Question

A sum amounts to Rs. 2916 in 2 years and Rs. 3194.28 in 3 years when compounded annually. Find the principal.

Answer 1

Hint: First, we should know the formula of the amount(A) of the compound interest(CI) with the principal(P) as $A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$ where r is rate of interest and n is the time in years. Then, we know in the compound interest for 3 years compounded annually, the amount after 2 years acts like the principal which gives the value of rate of interest r. Then, on getting the value of r, we can use the first condition given in the question for the first two years the amount is Rs.2916 compounded annually and get the original principal value.

Complete step-by-step answer:
In this question, we are supposed to find the principal of the compound interest.
So, we should know the formula of the amount(A) of the compound interest(CI) with the principal(P).
Then, the formula of amount A where r is rate of interest and n is the time in years is given by:
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$
So, we know in the compound interest for 3 years compounded annually, the amount after 2 years acts like the principal.
So, by using the above condition, we get:
$3194.28=2916{{\left( 1+\dfrac{r}{100} \right)}^{1}}$
Now, we need to solve the above equation to get the value of r as:
$\dfrac{3194.28}{2916}={{\left( 1+\dfrac{r}{100} \right)}^{1}} \\ $
$ \Rightarrow 1.0954 =\left( 1+\dfrac{r}{100} \right) \\ $
$ \Rightarrow 1.0954-1=\dfrac{r}{100} \\ $
$ \Rightarrow 0.0954=\dfrac{r}{100} \\ $
$ \Rightarrow r= 9.5% \\ $
Now, after getting the value of r as $9.5\%$, we can use the first condition given in the question for the first two years the amount is Rs.2916 compounded annually as:
$2916=P{{\left( 1+\dfrac{9.5}{100} \right)}^{2}}$
So, solve the above expression to get the value of original principal as:
$\begin{align}
  & 2916=P{{\left( \dfrac{100+9.5}{100} \right)}^{2}} \\
 & \Rightarrow 2916=P{{\left( \dfrac{109.5}{100} \right)}^{2}} \\
 & \Rightarrow P=\dfrac{2916\times 100\times 100}{109.5\times 109.5} \\
 & \Rightarrow P=2431.975 \\
 & \Rightarrow P\approx 2432 \\
\end{align}$
So, it gives the value of the principal as Rs. 2432.
Hence, the principal is Rs. 2432.

Note: Another approach to solve this type of question is that we can divide the conditional equations of the amount after 3 years and 2 years.
So, the amount after 2 years is given by:
$2916=P{{\left( 1+\dfrac{r}{100} \right)}^{2}}$
Similarly, the amount after 3 years is given by:
$3194.28=P{{\left( 1+\dfrac{r}{100} \right)}^{3}}$
Now, we can divide the amount after 3 years with amount after 2 years to get the value of r as:
$\dfrac{3194.28}{2916}=\dfrac{P{{\left( 1+\dfrac{r}{100} \right)}^{3}}}{P{{\left( 1+\dfrac{r}{100} \right)}^{2}}}$
So, by solving this we get r as 9.5% and further solve in the same way as done above.