This is an Overlapping Sets problem that contains an increased level of difficulty because the variable x is used to represent two completely different things (a number of students and a percent of some other group of students). Keep in mind that x can represent two different things, but is the same number in both cases.
An overlapping sets problem in which everything in the set can be categorized as being a member of one of two groups and also one of two other groups (for instance, everyone is a man or woman and also a junior or a senior, or all the vehicles are either cars or trucks and are also manual or automatic transmission) fits well in a type of chart called a double-set matrix.
Now, let’s place the numbers from the problem into the chart. The overall total is x. The total of honors students is 36. The number of males in the honors program is 15. We are also told that x% of the 35 female students are in the honors program. So, [(x/100)•35] can be placed into the chart.
Since this is an additive chart (all rows and columns can be added), let’s sum the top row. We'll solve the resulting equation for x:
Now that we have x, we can fill in the rest of the chart. The number of female honors students, [(x/100)•35] = 60% of 35 = 21. The total is simply 60. Before we unnecessarily complete the entire chart, however, let’s determine which information we actually need.
MANY people misread the final question in problems like this one. The question "What fraction of all the non-honors students are female?" means that all the "non-honors" students go on the bottom and the "female non-honors" students on the top. (Note that the sentence pattern "What fraction of m is n?" always means n/m, not m/n.) Let’s indicate with bold text the boxes we want (on paper, you could circle these boxes).
Subtract 36 from 60 to get 24. Subtract 21 from 35 to get 14.
The answer is 14/24 or 7/12.
The correct answer is B.
Very interesting method. I'll have to study it further to see whether it really saves time.