Question

Find the amount and the compound interest on Rs. 12800 for 1 year at $7\dfrac{1}{2}\% $ per annum compounded semi-annually.

Answer 1

Hint: Since, the interest is compounded semi-annually, determine the rate of interest for half year and respective time for the given time. Then substitute these values in the formula of compound interest, $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$ and then simplify it to get the amount. Subtract the principal amount from the total amount to find the compound interest.

Complete step-by-step answer:
If $7\dfrac{1}{2}\% $ is the interest of 1 year, then the interest of one quarter can be calculated by dividing it by 2.
Then, the interest for one quarter is \[7\dfrac{1}{2} = \dfrac{{\dfrac{{15}}{2}}}{2} = \dfrac{{15}}{4}\% \]
Also, there are 2 half-years in a year, so time will be taken as 2.
We know that amount on a principal amount $P$ at an interest of $r\% $ and time $t$ years when interest is compounded annually is given by $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$
Now substitute 12800 for $P$ , $\dfrac{{15}}{4}$ for \[r\] and 2 for $t$ in the above formula.
$
  A = 12800{\left( {1 + \dfrac{{\dfrac{{15}}{4}}}{{100}}} \right)^2} \\
   \Rightarrow A = 12800{\left( {\dfrac{{400 + 15}}{{400}}} \right)^2} \\
$
On simplifying the expression, we will get,
$
  A = 12800{\left( {\dfrac{{415}}{{400}}} \right)^2} \\
   \Rightarrow A = 13778 \\
$
The total amount is Rs. 13778.
We will now calculate the compound interest by subtracting principal amount from the total amount.
$13778 - 12800 = 978$
Hence, the compound interest is Rs. 978.

Note: When we have to calculate the compound interest semi-annually or half yearly and the rate of interest is given annually, then we divide the rate of interest by 2 and multiply the given time by 2. In this interest after every half-year gets added to the principal amount for the next half.