Question

At what rate percent per annum will a sum of Rs. 2000 amount to Rs. 2205 in 2 years, compounded annually ?

Answer 1

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).