Question

At what rate percent per annum will a sum of $Rs\,1000$ amount to $Rs\,1102.50$ in two years at compound interest?

Answer 1

Hint: The problem can be solved easily with the concept of compound interest. Compound interest is the interest calculated on the principal and the interest of the previous period. The amount in compound interest to be cumulated depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .

Complete step-by-step answer:
In the given problem,
Principal $ = P = Rs\,1000$
Rate of interest $ = R\% $
Time Duration $ = 2\,years$
In the question, the period after which the compound interest is compounded or evaluated is now given. So, we assume that the compound interest is compounded annually by default.
So, Number of time periods$ = n = 2$
Also, the amount after two years is given to us as $Rs\,1102.50$.
Now, The amount A to be paid after a certain number of time periods n on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .
Hence, Amount $ = A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Now, substituting all the values that we have with us in the formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$, we get,
$ \Rightarrow 1102.50 = 1000{\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Shifting all the terms without variables to the left side of the equation, we get,
$ \Rightarrow \dfrac{{1102.50}}{{1000}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Now, simplifying the calculations, we get,
$ \Rightarrow {\left( {1 + \dfrac{R}{{100}}} \right)^2} = 1.1025$
Taking square root on both sides of equation, we get,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \sqrt {1.1025} $
Computing the square root of number,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = 1.05$
Now, shifting all the constants to right side of equation, we get,
$ \Rightarrow \left( {\dfrac{R}{{100}}} \right) = 1.05 - 1$
Multiplying both sides by $100$, we get,
$ \Rightarrow R = 0.05 \times 100$
Simplifying the calculations,
$ \Rightarrow R = 5$
Therefore, the sum of $Rs\,1000$ will amount to $Rs\,1102.50$ in two years at $5\% $ per annum compound interest compounded annually.
So, the correct answer is “5%”.

Note: Time duration is not always equal to the number of time periods. The equality holds only when the compound interest is compounded annually. If the compound interest is compounded half yearly, then the number of time periods doubles in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest.