Question

What sum of money will amount to $ Rs.4374 $ in $ 3 $ years at $ 12\dfrac{1}{2}\% $ per annum, compounded annually?

Answer 1

Hint: We have to find the principal amount for which is $ 12\dfrac{1}{2}\% $ interest is added per annum for $ 3 $ years will add the sum to a final amount of $ Rs.4374 $ . So, using the compound interest formula which is
Final Amount = $ principal \times {\left( {1 + \dfrac{{rate}}{{100}}} \right)^{time}} $ we will find the principal amount..

Complete step-by-step answer:
Interest rate = $ 12\dfrac{1}{2}\% $ per annum
Final amount = $ Rs.4374 $
Time = $ 3 $ years
The formula for compound interest is
Total Amount = $ principal \times {\left( {1 + \dfrac{{rate}}{{100}}} \right)^{time}} $ .
Substituting all the values given in the question, the formula becomes,
 $ 4374 $ = $ principal \times {\left( {1 + \dfrac{{12\dfrac{1}{2}}}{{100}}} \right)^3} $
Solving the rate of interest part,
 $ 4374 $ = $ principal \times {\left( {1 + \dfrac{{\dfrac{{12 \times 2 + 1}}{2}}}{{100}}} \right)^3} $
First, we will multiply and add the numbers,
 $ 4374 $ = $ principal \times {\left( {1 + \dfrac{{\dfrac{{25}}{2}}}{{100}}} \right)^3} $
 $ 4374 $ = $ principal \times {\left( {1 + \dfrac{{12.5}}{{100}}} \right)^3} $
Cross multiplying and making the denominator equal,
 $ 4374 $ = $ principal \times {\left( {\dfrac{{100 + 12.5}}{{100}}} \right)^3} $
 $ 4374 $ = $ principal \times {\left( {\dfrac{{112.5}}{{100}}} \right)^3} $
 $ 4374 $ = $ principal \times {\left( {1.125} \right)^3} $
Taking the cube of $ 1.125 $ ,
 $ 4374 $ = $ principal \times 1.423 $
Dividing the final amount $ 4374 $ by $ 1.423 $ ,
 $ principal = \dfrac{{4374}}{{1.423}} $
Finally, we get the principal amount as,
 $ principal = 3073.78 $
The principal amount is $ Rs.3073.78 $
Therefore, $ Rs.3073.78 $ is compounded annually for $ 3 $ years at a rate of interest of $ 12\dfrac{1}{2}\% $ per annum to get the final amount as $ Rs.4374 $ .
We can cross-check the principal amount by using the same compound interest formula where we substitute principal amount as $ Rs.3073.78 $ , rate of interest as $ 12\dfrac{1}{2}\% $ and time as $ 3 $ years,
Final Amount = $ principal \times {\left( {1 + \dfrac{{rate}}{{100}}} \right)^{time}} $
Final Amount = $ 3073.78 \times {\left( {1 + \dfrac{{12.5}}{{100}}} \right)^3} $
Final Amount = $ 3073.78 \times {\left( {1.125} \right)^3} $
Final Amount = $ 3073.78 \times 1.423 $
Final Amount = $ 4373.98 \cong 4374 $
Therefore, $ Rs.3073.78 $ is the correct principal amount.
So, the correct answer is “ $ Rs.3073.78 $ ”.

Note: Compound interest is the interest calculated on the predominant and the interest accrued over the preceding period. It is distinct from easy interest, where interest isn't introduced to the principal while calculating the interest at some point of the following duration. Compound interest unearths its usage in the maximum of the transactions in the banking and finance sectors and different regions.