The numerator of a fraction is 3 more than the denominator. When 5 is added to the numerator and 2 is subtracted from the denominator, the fraction becomes $$\frac{8}{3}.$$ When the original fraction is divided by $$5\frac{1}{2},$$  the fraction so obtained is:

$$\eqalign{ & {\text{Let denominator}} = x \cr & {\text{Numerator}} = x + 3 \cr & {\text{Fraction become}} = \frac{{x + 3}}{x} \cr & {\text{According to the question,}} \cr & \frac{{x + 3 + 5}}{{x - 2}} = \frac{8}{3} \cr & 3x + 24 = 8x - 16 \cr & 5x = 40 \cr & x = 8 \cr & \therefore {\text{Denominator}} = 8 \cr & {\text{Numerator}} = 8 + 3 = 11 \cr & {\text{Fraction}} = \frac{{11}}{8} \cr & {\text{Then, }}\frac{{11}}{8} \div \frac{{11}}{2} = \frac{{11}}{8} \times \frac{2}{{11}} = \frac{1}{4} \cr} $$