solution -
Now, according to the question we are informed that the compound interest is on Rs. 12000. So, it will become our principal. That is, P = 12000. Now we are given the interest at 5 % p.a. this means that this is the rate at which the interest is going to be counted on. Thus, we have R = 5 %. Clearly the time given to us is 3 years. So, this will be regarded as T = 3. Now we will use the formula so that we can find the amount. For this we will apply the formula given by $A=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}$. Now we will substitute all the values into the formulas. Therefore, we get
$\begin{align}
  & A=P{{\left( 1+\dfrac{R}{100} \right)}^{T}} \\
 & \Rightarrow A=12000{{\left( 1+\dfrac{5}{100} \right)}^{3}} \\
\end{align}$
After solving it further we will get,
$\begin{align}
  & A=12000{{\left( 1+\dfrac{5}{100} \right)}^{3}} \\
 & \Rightarrow A=12000{{\left( 1+\dfrac{1}{20} \right)}^{3}} \\
 & \Rightarrow A=12000{{\left( \dfrac{20+1}{20} \right)}^{3}} \\
 & \Rightarrow A=12000{{\left( \dfrac{21}{20} \right)}^{3}} \\
 & \Rightarrow A=12000\times \left( \dfrac{21}{20} \right)\times \left( \dfrac{21}{20} \right)\times \left( \dfrac{21}{20} \right) \\
\end{align}$
After simplifying it into its simplest form we will get
$\begin{align}
  & A=3\times \left( \dfrac{21}{1} \right)\times \left( \dfrac{21}{1} \right)\times \left( \dfrac{21}{2} \right) \\
 & \Rightarrow A=3\times 21\times 21\times \left( \dfrac{21}{2} \right) \\
 & \Rightarrow A=1323\times \left( \dfrac{21}{2} \right) \\
\end{align}$
Now, we will use BODMASS rule here in which we will divide first and then multiply the numbers. So, we get
$\begin{align}
  & \Rightarrow A=1323\times \left( \dfrac{21}{2} \right) \\
 & \Rightarrow A=1323\times 10.5 \\
 & \Rightarrow A=13891.5 \\
\end{align}$
Therefore, the amount after 3 years will become Rs. 13891.5. Now we will find the compound interest on it. This can be done by the formula given by $\text{C}\text{.I}\text{.=Amount - Principal}$. After substituting the amount of Rs. 19891.5 and principal as Rs. 12000 into the formula we get,
$\begin{align}
  & \text{C}\text{.I}\text{.=Amount - Principal} \\
 & \Rightarrow \text{C}\text{.I}\text{.=13891}\text{.5 - 12000} \\
 & \Rightarrow \text{C}\text{.I}\text{.=1891}\text{.5} \\
\end{align}$
Thus, the compound interest on Rs. 12000 at 5 % p.a. for 3 years will be Rs.1891.5.

Note: Alternatively we can solve the question by using the simple interest method. But we will not get the same answer always instead we will get an approximate result. The method is done below.
For this we will consider the principal as Rs. 12000. Now, we will find simple interest on it by using the formula given by $\text{S}\text{.I}\text{.=}\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ T}}{\text{100}}$ where P is the principal, R is the rate and T is the time. In this case the time is taken individually. That is the 3 years will be split into 3 cases in which we will have time as 1 year each. Therefore we have
$\begin{align}
  & \text{S}\text{.I}{{\text{.}}_{1}}\text{=}\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ T}}{\text{100}} \\
 & \Rightarrow \text{S}\text{.I}{{\text{.}}_{1}}\text{=}\dfrac{\text{12000 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 1}}{\text{100}} \\
 & \Rightarrow \text{S}\text{.I}{{\text{.}}_{1}}\text{=}\dfrac{\text{120 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 1}}{\text{1}} \\
 & \Rightarrow \text{S}\text{.I}{{\text{.}}_{1}}\text{=600} \\
\end{align}$
So, the S.I. at 5 % for first year is Rs. 600. Now we will find the amount here. For this we will use the formula given by $\text{Amount at the end of 1 year = Principal+S}\text{.I}{{\text{.}}_{1}}$. Therefore we get
 $\begin{align}
  & \text{Amount at the end of 1 year = 12000+600} \\
 & \Rightarrow \text{Amount at the end of 1 year = 12600} \\
\end{align}$
Now we will follow the same steps for the next two years. That is for the second year we have principal as Rs 12600 now. Therefore, we will actually have
$\begin{align}
  & \text{S}\text{.I}{{\text{.}}_{2}}=\dfrac{12600\times 5\times 1}{100} \\
 & \Rightarrow \text{S}\text{.I}{{\text{.}}_{2}}=\dfrac{126\times 5\times 1}{1} \\
 & \Rightarrow \text{S}\text{.I}{{\text{.}}_{2}}=630 \\
\end{align}$
 Therefore, the $\text{Amount at the end of 2 year = Principal+S}\text{.I}{{\text{.}}_{2}}$. Thus, we get
$\begin{align}
  & \text{Amount at the end of 2 year = 12600+630} \\
 & \Rightarrow \text{Amount at the end of 2 year = 13230} \\
\end{align}$
Similarly for third year we get principal as Rs. 13230 and the $\text{Amount at the end of 3 year = Principal+S}\text{.I}{{\text{.}}_{3}}$ where the simple interest is carried out as
$\begin{align}
  & \text{S}\text{.I}{{\text{.}}_{3}}=\dfrac{13230\times 5\times 1}{100} \\
 & \Rightarrow \text{S}\text{.I}{{\text{.}}_{3}}=\dfrac{1323\times 1\times 1}{2} \\
 & \Rightarrow \text{S}\text{.I}{{\text{.}}_{3}}=661.5 \\
\end{align}$
Therefore, we have $\text{Amount at the end of 3 year = 13230+661}\text{.5}$. This, results into $\text{Amount at the end of 3 year = 13891}\text{.5}$.
Therefore, the total interest given here is given by $\text{S}\text{.I}{{\text{.}}_{\text{1}}}\text{+S}\text{.I}{{\text{.}}_{\text{2}}}\text{+S}\text{.I}{{\text{.}}_{\text{3}}}$. Therefore, we get it as Rs. 600 + Rs. 630 + Rs. 661.5. Thus we get Rs. 1891.5.
Thus, the compound interest on Rs. 12000 at 5 % p.a. for 3 years will be Rs.1891.5.

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