Question

Find the compound interest on \[Rs.25,000\] at the rate of \[12%\] per annum for 3 years.

Answer 1

Hint: From the question, it is given that the initial amount is equal to \[Rs.25,000\], the interest for the initial amount is equal to \[12%\] and the time for which the annual interest is calculated is equal to 3 years. We know that if P is the initial amount, r is the interest per year and A is the total amount obtained after t years, then \[A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\]. Now from the question, we should write the value of A, P, r and t. By using this formula, we should find the value of A. If A is the compound amount after t years, P is the initial amount and C.I is the compound interest for 3 years, then \[C.I=A-P\]. So, by the values of A and P, we should find the values of C.I.

Complete step-by-step answer:
Before solving the question, we should know that if P is the initial amount, r is the interest per year and A is the total amount obtained after t years, then \[A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\].
From the question, it is given that the initial amount is equal to \[Rs.25,000\], the interest for the initial amount is equal to \[12%\] and the time for which the annual interest is calculated is equal to 4 years.
We know that if P is the initial amount, r is the interest per year and A is the total amount obtained after t years, then \[A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\].
So, it is clear that
\[\begin{align}
  & P=25,000....(1) \\
 & r=12........(2) \\
 & t=3........(3) \\
\end{align}\]
Now from the values of P, r and t, we have to find the value of A.
\[\begin{align}
  & \Rightarrow A=\left( 25000 \right){{\left( 1+\dfrac{12}{100} \right)}^{3}} \\
 & \Rightarrow A=\left( 25000 \right){{\left( \dfrac{112}{100} \right)}^{3}} \\
 & \Rightarrow A=\left( 25000 \right)\left( \dfrac{1404928}{1000000} \right) \\
 & \Rightarrow A=\left( 25 \right)\left( \dfrac{1404928}{1000} \right) \\
 & \Rightarrow A=35123.200.....(4) \\
\end{align}\]
If A is the compound amount after t years, P is the initial amount and C.I is the compound interest for 3 years, then
\[C.I=A-P.....(5)\]
Now let us substitute equation (1) and equation (4) in equation (5), then we get
\[\begin{align}
  & \Rightarrow C.I=35,123.2-25000 \\
 & \Rightarrow C.I=10,123.2.....(6) \\
\end{align}\]
So, from equation (6) we can say that the value of C.I is equal to Rs.10,123.2.

Note: Some students have a misconception that if P is the initial amount, r is the interest per year and A is the total amount obtained after t years, then \[A=P{{\left( 1+r \right)}^{t}}\]. If this formula is applied, then the value of A is obtained as follows.
\[\begin{align}
  & \Rightarrow A=\left( 25000 \right){{\left( 1+12 \right)}^{3}} \\
 & \Rightarrow A=\left( 25000 \right){{\left( 13 \right)}^{3}} \\
 & \Rightarrow A=\left( 25000 \right)\left( 2197 \right) \\
 & \Rightarrow A=\left( 25000 \right)\left( 2197 \right) \\
 & \Rightarrow A=54,925,000.....(1) \\
\end{align}\]
Now from equation (1), it is clear that the value of A is equal to 54,925,000. But we know that the value of A is equal to Rs.35,123.2.
So, this misconception should get avoided.