Finding the fraction of a mixture using Ratio

Question

This question is actually a GRE Question which comes under ratio topic and I have no clue on how to solve it

Question

Party Cranberry is 3 parts cranberry juice and 1 part seltzer. Fancy Lemonade is 1 part lemon juice and 2 parts seltzer. An amount of Party Cranberry is mixed with an equal amount of Fancy Lemonade

A) The fraction of the resulting mix that is cranberry juice

B) The fraction of the resulting mix that is seltzer

Which one is greater? A or B?

Answer 1

Using ratios, let's first create a "Part : Part : Whole" relationship for each glass. Whole is the sum of the parts in each glass.

Party Cranberry would be $\quad$ 3 : 1 : 4

Fancy Lemonade would be $\enspace$ 1 : 2 : 3

Now, one glass of Party Cranberry is mixed with an equally sized glass of Fancy Lemonade. Think about the above ratios using cups. Party Cranberry is 3 cups cranberry and 1 cup seltzer, filling up a 4-cup glass. Fancy Lemonade is 1 cup lemon and 2 cups seltzer, filling up a 3-cup glass. So... in order to fill up two equal-sized glasses for each beverage, each glass would have to be a multiple of 4 and 3. Let's say 12 cups.

Now, multiply each ratio by a number so the the "Wholes" for each are 12.

Party Cranberry would be $\quad$ 9 : 3 : 12

Fancy Lemonade would be $\enspace$ 4 : 8 : 12

We now have two 12-cup glasses completely filled with each beverage. If we dump this mixture into a 24-cup glass, how much will be cranberry and how much will be seltzer?

Answer 2

Instead of making confusing countings it's easy to solve the problem in the following way: since you mix an equal amount of the two drinks, in the mix you obtain: 3 parts of cranberry juice, 1 part of seltzer from the first drink and 1 part lemon juice,2 parts seltzer from the second. Totally you have 7 parts (you can see them as units) of which 3 of cranberry juice,3 of seltzer. It is clear that the fractions A) and B) are then equal. You can expand this method if you mix amounts of the first and second drink in ratio of $\alpha:\beta$ (that is $\alpha$ units of the first drink,$\beta$ of the second).Then you'll have $3\alpha$ parts of cranberry juice and $\alpha$ parts of seltzer from the first drink,$\beta$ parts of lemon juice and $2\beta$ part of seltzer from the second,for a total of $4\alpha + 3\beta$ parts.