Find the amount and the compound interest on Rs. 160000 for 2 year at $10\%$ per annum compound interest half-yearly.

Hint: Here we will convert the given rate and year for half-yearly as we have to find the amount for half-yearly. We will then calculate the amount accumulated during this period using the calculated rate and given principal. Then we will put these values in the formula of the Compound Interest and solve it to get the required answer.

Formula Used:
We will use the following formulas:
1. Compound Interest = Amount – Principal
2. Amount \[ = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\]
Here, \[P = \] Principal Amount, \[R = \] Interest Rate and \[n = \] Time

Complete step-by-step answer:
The data given to us are,
Principal Amount \[ = {\rm{Rs}}.160000\]…..\[\left( 1 \right)\]
Rate per annum \[ = 10\% \]
$\therefore $ Rate per half yearly \[ = 5\% \]….\[\left( 2 \right)\]
Time \[ = 2year\]
As 1 year have two half year
$\therefore $ Time for half yearly \[ = 2 \times 2 = 4\] half year …..\[\left( 3 \right)\]
Substituting the value from equation \[\left( 1 \right)\] and \[\left( 2 \right)\] in formula Amount\[ = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\], we get
Amount \[ = 160000{\left( {1 + \dfrac{5}{{100}}} \right)^4}\]
Taking LCM inside the bracket, we get
\[ \Rightarrow \] Amount \[ = 160000{\left( {\dfrac{{100 + 5}}{{100}}} \right)^4}\]
Adding the terms, we get
\[ \Rightarrow \] Amount \[ = 160000{\left( {\dfrac{{105}}{{100}}} \right)^4}\]
Dividing the terms inside the bracket, we get
\[ \Rightarrow \] Amount \[ = 160000{\left( {1.05} \right)^4}\]
Simplifying the equation, we get
\[ \Rightarrow \] Amount \[ = {\rm{Rs}}.194481\]….\[\left( 4 \right)\]
So, we get our amount as Rs. 194481.
Next, we will find the Compound interest.
Substituting the value from equation \[\left( 1 \right)\] and \[\left( 4 \right)\] in the formula Compound Interest = Amount – Principal, we get
Compound Interest \[ = 194481 - 160000\]
Subtracting the terms, we get
\[ \Rightarrow \] Compound Interest \[ = \] Rs. 34481

Therefore, the amount after the compound interest of Rs. 34481 on the principal amount Rs. 160000 for 2 years at 10% per annum compounded half-yearly is Rs. 1994481.

Note:
Compound interest is calculated on the principal amount plus the interest in the previous year. It is because of the reinvestment of interest rather than paying it out. Compound interest increases the money at such a rate that it is sometimes also known as exponential growth. The compound Interest can be calculated monthly or per day as well and is often called interest on interest.