Question

At what rate per annum, will Rs. 5120 amount to Rs. 7290 in 3 years when compounded annually?

Answer 1

Hint: We are given that the principal amount is Rs. 5120, the total amount after 3 years is Rs. 7290 and the time is 3 years. Substitute the given values in the formula $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$, where $A$ is the total amount, $P$ is the principal amount, $t$ is the time and $r$ denotes the rate of interest. Solve the equation to find the value of $r$.

Complete step by step answer:

We are given that the principal amount is Rs. 5120 and the total amount after three years is 7290, when interest is compounded annually.
We know that the amount after compound interest for a given principal amount can be calculated using the formula,
$A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$, where $A$ is the total amount, $P$ is the principal amount, $t$ is the time and $r$ denotes the rate of interest.
On substituting the values $A = 7290$, $P = 5120$ and $t = 3$ in the above formula, we get $7290 = 5120{\left( {1 + \dfrac{r}{{100}}} \right)^3}$
Divide the equation throughout by 5120
$
  \dfrac{{7290}}{{5120}} = {\left( {1 + \dfrac{r}{{100}}} \right)^3} \\
   \Rightarrow \dfrac{{729}}{{512}} = {\left( {1 + \dfrac{r}{{100}}} \right)^3} \\
$
Now, we know that $729 = {9^3}$ and $512 = {8^3}$
$\dfrac{{{9^3}}}{{{8^3}}} = {\left( {1 + \dfrac{r}{{100}}} \right)^3}$
${\left( {\dfrac{9}{8}} \right)^3} = {\left( {1 + \dfrac{r}{{100}}} \right)^3}$
If the exponents are equal, their bases must be equal.
$\dfrac{9}{8} = 1 + \dfrac{r}{{100}}$
Solve the above equation to find the value of $r$
$
  \dfrac{9}{8} - 1 = \dfrac{r}{{100}} \\
   \Rightarrow \dfrac{1}{8} = \dfrac{r}{{100}} \\
   \Rightarrow r = \dfrac{{100}}{8} \\
   \Rightarrow r = 25 \\
$
Hence, the rate of interest is 25%.

Note: One can proceed this question by not simplifying $\dfrac{{7290}}{{5120}}$ as $\dfrac{{729}}{{512}}$ and then to ${\left( {\dfrac{9}{8}} \right)^3}$, which can make the calculations difficult. Also, when compound interest is calculated, the principal amount for every year is different. The principal amount for the following year will be the addition of the principal amount and the interest of the previous months.