Question

At what rate per annum will Rs. $640$ amount to $774.40$ in $2$ years, when interest is being compounded annually?

Answer 1

Hint: We have given amount, principal amount and time. We have to calculate the rate of interest. The rate of interest is compounded annually. To calculate this there is a relation between Amount, Principal amount, Rate of interest and time. Which is give as
\[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\]
How $A$ represent Amount, $P$ represent Principal amount and $r$ represent time in years, $R$ is rate of interest.

Complete step-by-step answer:
We have given that
Amount $(A)$ = $774.40$
Principal $(P)$= $640$
Time $(n)$ = $2$years
Now we apply formula of Amount
$Amount = \Pr incipal{\left( {1 + \dfrac{R}{{100}}} \right)^n}$
$\Rightarrow$ $774.40 = 640{\left( {1 + \dfrac{R}{{100}}} \right)^2}$
 $640$ is multiplied with the bracket.
We can take it in divide with $774.40$
$\dfrac{{774.40}}{{640}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
$\Rightarrow$ $\dfrac{{774.40}}{{640 \times 100}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
$\Rightarrow$ $\dfrac{{7744}}{{6400}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Take square root on both sides. The square root on the right hand side cancels the square.
Therefore $\sqrt {\dfrac{{7744}}{{6400}}} = \left( {1 + \dfrac{R}{{100}}} \right)$
Square root of $7744 = 88$
Square root of $6400 = 80$
Therefore $\dfrac{{88}}{{80}} = \left[ {1 + \dfrac{R}{{100}}} \right]$
$\Rightarrow$ $\dfrac{{11}}{{10}} = 1 + \dfrac{R}{{100}}$
$\Rightarrow$ $\dfrac{{11}}{{10}} - 1 = \dfrac{R}{{100}} \to \dfrac{R}{{100}} = \dfrac{{11 - 10}}{{10}} = \dfrac{1}{{10}}$
$\Rightarrow$ $R = \dfrac{{100}}{{10}}$
$\Rightarrow$ $R = 10\% $
So rate of interest compounded annually is $10\% $

Note: Principal amount: Principal amount is total amount of money borrowed, not including any interest as dividends.
Rate of interest: An interest state is the amount of interest due per period, as a portion of amount lent deposited or borrowed.
Compound interest: Compound interest in the addition of the interest to the principal sum of a loan or deposit or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the neat period is then earned on the principal sum plus previously accumulated interest.