Find the compound interest for Rs.62500 for \[1\dfrac{1}{2}\] years at \[8\%\] per annum, compounded half yearly.

Hint: When the interest is compounded half-yearly, then the time period for which the amount is borrowed gets doubled and the rate at which the interest is calculated is halved. Use the compound interest formula to get the desired answer.

Complete step-by-step answer:
Assume \[P\], \[r\] and \[t\] denote the principal amount, rate of interest and the time period respectively.
Given the amount borrowed is Rs.62500 for \[1\dfrac{1}{2}\] years at \[8\%\] per annum.
So, \[P = 62500\], \[r = 8\], and \[t = 1\dfrac{1}{2} = \dfrac{3}{2}\].
Since the amount is compounded half yearly, so the rate of interest gets halved and the time period gets doubles, i.e.,
\[r = \dfrac{8}{2} = 4\] and \[t = \dfrac{3}{2}\times 3 = 3\]
The formula for calculating the compound interest is given by,
\[CI = P\left[\left(1+\dfrac{r}{100}\right)^{n}-1\right]\]
Substituting the values into the formula, the compound interest is calculated as,
\[\begin{align*}CI &= 62500\left[\left(1+\dfrac{4}{100}\right)^{3}-1\right]\\ &= 62500\left[\left(\dfrac{104}{100}\right)^{3}-1\right]\\ &= 62500(1.12-1)\\ &= 7500\end{align*}\]
So, the compound interest is Rs7500.

Note: The compound interest can also be calculated using the formula for the simple interest for each successive year. The interest for the first year when added to the principal amount becomes the principal for the second year and so on. The compound interest is equal to the difference in the final amount and the initial amount.