Question

At what rate \[\% \] will a sum of Rs. 7500 amount to Rs. 8427 in 2 years, compounded annually?

Answer 1

Hint:
Here, we need to find the rate of compound interest. We will use the formula for amount when a principal is compounded for a period of time. Then, we will simplify the equation to find the required rate of interest.
Formula used: The amount \[A\] of an investment after \[t\] years is given by $A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$, where $P$ is the amount invested, $n$ is the number of compounding periods in a year and $r$ is the interest rate compounded annually.

Complete step by step solution:
The amount \[A\] of an investment after \[t\] years is given by $A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$, where $P$ is the amount invested, $n$ is the number of compounding periods in a year and $r$ is the interest rate compounded annually.
We will use this formula to form an equation and solve it to find the rate.
Since the sum is compounded annually, the number of compounding periods in a year is 1.
Substituting \[n = 1\], \[t = 2\], \[P = {\text{Rs}}{\text{. }}7500\] and \[A = {\text{Rs}}{\text{. }}8427\] in the formula, we get
$ \Rightarrow 8427 = 7500{\left( {1 + \dfrac{{r\% }}{1}} \right)^{1 \times 2}}$
We have formed a linear equation in one variable in terms of \[r\]. We will solve this equation to find the value of \[r\], and hence, the rate of interest compounded annually.
Simplifying the expression, we get
$ \Rightarrow 8427 = 7500{\left( {1 + \dfrac{r}{{100}}} \right)^2}$
Dividing both sides of the equation by 7500, we get
$ \Rightarrow \dfrac{{8427}}{{7500}} = \dfrac{{7500{{\left( {1 + \dfrac{r}{{100}}} \right)}^2}}}{{7500}} \\
   \Rightarrow 1.1236 = {\left( {1 + \dfrac{r}{{100}}} \right)^2} \\
\text{Taking the square root of both sides of the equation, we get}
   \Rightarrow \sqrt {1.1236} = \sqrt {{{\left( {1 + \dfrac{r}{{100}}} \right)}^2}} \\
   \Rightarrow 1.06 = 1 + \dfrac{r}{{100}} \\
\text{Subtracting 1 from both sides of the equation, we get}
   \Rightarrow 1.06 - 1 = 1 + \dfrac{r}{{100}} - 1 \\
   \Rightarrow 0.06 = \dfrac{r}{{100}} \\ $
Multiplying both sides of the equation by 100, we get
\[ \Rightarrow 0.06 \times 100 = \dfrac{r}{{100}} \times 100\]
Therefore, we get the rate as
$ \Rightarrow r = 6\% $

Thus, we get that the sum of Rs. 7500 amounts to Rs. 8427 in 2 years, compounded annually at the rate $6\% $ per annum.

Note:
We have formed a linear equation in one variable in terms of \[r\] in the solution using the formula $A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$. A linear equation in one variable is an equation that can be written in the form \[ax + b = 0\], where \[a\] is not equal to 0, and \[a\] and \[b\] are real numbers. For example, \[x - 100 = 0\] and \[100P - 566 = 0\] are linear equations in one variable \[x\] and \[P\] respectively. A linear equation in one variable has only one solution.