Source: Official Guide for GMAT Review 2016 Problem Solving; #192

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# Of the 300 subjects who participated in

Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subject experienced exactly two of these effects, how many subjects experienced only on of these effects?

### 1 Explanation

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Of the 300 subjects
40% had sweaty palms: 0.4 * 300 = 120
30% experienced vomiting: 0.3*300 = 90
75% experienced dizziness: 0.75*300 = 225

These three groups make up 120 + 90 + 225 = 435 (while there are only 300 subjects...)

35% experienced exactly two effects: 0.35*300 = 105

Since people that experience sweaty palms and vomiting are counted in the sweaty palms group AND vomiting group, they are counted twice. In this way, people experiencing two effects are counted two times, while people experiencing all three effects are counted three times.

We can set up the following equation to find the number of people experiencing three effects:

300 subjects = 120 + 90 + 225 - 105 - 2x
300 = 330 - 2x
x = 15

We only want to count people experiencing one effect. Subtraction of all people experiencing two and three effects from the three groups leads to our answer.

Answer = 120 + 90 + 225 - 2 * 105 - 3 * 15
D

Nov 29, 2016 • Comment

Cydney Seigerman, Magoosh Tutor

Thanks for sharing your solution, Maarten! Nice work :D

Marius Scholinz

Could you please elaborate why people experiencing three effects can be represented as 2x? Thanks!

Cydney Seigerman, Magoosh Tutor

Happy to clarify, Marius :)

To solve this question, we can use the formula for 3 overlapping sets:

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

As you can see, we multiply the group that includes all three by 2.

For a more in-depth look into this equation, I recommend this GMAT Forum thread:

I hope this helps :)

Arthi Yerramilli

If the "neither" pool had been >0, how would this impact calculating the final answer? I understand that we'd add it in the first part to arrive at the number of people experiencing 3 symptoms, but to find the number of people who ONLY experienced 1 symptom, would we ignore that number (I.e., still only do Group 1 +Group 2 +Group 3 - (both) - 2*(all 3)? Thanks in advance!

David Recine, Magoosh Tutor

Great question. If-- say-- 10% of the group experienced no symptoms, that would get us a fourth group, a Group 4, which would actually be subtracted form the total of 300 int he first step. This is because the group we're really dealing with is symptom sufferers, and if not everyone in the 300 is a symptom sufferer, then we're actually working with a total number of people that is less than 300.

So in the first step, we'd have 300 - (Group 4) (Group 1) + (Group 2) + (Group 3) - (Group experiencing 2 effects) - 2x.

This would ultimately come out to:
270 = 120 + 90 + 225 - 105 - 2x
270 = 330 - 2x
x = 30.

And then in the second step, you could leave out the (Group 4) figure, although you'd now have an x value of 30 as follows:

120 + 90 + 225 - (2 * 105) - (3 * 30)