What is the difference between the compound interests on $\text{Rs. }5000$ for $1\frac{1}{2}$ years at $4\%$ per annum compounded yearly and half-yearly?

Let's calculate the compound interest ($\text{CI}$) for both scenarios separately with Principal $P=\text{Rs. }5000$, Rate $R=4\%$, and Time $T=1.5\text{years}$.
Case 1: Compounded Yearly

* Interest for the $1^{\text{st}}$ year:

$$I_1=5000\times \frac{4}{100}=\text{Rs. }200$$

* Amount after $1$ year $=5000+200=\text{Rs. }5200$.
* Interest for the remaining $\frac{1}{2}$ year is computed on this accumulated amount:

$$I_2=5200\times \frac{4}{100}\times \frac{1}{2}=\text{Rs. }104$$

* Total ${\text{CI}}_{\text{yearly}}=200+104=\text{Rs. }304$

Case 2: Compounded Half-Yearly

* Time $=1\frac{1}{2}\text{ years}=3\text{ half-years}$
* Rate per half-year $=\frac{4\%}{2}=2\%$
* Amount ($A$) after $3$ half-years is:

$$A=5000\times {\left(1+\frac{2}{100}\right)}^3=5000\times (1.02{)}^3$$

$$A=5000\times 1.061208=\text{Rs. }5306.04$$

* Total ${\text{CI}}_{\text{half-yearly}}=5306.04-5000=\text{Rs. }306.04$$\ast \ast Differencebetweenthetwointerests:\ast \ast$$\text{Difference}={\text{CI}}_{\text{half-yearly}}-{\text{CI}}_{\text{yearly}}=306.04-304=\text{Rs. }2.04$ $