The simple interest on a certain sum is $50\%$ of the sum in $8$ years at a certain rate per cent p.a. What will be the interest (in ₹) on a sum of ₹$7,200$ at the same rate in $5\frac{2}{3}$ years at simple interest?

Let the principal sum be $P$.The simple interest ($SI$) in $8$ years is $50\%$ of $P$, which is $0.5P$.Using the Simple Interest formula:
$$SI=\frac{P\times R\times T}{100}$$
$$0.5P=\frac{P\times R\times 8}{100}$$
$$0.5=\frac{8R}{100}⟹R=\frac{50}{8}=6.25\%\text{ p.a.}$$
Now, we need to calculate the simple interest on a new sum of ₹$7,200$ at the same rate for $5\frac{2}{3}=\frac{17}{3}$ years:
$$S{I}_{\text{new}}=\frac{7200\times 6.25\times \frac{17}{3}}{100}$$
$$S{I}_{\text{new}}=\frac{7200\times \frac{25}{4}\times \frac{17}{3}}{100}$$
$$S{I}_{\text{new}}=72\times \frac{25}{4}\times \frac{17}{3}=18\times 25\times \frac{17}{3}=6\times 25\times 17=150\times 17=2550$$
Thus, the interest is ₹$2,550$.