A recipe calls for 1 cup of milk for every $2\frac{1}{2}$ cups of flour to make a cake that would feed 6 persons. How many cups of both flour and milk will be needed to make a similar cake for 8 persons?

Given, milk needed for making cake = 1 cup
and flour needed for making cake $=2\frac{1}{2}$ cups
= $\frac{5}{2}$ cups
Then, total amount needed = Milk + Flour
= $(1+\frac{5}{2})=\frac{7}{2}$ cups
So, $\frac{7}{2}$ cups of milk and flour are needed to make cake for 6 persons.
Let the needed amount of cups of milk and flour to make cake for 8 persons = x
(where x is multiple of cups)
So, Cups : Persons :: Cups : Persons
$\frac{7}{2}:6::x:8\frac{\frac{7}{2}}{6}=\frac{x}{8}6\times x=\frac{7}{2}\times 8x=\frac{7}{2}\times 8\times \frac{1}{6}x=\frac{14}{3}=4\frac{2}{3}$
Hence, the cups needed for 8 persons is $4\frac{2}{3}$