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A person invests Rs.5000 for three years at a certain rate of interest compounded annually. At the end of two years this sum amounts to Rs.6272. Calculate :
(i) the rate of interest per annum.
(ii) The amount at the end of the third year.

Last updated date: 20th Jun 2024
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Hint: Compound interest is the additional interest to the principal amount which can also be said as interest on interest. It can be calculated using the below mentioned formula. Substitute the given values of Principal Amount P, time period T and final amount A to get the rate of interest R. And for the second sub question, substitute the values of obtained R, P and T to get the value of A.
 $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $ , where A is the final amount after implying the compound interest, P is the principal amount, T is the Time period and R is the rate of interest.

Complete step-by-step answer:
We are given that a person invests Rs.5000 for three years for a certain compound interest.
We have to calculate the interest rate when the amount is Rs. 6272 after two years and the amount at the end of the third year.
We know that the final amount for a compound interest can be calculated using $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $
 $ P = 5000,T = 2,A = 6272 $
 $ \Rightarrow 6272 = 5000{\left( {1 + \dfrac{R}{{100}}} \right)^2} $
 $ \Rightarrow \dfrac{{6272}}{{5000}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2} $
 $ \Rightarrow \sqrt {\dfrac{{6272}}{{5000}}} = 1 + \dfrac{R}{{100}} $
 $ \Rightarrow 1 + \dfrac{R}{{100}} = 1.12 $
 $ \Rightarrow \dfrac{R}{{100}} = 1.12 - 1 = 0.12 $
 $ \Rightarrow R = 0.12 \times 100 $
 $ \therefore R = 12\% $
Therefore, the rate of interest is 12 percent.
Using this interest rate, we can find the amount at the end of three years.
Principal amount is Rs.5000, rate of interest is 12, Time period is three years.
Final Amount is
 $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $
 $ A = 5000{\left( {1 + \dfrac{{12}}{{100}}} \right)^3} $
 $ \Rightarrow A = 5000{\left( {1.12} \right)^3} $
 $ \Rightarrow A = 7024.64 $
 $ \therefore A = Rs.7024.64 $
At the end of third year, the amount is 7024 rupees 64 paisa (Rs.7024.64).
So, the correct answer is “Rs.7024.64”.

Note: The interest can be either simple or compound. In simple interest, the interest amount does not change till the end of the return period whereas in compound interest, the interest amount gradually changes as the interest is imposed on the principal amount plus the previous accumulated interest combined.