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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 Rs. 6272. Calculate: (a) The rate of interest per annum.(b) The amount at the end of 3 years.  Hint: Start by using the formula that $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ for the first 2 years. In the formula A is the amount, P is the principal and r is the interest rate. For the first 2 years, amount, principal and time is known, so put the values and get r from it. Again apply the same formula for getting the amount at the end of 3 years.

Before starting with the question, let us know about interest.
Interest in the financial term is the amount that a borrower pays to the lender along with the repayment of the actual principal amount.
Broadly, there are two kinds of interest, first is the simple interest, and the other is the compound interest.
Let us apply the formula $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ , for the first 2 years. For the first 2 years, amount A is Rs. 6272, principal P is Rs. 5000 and time t is 2 years. So, if we put this in formula, we get
$6272=5000{{\left( 1+\dfrac{r}{100} \right)}^{2}}$
$\Rightarrow \dfrac{6272}{5000}=\dfrac{3136}{2500}={{\left( 1+\dfrac{r}{100} \right)}^{2}}$
Taking root of both the side gives:
$\sqrt{\dfrac{3136}{2500}}=\left( 1+\dfrac{r}{100} \right)$
$\Rightarrow \dfrac{56}{50}=\left( 1+\dfrac{r}{100} \right)$
$\Rightarrow \dfrac{r}{100}=\dfrac{6}{50}$
$\Rightarrow r=\dfrac{6}{50}\times 100=12\%$
Therefore, the rate of interest is 12%.
Now we will again use the same formula for 3 years. For this case, P=5000, r=12 and t=3.
$A=5000{{\left( 1+\dfrac{12}{100} \right)}^{3}}$
$A=5000{{\left( \dfrac{112}{100} \right)}^{3}}=\dfrac{5000\times 112\times 112\times 112}{100\times 100\times 100}=Rs.\text{ }7024.64$
Therefore, the amount at the end of 3 years is Rs. 7024.64.

Note: Don’t get confused and take 6272 to be the principal amount. Also, be careful with the calculations and solving part as there is a possibility of making a mistake in the calculations. It is recommended to learn all the basic formulas related to simple as well as compound interests as they are very much useful in the problems related to money exchange. Remember that the above formula is valid if and only if interest is compounded yearly else the formula becomes $A=P{{\left( 1+\dfrac{r}{100n} \right)}^{nt}}$ , where n is the number of times interest is compounded in a year.

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