Question

The sum of money at the compound interest amounts to Rs.10580 in 2 years and to Rs. 12167 in 3 years. The rate of interest per annum is
A.12%
B.14%
C.15%
D.$16\dfrac{2}{3}\% $

Answer 1

Hint: We will use the formula of compound interest $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$, where $A$ is the total amount, $P$ is the principal amount, \[r\] is the rate of interest and $t$ is the time to form equations according to given conditions. Then, we will solve the equations and find the value of $r$.

Complete step by step answer:

We are given that sum of money amounts to be Rs. 10580 in 2 years when the interest is compounded.
We know that $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$ , where $A$ is the total amount, $P$ is the principal amount, \[r\] is the rate of interest and $t$ is the time.
Then, we have $10580 = P{\left( {1 + \dfrac{r}{{100}}} \right)^2}$ ……. eq(1)
Also, we are given that the amount is 12167 after 3 years, then we have,
$12167 = P{\left( {1 + \dfrac{r}{{100}}} \right)^3}$ …….. eq(2)
We will solve the equation (1) and (2) by dividing the equation (2) by (1)
Then we get,
$\dfrac{{12167 = P{{\left( {1 + \dfrac{r}{{100}}} \right)}^3}}}{{10580 = P{{\left( {1 + \dfrac{r}{{100}}} \right)}^2}}}$
Solve the above equation to find the value of $r$
$
  1.15 = 1 + \dfrac{r}{{100}} \\
   \Rightarrow 1.15 - 1 = \dfrac{r}{{100}} \\
   \Rightarrow 0.15 = \dfrac{r}{{100}} \\
$
Multiply the equation throughout by 100
$r = 15$
Hence, option C is correct.

Note: In this question, the principal amount for the third year will be the total amount after 2 years as the interest is calculated is compound interest. Then, we can also calculate the interest of the third year by subtracting 10580 from 12167, which is interest is Rs. 1587. Then, we can say interest in Rs.10580 for the third year is Rs.1587.