Explanation

We have a situation here that has a relatively simple situation but a fairly large number of potential variables to work with - we have the girls, boys, and total in one group, the same in the other group, and the ratios. We are not keen on working with 6 or so variables, so we will try to get away with doing this purely by analysis of cases. So we will skip equation-writing and go straight to the statements, evaluating them separately at first.

Statement (1) looks insufficient because we want to know a ratio, and it's not giving ratio-type information, but rather an absolute difference. We will try to generate multiple possible cases. We start with 10 girls and 10 boys and assign them to Group B, since that has the smaller G/B ratio. Then Group A would have to have 20 boys and 10 girls. That doesn't match Statement (1), because the difference in the totals is only 10. To make the difference 30, we multiply all the numbers by 3.

Case I:

Group A: 60 girls, 30 boys, 90 total. G/B=2

Group B: 30 girls, 30 boys, 60 total. G/B= 1

This is a legal case. To generate another case, we'll start with a different G/B ratio in Group B... Say 1/2 in Group B. Group B: 10 girls, 20 boys, 30 total, G/B=1/2. Then Group A would be 20 girls, 20 boys, 40 total, for a G/B=1. Again, we multiply all the numbers to get the right total difference. Again we have a total difference of 10 and so need to multiply by 3 to fit Statement (1):

Case II:

Group A: 60 girls, 60 boys, 120 total. G/B=1

Group B: 30 girls, 60 boys, 90 total. G/B= 1/2

Now that we have achieved two cases--is the answer to the question in each case the same or different? In Case I, the ratio of the totals is 90/60 or 3/2. In Case II, it's 120/90 or 4/3. Therefore, we do not have sufficient information to answer the question definitively. Statement (1) is insufficient.

Statement (2) gives information about the G/B ratio in Group A. If the G/B ratio in Group A is 6/5, then the G/B ratio in Group B is 3/5, by the condition in the question. Coming up with cases for this statement is somewhat faster. We can do a case in which there are 6 girls, 5 boys in A, and 3 girls, 5 boys, in B. In another case, we can keep the ratios but double both numbers in Group A only. Both cases fit all the information given to us, but in the latter of the cases, the thing we are asked, the ratio of A total to the B total, has doubled. Therefore, we can't answer the question definitively with this information, and Statement (2) is insufficient.

Combining statements, we need to test cases that fit both conditions. It's for this type of circumstance that you want to keep your notes for each case organized and not erased, so that, if you have to combine statements, you don't have to redo any work. We can work from one of our cases from analyzing Statement (1) and say we have 60 girls in Group A; then we need to have 50 boys to get the G/B ratio specified by Statement (2). The G/B ratio in group B must be half, so we change those girls and boys to 30 and 50. Luckily, the total difference of 30 has been preserved, so this is a valid case:

Case III:

Group A: 60 girls, 50 boys, 110 total. G/B=6/5

Group B: 30 girls, 50 boys, 80 total. G/B= 3/5

In this case the ratio of the total in A to the total in B is . Can we generate another allowed case? If try to get a different case but keep the ratio, say by dividing by a number or multiplying by a number, that keeps the ratio but it messes the condition of Statement (1). And changing in some other way will alter the ratio. Therefore Case III is, in fact, the only possible case when the statements are combined. We therefore can know the ratio of the totals definitively, and the statements together are sufficient. The answer is (C).

If you also approached this question with analysis by cases and ran into problems, here are a few questions to ask yourself. First, were you writing down too much or too little? It can be easy to run into problems if you write nothing down, because there are a few things to keep track of and access later. Another consideration is the manner of choosing cases. Whenever you are dealing with ratios, you multiply or divide to maintain a ratio, and each act of doing so will change the sums and differences of the elements involved. Keeping that in mind will make the process of constructing cases more fluid. Lastly, consider whether you can go faster sometimes by going slower. For example, it's sensible that you might hit a situation such as Case III and stop and have to think about it. It's perfectly fine to spend 10 or 20 seconds in your final evaluation of Case III to determine the answer and it will generally be faster than trying to scribble down a bunch of further cases more quickly than you have time to digest them and see the patterns.

Again, the correct answer is (C).