Question

The sum of $\$ 8000$ was deposited at $8$ percent interest compounded annually. How much money was in the bank at the end of $2$ years ?

Answer 1

Hint: The problem can be solved easily with the concept of compound interest. Compound interest is the interest calculated on the principal and the interest of the previous period. The amount in compound interest to be paid depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A to be paid after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the

Formula used: $A = P{(1 + \dfrac{R}{{100}})^T}$ .

Complete step by step solution:
In the given problem,
Principal $ = \$ 8000$
Rate of interest$ = 8\% $ per annum
Time Duration $ = 2$ years
According to the question, the compound interest is compounded annually.
So, Number of time periods $ = 2$
Now, The amount A after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{(1 + \dfrac{R}{{100}})^T}$ .
Hence, Amount $ = P{(1 + \dfrac{R}{{100}})^T}$
Substituting the values,
$ \Rightarrow $ Amount $ = \left( {\$ 8000} \right){\left( {1 + \dfrac{8}{{100}}} \right)^2}$
$ \Rightarrow $ Amount $ = \left( {\$ 8000} \right){\left( {1 + \dfrac{2}{{25}}} \right)^2}$
Taking the LCM,
$ \Rightarrow $ Amount $ = \left( {\$ 8000} \right){\left( {\dfrac{{25 + 2}}{{25}}} \right)^2}$
$ \Rightarrow $ Amount $ = \left( {\$ 8000} \right){\left( {\dfrac{{27}}{{25}}} \right)^2}$
$ \Rightarrow $ Amount $ = \left( {\$ 8000} \right)\left( {\dfrac{{27 \times 27}}{{25 \times 25}}} \right)$
$ \Rightarrow $ Amount $ = \left( {\$ 8000} \right)\left( {\dfrac{{27 \times 27}}{{25 \times 25}}} \right)$
On further solving,
$ \Rightarrow $ Amount $ = \$ 9331.2$
So, The amount at the end of $2$ years on $\$ 8000$ at $8\% $ per annum compounded annually is $\$ 9331.2$ .

Note: Time duration is not always equal to the number of time periods. The equality holds only when the compound interest is compounded annually. If the compound interest is compounded half yearly, then the number of time periods double in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest.