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# 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. $\$ 10100$Last updated date: 20th Jun 2024 Total views: 414k Views today: 11.14k Answer Verified 414k+ views Hint: In this problem, given that the rate of interest is annual and the interest is compounded semi-annually (that is, 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 (that is,$2T$). We will find the amount (balance)$A$for$1$year by using the formula$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$. Complete step by step solution: Here given that the principal amount$P = $$\ 10,000, rate of interest R = 10\%  per annum and time T = 1 year. Also given that the interest is compounded semi-annually (that is, six months). So, the interest paid at the end of every six months is one-half of the rate of interest per annum. So, the rate of annual interest is R = \dfrac{{10}}{2}\% = 5\%  and the number of years is doubled. That is, T = 2 half years. Now we are going to find the amount for 1 year by using the formula A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} where P is principal amount, R is rate of annual interest and T is time in half years. Now we are going to substitute the values of P, R and T in the formula of amount for 1 year. Therefore, A = 10000{\left( {1 + \dfrac{5}{{100}}} \right)^2}  \Rightarrow  A = 10000{\left( {\dfrac{{100 + 5}}{{100}}} \right)^2}  \Rightarrow$$A = 10000{\left( {\dfrac{{105}}{{100}}} \right)^2} = 10000\left( {\dfrac{{105 \times 105}}{{100 \times 100}}} \right) \Rightarrow A = 105 \times 105 = \$11025$
Therefore, the balance after $1$ year will be $\$ 11025$. Therefore, option C is correct. Note: Simple interest is calculated only on the principal amount but compound interest is calculated on principal amount as well as previous year’s interest. If interest is paid only for$T = 1$year then there is no distinction between simple interest and compound interest. To find simple interest, we can use the formula$\dfrac{{PRT}}{{100}}$where$P$is principal amount,$R$is rate of interest and$T\$ is time in years.