Question

At what rate per cent will $Rs\,12,000$ yield $Rs\,1,891.50$ as compound interest in $3$ years?

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 paid depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A to be paid 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: $A = P{(1 + \dfrac{R}{{100}})^T}$ .

Complete step-by-step solution:
In the given problem,
Principal $ = P = Rs\,12,000$
Rate of interest $ = R$$\% $
Time Duration $ = 3\,years$
In the question, the period after which the compound interest is compounded or evaluated is now given. So, we take that the compound interest is compounded annually by default.
So, Number of time periods$ = n = 3$
Also, Compound interest$ = Rs\,1,891.50$
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: $A = P{(1 + \dfrac{R}{{100}})^T}$ .
Hence, Amount $ = P{(1 + \dfrac{R}{{100}})^T}$
Amount that we receive after $3$ years $ = $Principal$ + $Compound interest
Substituting the values of principal and compound interest as given to us in the question, we get,
$ \Rightarrow \,Amount = Rs\,12,000\, + \,Rs\,1,891.50 = \,Rs\,13,891.50$
So, Total amount $ = \,Rs\,13,891.50$
Now, substituting all the values that we have with us in the formula $A = P{(1 + \dfrac{R}{{100}})^T}$, we get,
$ \Rightarrow Rs\,13,891.50 = Rs\,12,000{\left( {1 + \dfrac{R}{{100}}} \right)^3}$
Shifting all the terms without variables to the left side of the equation, we get,
\[ \Rightarrow \dfrac{{13,891.50}}{{12,000}} = {\left( {1 + \dfrac{R}{{100}}} \right)^3}\]
Simplifying the calculation in left side of the equation, we get,
\[ \Rightarrow 1.157625 = {\left( {1 + \dfrac{R}{{100}}} \right)^3}\]
Taking cube root on both sides of the equation, we get,
\[ \Rightarrow {\left( {1.157625} \right)^{\dfrac{1}{3}}} = \left( {1 + \dfrac{R}{{100}}} \right)\]
Calculating the cube root, we get,
\[ \Rightarrow 1.05 = \left( {1 + \dfrac{R}{{100}}} \right)\]
\[ \Rightarrow \dfrac{R}{{100}} = 1.05 - 1\]
\[ \Rightarrow \dfrac{R}{{100}} = 0.05\]
Shifting all the constants to the right side of the equation, we get the value of R as:
\[ \Rightarrow R = 5\]
So, The principal amount of $Rs\,12,000$ compounds at $5\% $ per annum compounded annually so as to give $Rs\,1,891.50$ as compound interest in $3$ years.

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 double in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest.