Find the amount and compound interest for Rs$7500$ for a year at $8\% $ per annum compounded half yearly.

Hint: Here, the principle is Rs $7500$, the rate of interest is $8\% $ per annum and the time period for which we have to calculate the amount and compound interest is $1$ year that is two half years. We can calculate the amount and compound interest by using the formula:
(1) $A = P \times {\left( {1 + \dfrac{R}{{100}}} \right)^T}$.
(2) $C.I = A - P$.
Where $A$ is the amount, $P$ is the principle or sum of money, $R$ is the rate of interest, $T$ is the time period and $C.I$ is the compound interest.

Complete step-by-step solution:
Given: principle $P = 7500$, rate $R = 8\% $ per annum, $T = 2$ half year. And we have to calculate the amount and compound interest on the given sum of money.
Now, apply the formula for the amount that is $A = P \times {\left( {1 + \dfrac{R}{{100}}} \right)^T}$. Putting the numerical value of $P$, $R$ and $T$ in the given formula we get,
$
   \Rightarrow A = 7500 \times {\left( {1 + \dfrac{8}{{100}}} \right)^2} \\
   \Rightarrow A = 7500 \times {\left( {1 + \dfrac{2}{{25}}} \right)^2} \\
   \Rightarrow {\rm A} = 7500 \times {\left( {\dfrac{{27}}{{25}}} \right)^2} \\
   \Rightarrow A = 7500 \times \dfrac{{27}}{{25}} \times \dfrac{{27}}{{25}} \\
   \Rightarrow A = \dfrac{{5467500}}{{625}} \\
  \therefore A = 8748
 $
Thus, the amount is equal to Rs$8748$.
Now, apply the formula for compound interest that is $C.I = A - P$. Putting the value of the amount and principle in the given formula we get the compound interest.
$
   \Rightarrow C.I = 8478 - 7500 \\
  \therefore C.I = 978
 $
Thus, the compound interest is Rs$978$.

Hence, the amount and compound interest for the given sum at the rate of $8\% $ per annum for one year compounded half yearly is Rs$8478$ and Rs$978$ respectively.

Note: The compound interest on a given sum is calculated by simple interest method also. In this method firstly find the simple interest for Rs$7500$ at the rate of $8\% $ per annum for a time period of first half year then calculate the amount for first half year which is the principle for second half year and the rate of interest is $8\% $ per annum. Calculate the simple interest for the second half year and then calculate the amount which is the sum of principle and simple interest for the second half year. This amount is equal to the amount when the given sum is compounded half yearly for the same rate of interest then subtracting principle from the amount, we get compound interest.