marcusaurelius
Great, thank you. When I was reading through the problem I think I assumed they were asking for the number of males rather than the percentage.
I think you must have! Sorry if this repeats what others have said, but here's a solution that illustrates that pitfall, with a little additional commentary on Weighted Averages.
With just the info from the question stem, the overlapping sets chart looks like this. The question is “What is m/Total?” (or, “What is m/(m+f)?”)
------Have-----Haven’t----Total
M----0.72m-----0.28m-----m
F-----0.8f--------0.2f-------f
Total------------------------m+f
(1) Tells us that m+f = 840, but nothing about how the 840 splits between m and f.
(2) If 75% of students have applied to college, let’s put that expression in the chart:
---------Have--------Haven’t-------Total
M-------0.72m--------0.28m--------m
F--------0.8f-----------0.2f----------f
Total--0.75(m+f)---0.25(m+f)----m+f
We can pull two new equations out of the vertical Have and Haven’t columns:
0.72m + 0.8f = 0.75(m+f)
0.28m + 0.2f = 0.25(m+f)
Multiply everything by 100 to avoid dealing with decimals.
72m + 80f = 75(m+f)
28m + 20f = 25(m+f)
Move all m’s and f’s to one side:
-3m + 5f = 0
3m + -5f = 0
When we add the equations we get 0=0. Yikes! This is why you would conclude that (2) is insufficient!
All the math we’ve done so far is right, but there are two things we have overlooked:
a. We haven’t made full use of the fact that m and f are tightly related to one another, because there’s not a third gender (in this problem): m = total –f and f = total –m.
b. We were trying to solve for m and f. But our question is really about m/total.
For combo problems (where you are solving for not one variable, but an expression with 2 or more), the “trick” is to solve for the combo as directly as you can.So let’s back up to this:
---------Have--------Haven’t-------Total
M-------0.72m--------0.28m--------m
F--------0.8f-----------0.2f----------f
Total--0.75(m+f)---0.25(m+f)----m+f
Pull out that vertical sum from the “Have” column, and rewrite the f’s in terms of m.
0.72m +0.8f = 0.75(Total students)
0.72m + 0.8(Total students – m) = 0.75(Total students)
72m + 80(Total students) – 80m = 75(Total students)
5(Total students) = 8m
5/8 = m/Total students
5/8 of the students are male.
As others have noted, there is a “weighted average” insight we could have exploited. The “weighted average” formula here is:
(Fraction of the class that is male)*(male application rate) + (Fraction of the class that is female)*(female application rate = overall application rate.
Here’s another post I wrote on the subject:
weighted-average-problems-93260.html#p717555