Question

If \[\$ 10,000\] is invested at 10 percent annual interest, compounded semi-annually, what is the balance after 1 year?
A. $\$ 10100.25$
B. $\$ 10200.25$
C. $\$ 11025$
D. $\$ 10100$

Answer 1

Hint: It is given in the question that if $\$ 10,000$ is invested at 10 percent annual interest, compounded semi-annually.
Then, what is the balance after 1 year?
 The interest is compounded semi-annually i.e. six months.
That means the interest paid at the end of every six months is one half of the rate of interest per annum.
So, the rate of annual interest is $\dfrac{R}{2}\% $ and the number of years is doubled i.e. $T = 2$ half year
Now, we will find the amount A for 1 year by using the formula i.e. $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ where P is the principal amount, R is the rate of interest and T is time in half years.
Finally, we will put the value of P, R and T in the $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ to get the required answer.

Complete step-by-step answer:
It is given in the question that if $\$ 10,000$ is invested at 10 percent annual interest, compounded semi-annually.
Then, what is the balance after 1 year?
 The interest is compounded semi-annually i.e. six months.
That means the interest paid at the end of every six months is one half of the rate of interest per annum.
So, the rate of annual interest is $\dfrac{R}{2}\% $ and the number of years is doubled i.e. $T = 2$ half year
Now, we will find the amount A for 1 year by using the formula
i.e. $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ where P is the principal amount, R is the rate of interest and T is time in half years.
Now, substitute the value of P, R and T in the above equation.
 $\Rightarrow A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
 $\Rightarrow A = 10000{\left( {1 + \dfrac{5}{{100}}} \right)^2}$
 $\Rightarrow A = 10000{\left( {\dfrac{{100 + 5}}{{100}}} \right)^2}$
 $\Rightarrow A = 10000{\left( {\dfrac{{105}}{{100}}} \right)^2}$
 $\Rightarrow A = 10000\left( {\dfrac{{105 \times 105}}{{100 \times 100}}} \right)$
 $\Rightarrow A = 105 \times 105$
 $\Rightarrow A = \$ 11025$
Therefore, the balance after 1 year will be $\$ 11025$ .
Therefore, option C is correct.

Note: Since, simple interest is calculated only on the principal amount but compound interest is calculated on principal amount as well as previous year’s interest.
To find simple interest, we will use the formula $\dfrac{{PRT}}{{100}}$ where P is the principal amount, R is the rate of interest and T is the time of year.