An interesting ratio and proportion problem from Hall and Knight

Q:  Two casks A and B were filled with 2 kinds of sherry, mixed in the cask A in the ratio of 2:7, and in the cask B in the ratio of 1:5. What quantity must be taken from each to form a mixture which shall consist of 2 gallons of one kind and 9 gallons of the other,

                                                                     SOLUTION.

Let us assume that we take ‘a’  gallons from the first cask and ‘b’ from the other cask. The, considering the first cask, we have :

                                             2a/9 parts(or, units) of First type of sherry,S1,and

                                             7a/9 parts(or, units) of First type of sherry,S2 .

And, considering the second cask, we have:

                                             b/6 parts (or, units) of First type of sherry, S1,and

                                             5b/6 parts (or, units) of First type of sherry, S2.

 In the mixture, therefore, we have 2a/9+b/6 units of S1 and 7a/9+5b/6 units of S2.

Now, 2a/9+b/6 = (4a+3b) / 18 and 7a/9+5b/6 = (14a+15b) / 18

Therefore, as per the question, the ratio of S1 to S2 in the mixture is,

                                              --    (  (4a+3b) / 18  )  /   ( (14a+15b) / 18 )     =  2 / 9

                                              --   (4a+3b) / (14a+15b)  =  2 / 9

From which,

                                              --        36a+27b = 28a+30b

                                             --           8a = 3b

From which, we have finally,

                                                              a/b = 3/8

Thus, we have to take 3 gallons from cask A and 8 gallons from cask B to form the mixture.