Question

A sum of money invested at compounded interest amounts to Rs. 800 in 3 years and to Rs. 840 in 4 years. The rate of interest per annum is
A. \[3\% \]
B. \[4\% \]
C. \[5\% \]
D. \[6\% \]

Answer 1

Hint: Here, we are given different amounts which we receive after \[n\] years, when a sum of money is invested at compound interest. We will use the formula of compound interest (C.I.) and find two equations. We will substitute the given values, which when divided together, would leave only the required variable whose value has to be found. Solving further, we would get the required value of rate of interest per annum.

Formula Used:
We will use the formula of Amount, \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] , where \[A\] is the final amount, \[P\] is the principal, \[r\] is the rate of interest per annum and \[n\] is the number of years.

Complete step-by-step answer:
Compound Interest is the addition of interest in the principal amount or in simple terms we can say, reinvesting of interest.
Now, according to the question, a sum of money (i.e. Principal) is invested at compounded interest amounts to Rs. 800 in 3 years.
Hence, the given total amount, \[A = {\rm{Rs}}.800\]
Number of years, \[n = 3\]
Substituting \[n = 3\] and \[A = 800\] in\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\], we get
\[800 = P{\left( {1 + \dfrac{r}{{100}}} \right)^3}\]……………………………………..\[\left( 1 \right)\]
Similarly, it is given that, a sum of money invested at compounded interest amounts to Rs. 840 in 4 years.
Here, total amount \[A = {\rm{Rs}}.840\]
And number of years, \[n = 4\]
Substituting \[n = 4\] and \[A = 840\] in\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\], we get
\[840 = P{\left( {1 + \dfrac{r}{{100}}} \right)^4}\]……………………………………\[\left( 2 \right)\]
Since, we are required to find the rate of interest, we will divide the equation \[\left( 2 \right)\] by equation \[\left( 1 \right)\]. Therefore, we get
\[\dfrac{{840}}{{800}} = \dfrac{{P{{\left( {1 + \dfrac{r}{{100}}} \right)}^4}}}{{P{{\left( {1 + \dfrac{r}{{100}}} \right)}^3}}}\]
Now, further simplifying this equation and cancelling out the similar terms, we get
\[ \Rightarrow \dfrac{{21}}{{20}} = \left( {1 + \dfrac{r}{{100}}} \right)\]
Subtracting 1 from both the sides, we get,
\[ \Rightarrow \dfrac{{21}}{{20}} - 1 = \dfrac{r}{{100}}\]
Taking LCM on LHS, we get
\[ \Rightarrow \dfrac{{21 - 20}}{{20}} = \dfrac{r}{{100}}\]
\[ \Rightarrow \dfrac{1}{{20}} = \dfrac{r}{{100}}\]
Multiplying both sides by 100, we get
\[ \Rightarrow \dfrac{1}{{20}} \times 100 = \dfrac{r}{{100}} \times 100\]
Simplifying further, we get,
\[ \Rightarrow r = 5\]
Hence, the required rate of interest per annum, \[r\] is equal to \[5\% \].
Therefore, option C is the correct answer.

Note: We can make a mistake by getting confused between the principal and the amount. Principal is the sum which is invested for a given period of time, whereas amount is the total which is receivable based on the principal invested, time period and the rate of interest.
Now, in this question, as it was given: ‘A sum of money invested at compounded interest amounts to’
Here the amount given is not the principal but the amount accumulated after compounding the principal on the interest. Hence, we have used the formula of CI rather than Simple interest (SI).