At what rate percent per annum will a sum of Rs. 2000 amount to Rs. 2205 in 2 years, compounded annually ?
ANSWER
Verified
Hint: We have to only use the compound interest formula i.e. \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\], where A is the amount after T years, P is the principal amount, R is the rate of interest and T is the time period.
Complete step-by-step solution -
As we know that the amount after two years will be equal to Rs. 2205. The principal amount at the starting is equal to Rs. 2000. And the time period is 2 years. So, R be the rate of interest on which the principal amount is compounded annually. So, now we can apply the formula of compound interest i.e. \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\] and then find the value of R by manipulating that equation. So, putting values of A, P and T in the compound interest formula. We get, \[2205 = 2000{\left( {1 + \dfrac{R}{{100}}} \right)^2}\] Now dividing both sides of the above equation by 2000. We get, \[\dfrac{{2205}}{{2000}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}\] \[\dfrac{{441}}{{400}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}\] Now taking the square root on both sides of the above equation. We get, \[\sqrt {\dfrac{{441}}{{400}}} = \dfrac{{21}}{{20}} = \left( {1 + \dfrac{R}{{100}}} \right)\] Now subtracting 1 to both sides of the above equation. We get, \[\dfrac{{21}}{{20}} - 1 = \dfrac{1}{{20}} = \dfrac{R}{{100}}\] On multiplying both sides of the above equation by 100. We get, R = 5% Hence, the rate of interest will be equal to 5%.
Note: Whenever we come up with this type of problem the we had to only use compound interest formula i.e. \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\] And after that dividing both sides of the equation by p and then taking square root to both the sides and after that subtracting 1 from both sides and multiplying by hundred. We will get the required value of R (i.e. rate of interest at which principal amount is compounded annually).