Two vessels contain a mixture of spirit and water. In the first vessel the ratio of spirit to water is 8 : 3 and in the second vessel the ratio is 5 : 1 . A 35 litre cask is filled from these vessels so as to contain a mixture of spirit and water in the ratio of 4 : 1. How many litres are taken from the first vessel?
11 litres
22 litres
16.5 litres
17.5 litres
Solution
Let the quantity taken from the first vessel be $n$
So, the quantity taken from the second vessel is $35-n$
Using the rule of allegation, the ratio of the two quantities is given by
$\dfrac { \frac{5}{5+1} - \frac{4}{4+1}} { \frac{4}{4+1} - \frac{8}{8+3}} = \dfrac { \frac{5}{6} - \frac{4}{5}} {\frac{4}{5} - \frac{8}{11}}$
$ = \dfrac { \frac{25-24}{30}} {\frac{44-40}{55}} = \dfrac{1}{30} \times \dfrac{55}{4}= \dfrac{11}{24}$
So, $n=11$ and $35-n=24$
The required quantity is 11 litres