OG#178

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

s = sweaty palm only
v = vomiting only
d = dizziness only

svd = sweaty palm, vomitting and dizziness

sv = sweaty palm and vommiting
sd = sweaty palm and dizziness
vd = vommiting and dizziness

General formula
total = s+v+d + sv+sd+vd + svd [1]
100% = s+v+d + sv+sd+vd + svd

all s = s + sv + sd + svd
all v = v + sv + vd + svd
all d = d + sd + vd + svd

all (s+v+d) = s+v+d + sv+sd+vd + svd + sv+sd+vd + 2svd
40+30+75 = s+v+d + sv+sd+vd + svd + sv+sd+vd + 2svd
145 = s+v+d + sv+sd+vd + svd + sv+sd+vd + 2svd [2]
100 = s+v+d + sv+sd+vd + svd [1]

Subtracting equation 1 from equation 2
45 = sv+sd+vd + 2svd
45 = 35 + 2svd
svd = (45 - 35) / 2 = 5

100 = s+v+d + (sv+sd+vd) + svd
100 = s+v+d + 35 + 5 = s+v+d + 40
60 = s+v+d

60% of subjects expierence only 1 side effect
60% of 300subjects = 0.6 * 300 = 180

wied81 wrote: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 subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects?

A) 105

B) 125

C) 130

D) 180

E) 195

Here is one formula for 3 overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
Let T = 100%.
Sweaty palms = 40.
Vomiting = 30.
Dizziness = 75.
Exactly 2 of the groups = 35.
Let x = the percentage in all 3 groups.

Plugging these values into the formula, we get:
100 = 40 + 30 + 75 - 35 - 2x
100 = 110 - 2x
x=5.

Since 35% are in 2 of the groups and 5% are in all 3 groups, the percentage in exactly one of the groups = 100-35-5 = 60.
Number in exactly one of the groups = .6(300) = 180.

The correct answer is D.

Everytime This question comes to my mind -

why the common are of all the three circles in Venn diagram is not subtracted three times.

since it is shared three times among SV, SD & DV.

Hi Mitch ,

Can you please check my solution.

Total = 300

Sweaty Palm = 40/100*300 = 120
Vomiting = 30/100*300 = 90
Dizziness = 75/100*300 = 225

and exactly two = 35/100*300= 105
and let all three be x

So by using the formula T = A + B + C - (AB + AC + BC) - 2(ABC)

300=120+90+225-105-2(x)

so x= 50

now what should I do ??

Please advise and correct me if I am wrong.

Thanks