Question

At what rate per cent per annum will $Rs\,6000$ amount to $Rs\,6615$ in two years when interest is compounded annually?

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\,6,000$
Rate of interest $ = R\% $
Time Duration $ = 2\,years$
In the question, the period after which the compound interest is compounded or evaluated is given as a year.
So, Number of time periods $ = n = 2$
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: $A = P{(1 + \dfrac{R}{{100}})^T}$ .
Hence, Amount $ = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Substituting the values of known quantities, we get,
$ \Rightarrow 6615 = 6000{\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Shifting the terms in the equation, we get,
$ \Rightarrow {\left( {1 + \dfrac{R}{{100}}} \right)^2} = \dfrac{{6615}}{{6000}}$
Cancelling common factors in numerator and denominator, we get,
 $ \Rightarrow {\left( {1 + \dfrac{R}{{100}}} \right)^2} = \dfrac{{441}}{{400}}$
Taking square root on both sides of equation, we get,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \sqrt {\dfrac{{441}}{{400}}} $
We know that square roots of $441$ and $400$ are $21$ and $20$ respectively. So, we get,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \dfrac{{21}}{{20}}$
Isolating the variable R. we get,
$ \Rightarrow \dfrac{R}{{100}} = \dfrac{{21}}{{20}} - 1$
Taking LCM of fractions,
$ \Rightarrow \dfrac{R}{{100}} = \dfrac{{21 - 20}}{{20}}$
Multiplying both sides of equation by $100$,
$ \Rightarrow R = \dfrac{1}{{20}} \times 100$
Simplifying the calculations, we get,
$ \Rightarrow R = 5$
So, the rate of interest per annum for which $Rs\,6000$ amounts to $Rs\,6615$ in two years is $5\% $.
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. Care should be taken while doing calculations.