The difference between the compound interest and simple interest on a certain sum at 15% per annum for 3 years is Rs 283.50.

Given;

$\mathrm{CI}-\mathrm{SI}=\mathrm{Rs} 283.50$

$\mathrm{R}=15 \%$

$\mathrm{n}=3$ years

Let the sum be Rs $\mathrm{x}$.

We know that:

$\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$

$=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$

$=\mathrm{x}\left(1+\frac{15}{100}\right)^{3}$

$=\mathrm{x}(1.15)^{3}$       ...(1)

Also,

$\mathrm{SI}=\frac{\mathrm{PRT}}{100}=\frac{\mathrm{x}(15)(3)}{100}=0.45 \mathrm{x}$

$\mathrm{A}=\mathrm{SI}+\mathrm{P}=1.45 \mathrm{x} \quad \ldots(2)$

Thus, we have :

$\mathrm{x}(1.15)^{3}-1.45 \mathrm{x}=283.50 \quad[$ From $(1)$ and $(2)]$

$1.523 \mathrm{x}-1.45 \mathrm{x}=283.50$

$0.070875 \mathrm{x}=283.50$

$\mathrm{x}=\frac{283.50}{0.070875}$

=4,000

Thus, the sum is Rs 4,000 .