Question

At what rate of interest will Rs.20000 become Rs.24200 after 2 years when interest is compounded annually?
A). $$5\% $$
B). $$6\% $$
C). $$10\% $$
D). $$15\% $$

Answer 1

Hint: Here in this question, we need to find the rate of interest. For this, we have to use a formula of compound interest $$A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$$ on substituting the given principal amount, final amount and time period and on further simplify by using a basic arithmetic operation to get the required solution.

Complete step-by-step solution:
Compound interest is the addition of interest to the principal sum of a loan or deposit over the period.
The formula to find a compound interest is:
$$A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$$
Where,
A = final amount
P = principal amount
r = rate of interest
n = number of times interest is compounded per year.
The interest rate is defined as the proportion of an amount loaned which a lender charges as interest to the borrower, normally expressed as an annual percentage.
Now consider the given question:
Principle amount $$P = 20000$$
Final amount $$A = 24200$$
number of times interest is compounded per year $$n = 2$$
then by the formula of compound interest
$$ \Rightarrow \,\,24200 = 20000{\left( {1 + \dfrac{r}{{100}}} \right)^2}$$
Divide 20000 on both sides, then we have
$$ \Rightarrow \,\,\dfrac{{24200}}{{20000}} = {\left( {1 + \dfrac{r}{{100}}} \right)^2}$$
$$ \Rightarrow \,\,1.21 = {\left( {1 + \dfrac{r}{{100}}} \right)^2}$$
Taking square root on both sides
$$ \Rightarrow \,\,\sqrt {1.21} = 1 + \dfrac{r}{{100}}$$
$$ \Rightarrow \,\,1.1 = 1 + \dfrac{r}{{100}}$$
Subtract 1 on both the sides, then we have
$$ \Rightarrow \,\,1.1 - 1 = \dfrac{r}{{100}}$$
$$ \Rightarrow \,\,0.1 = \dfrac{r}{{100}}$$
Multiply 100 on both sides, then we get
$$ \Rightarrow \,\,r = 0.1 \times 100$$
$$\therefore \,\,\,\,\,r = 10\% $$
Hence, the required rate of interest is $$10\% $$.
Therefore, option (c) is the correct answer.

Note: Remember, the principal amount ‘P’ is the initial size of amount or money the borrower has borrowed from the lender. Final amount ‘A’ is the amount or money accumulated after n years, including the interest. The final amount is less than the principal amount.