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. $$A=P{\left(1+{\displaystyle \frac{R}{100}}\right)}^T$$, 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. $$A=P{\left(1+{\displaystyle \frac{R}{100}}\right)}^T$$ 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,
$$2205=2000{\left(1+{\displaystyle \frac{R}{100}}\right)}^2$$
Now dividing both sides of the above equation by 2000. We get,
$${\displaystyle \frac{2205}{2000}}={\left(1+{\displaystyle \frac{R}{100}}\right)}^2$$
$${\displaystyle \frac{441}{400}}={\left(1+{\displaystyle \frac{R}{100}}\right)}^2$$
Now taking the square root on both sides of the above equation. We get,
$$\sqrt{{\displaystyle \frac{441}{400}}}={\displaystyle \frac{21}{20}}=\left(1+{\displaystyle \frac{R}{100}}\right)$$
Now subtracting 1 to both sides of the above equation. We get,
$${\displaystyle \frac{21}{20}}-1={\displaystyle \frac{1}{20}}={\displaystyle \frac{R}{100}}$$
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. $$A=P{\left(1+{\displaystyle \frac{R}{100}}\right)}^T$$ 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).