Question

Find the sum which amounts to $Rs\,1352$ in two years at $4\% $ 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: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ . So, we will substitute the known values in the formula and find the principal amount.

Complete step by step answer:
In the given problem,
Principal $ = P$
Rate of interest $ = 4\% $
Time Duration $ = 2\,years$
In the question, the period after which the compound interest is compounded or evaluated is not 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 $Rs\,1352$.

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 1352 = P{\left( {1 + \dfrac{4}{{100}}} \right)^2}$
Taking LCM in right side of equation, we get,
$ \Rightarrow 1352 = P{\left( {\dfrac{{104}}{{100}}} \right)^2}$

Shifting all constant terms to the left side of the equation, we get,
$ \Rightarrow 1352 \times \dfrac{{{{100}^2}}}{{{{104}^2}}} = P$
$ \Rightarrow P = 1352 \times \dfrac{{{{100}^2}}}{{{{104}^2}}}$
Cancelling common factors in numerator and denominator, we get,
$ \Rightarrow P = 1 \times \dfrac{{{{100}^2}}}{8}$
Now, simplifying the calculations, we get,
$ \Rightarrow P = \dfrac{{100 \times 100}}{8}$
$ \Rightarrow P = \dfrac{{25 \times 50}}{1}$
$ \therefore P = Rs\,1250$
Therefore, the principal amount is $Rs\,1250$.

Hence, the sum $Rs\,1250$ will amount to $Rs\,1352$ in two years at $4\% $ compound interest.

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.