Question

In what time will $Rs\,2000$ amount to $Rs\,2205$ at $5\% $ per annum, compounded annually?

Answer 1

Hint: We have to find the time in this question. So we will use the formula of Amount i.e.
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$ , where A is the amount,
$P$ is the principal,
$R$ is the rate and
 $n$ is the number of years. We will put the values in the formula and then we solve it.

Complete answer:Here we have
We should know that compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interests from the previous period.
Here we have been given
 $P = Rs\,2000$
Rate:
$R = 5\% $ and,
$A = Rs\,2205$
We know the formula
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$
By putting the values in the formula we get:
$2205 = 2000{\left( {1 + \dfrac{5}{{100}}} \right)^n}$
We will take $2000$ to the left hand side i.e.
$\dfrac{{2205}}{{2000}} = {\left( {\dfrac{{100 + 5}}{{100}}} \right)^n}$
On simplifying the value we can write
$\dfrac{{441}}{{400}} = {\left( {\dfrac{{105}}{{100}}} \right)^n}$
We can write the value in simpler form in the right hand side i.e.
$\dfrac{{441}}{{400}} = {\left( {\dfrac{{21}}{{20}}} \right)^n}$
We know that
 $441 = {(21)^2}$
and $400$ can be written as
 ${(20)^2}$
So we can write the expression as
${\left( {\dfrac{{21}}{{20}}} \right)^2} = {\left( {\dfrac{{21}}{{20}}} \right)^n}$
Since the base on both side of the equation is same, we can eliminate them, so it gives us
$n = 2$ .
Hence it gives us time i.e. $2$ years.

Note:
We should note that in the above solution we have used the exponential formula i.e. If we have
${(P)^a} = {(P)^b}$ , then we can write them as $a = b$ .
Hence powers are equal if their bases are equal.