Question

What is the sum that should be invested if the final amount is equal to $ 2809 $ and compound interest for $ 2 $ years at $ 6\% $ per annum?
(A) Rs. $ 1575 $
(B) Rs. $ 2000 $
(C) Rs. $ 2500 $
(D) Rs. $ 2250 $

Answer 1

Hint: In the given problem, the rate of interest is annual and the interest is compounded for $ 2 $ years. We have to find the principal amount (invested amount). For this, we will use the formula $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $ where $ P $ is principal amount, $ A $ is the final amount, $ R $ is the rate of interest per annum and $ T $ is time in years.

Complete step-by-step answer:
Here given that the final amount $ A = $ Rs. $ 2809 $ , rate of interest $ R = 6\% $ per annum and time $ T = 2 $ years. Also given that the interest is compounded for $ 2 $ years. Now we are going to find the principal amount $ P $ for $ 1 $ year by using the formula $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $ where $ P $ is principal amount, $ R $ is the rate of annual interest and $ T $ is time in years.
Now we are going to substitute the values of $ A $ , $ R $ and $ T $ in the above formula. So, we can write
 $ 2809 = P{\left( {1 + \dfrac{6}{{100}}} \right)^2} $ . Let us simplify this equation. So, we can write
 $ 2809 = P{\left( {\dfrac{{100 + 6}}{{100}}} \right)^2} $
 $ \Rightarrow 2809 = P{\left( {\dfrac{{106}}{{100}}} \right)^2} $
 $ \Rightarrow 2809 = P{\left( {\dfrac{{53}}{{50}}} \right)^2} $
Let us solve the above equation for $ P $ . So, we can write
 $ P = 2809 \times {\left( {\dfrac{{50}}{{53}}} \right)^2} $
 $ \Rightarrow P = 2809 \times \dfrac{{2500}}{{2809}} $
 $ \Rightarrow P = $ Rs. $ 2500 $
Therefore, we can say that Rs. $ 2500 $ should be invested if the final amount is equal to $ 2809 $ and compound interest for $ 2 $ years at $ 6\% $ per annum.

Note: Simple interest is calculated only on the principal amount but compound interest is calculated on principal amount as well as previous year’s interest. If interest is paid only for $ T = 1 $ year then there is no distinction between simple interest and compound interest.