The ratio of alcohol to water in a vessel of capacity 80 litres is 5:3, and two draws of 20 litres each are planned. After the $1^{st}$ draw, 20 litres of water are poured into the vessel, mixed well, and then the $2^{nd}$ draw ensues. The liquid drawn in 2 lots is also mixed well in another vessel. What's the ratio of alcohol to water in this second vessel?

Vessel capacity = $80$ litres, Alcohol : Water = $5 : 3$

$\text{Alcohol} = \frac{5}{8} \times 80 = 50$ L

$\text{Water} = \frac{3}{8} \times 80 = 30$ L

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The vessel is well-mixed, so the 20 L drawn also has ratio $5:3$.

$\text{Alcohol drawn} = \frac{5}{8} \times 20 = 12.5$ L

$\text{Water drawn} = \frac{3}{8} \times 20 = 7.5$ L

Remaining in the vessel:

$\text{Alcohol} = 50 - 12.5 = 37.5$ L

$\text{Water} = 30 - 7.5 = 22.5$ L

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20 L of pure water is added and mixed well.

$\text{Alcohol} = 37.5$ L

$\text{Water} = 22.5 + 20 = 42.5$ L

$\text{Total} = 80$ L

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The second draw of 20 L is taken from this well-mixed vessel.

$\text{Alcohol drawn} = \frac{37.5}{80} \times 20 = 9.375$ L

$\text{Water drawn} = \frac{42.5}{80} \times 20 = 10.625$ L

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Combining both draws in the second vessel:

$\text{Total Alcohol} = 12.5 + 9.375 = 21.875$ L

$\text{Total Water} = 7.5 + 10.625 = 18.125$ L

$\frac{21.875}{18.125} = \frac{21.875 \times 8}{18.125 \times 8} = \frac{175}{145} = \frac{35}{29}$

Therefore, the ratio of alcohol to water in the second vessel $= 35 : 29$.