Two vessels, A and B, contain a mixture of milk and water in the ratio of 1 : 4 and 5 : 4, respectively. In what ratio should the mixtures in vessels A and B be mixed, such that the resultant mixture contains half milk and half water?

Mixture in A: milk : water = 1 : 4
So, milk fraction in A = $ \frac{1}{5} $

Mixture in B: milk : water = 5 : 4
So, milk fraction in B = $ \frac{5}{9} $

Required mixture has half milk ⇒ milk fraction = $ \frac{1}{2} $

Using allegation:

Ratio of A : B
$= \left( \frac{5}{9} - \frac{1}{2} \right) : \left( \frac{1}{2} - \frac{1}{5} \right)$

$= \left( \frac{10 - 9}{18} \right) : \left( \frac{5 - 2}{10} \right)$
$= \frac{1}{18} : \frac{3}{10}$

Multiply by LCM (90):

$= 5 : 27$

Answer: 5 : 27