Question

A certain sum of money is placed on compound interest amounts to \[Rs.4000\] in $3$ year and \[Rs.5000\] in $5$ years. Calculate the rate of interest.

Answer 1

Hint:To solve this problem we should know about the basics about the compound interest.
Compound interest: It is the interest on a loan or deposit calculated based on the both the initial principal and the interest from previous periods.
Formula: $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$
Here,
$A$ is the final amount after a given time-period.
$P$ is principal value.
$r$ is the rate of interest per year.
$t$ is the time period of the loan.

Complete step by step answer:
As we have to find the rate of interest for given data.
Let, rate of interest is $r$ . It is the rate of interest per year.
As given, the amount changes to \[Rs.4000\] in $3$ year and \[Rs.5000\] in $5$ years. Calculate the rate of interest.
Let the principal amount is $P$ for calculation convenience.
Let take first case:
 $A = Rs.4000$
 $t = 3year$
So, keeping it in formula as we have,
 $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$
Keeping value in the given equation. We get,
 $ \Rightarrow 4000 = P{\left( {1 + \dfrac{r}{{100}}} \right)^3}$ ………. $(1)$
Let take second case:
 $A = Rs.5000$
 $t = 5year$
Keeping this value in compound interest formula. We get,
 $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$
 $ \Rightarrow 5000 = P{\left( {1 + \dfrac{r}{{100}}} \right)^5}$ ………. $(2)$
Dividing $(2)$ from $(1)$ . We get,
 \[ \Rightarrow \dfrac{{5000}}{{4000}} = \dfrac{{P{{\left( {1 + \dfrac{r}{{100}}} \right)}^5}}}{{P{{\left( {1 + \dfrac{r}{{100}}} \right)}^3}}}\]
By solving it,
 \[ \Rightarrow \dfrac{5}{4} = {\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
 \[ \Rightarrow 1.25 = {\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
Taking under root on both side. We get,
  \[ \Rightarrow \sqrt {1.25} = 1.118 = \left( {1 + \dfrac{r}{{100}}} \right)\]
 \[ \Rightarrow 1.118 - 1 = 0.118 = \dfrac{r}{{100}}\]
 \[ \Rightarrow 0.118 \times 100 = r = 11.8\% \]
Hence, the rate of interest for a given date is $r = 11.8\% $ .

Note: Compound interest used in our daily life. It is used in calculation of money in savings account, fixed deposit credit card etc. Compound interest makes a sum of money grow faster than simple interest. The simple interest rate is the interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is known as the nominal interest rate.