Question

In what time will Rs. 8000 amount change to Rs. 8820 if it is compounded at a rate of 5 % annually.

Answer 1

Hint: The formula for compound interest is \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] where A is the final amount, P is the principal amount, r is the interest rate per year, n is the number of years. Apply this to the given data of the question to get the answer.

Complete step by step answer:-
Compound interest is the interest that is calculated on the principal amount along with the interest accumulated over the previous period or year.
The formula to calculate compound interest with a principal amount P, at an annual rate r for n years is given as follows:
\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}............(1)\]
In this problem, the principal amount P is given as Rs. 8000.
\[P = 8000...........(2)\]
The amount is compounded at a rate of 5 % annually.
\[r = 100...........(3)\]
We need to find the number of years n to accumulate an amount of Rs. 8820.
\[A = 8820..........(4)\]
Using equations (2), (3), and (4) in equation (1), we have:
\[8820 = 8000{\left( {1 + \dfrac{5}{{100}}} \right)^n}\]
Simplifying the terms in the bracket we get:
\[8820 = 8000{\left( {1.05} \right)^n}\]
Taking 8000 to the other side and dividing with 8820, we get:
\[\dfrac{{8820}}{{8000}} = {\left( {1.05} \right)^n}\]
\[1.1025 = {\left( {1.05} \right)^n}\]
Apply log to both the sides of the equation, to get as follows:
\[\log (1.1025) = \log ({1.05^n})\]
We know that the value of \[\log {x^n}\] is equal to \[n\log x\].
\[\log (1.1025) = n\log (1.05)\]
We know that the square of 1.05 is 1.1025, then, we have:
\[\log {\left( {1.05} \right)^2} = n\log (1.05)\]
\[2\log \left( {1.05} \right) = n\log (1.05)\]
Solving for n, we have:
\[n = 2\dfrac{{\log (1.05)}}{{\log (1.05)}}\]
\[n = 2\]
Hence, it takes 2 years for the amount Rs. 8000 to change to Rs. 8820 when it is compounded at the rate of 5 % annually.