Three Groups Problem

[AndyMichael89]

A number of individuals has been surveyed - each ownes at least one car of three differend car brands:

car brand B: owned by 45
car brand M: owned by 38
car brand P: owned by 27

Follwoing overlaps:
BM: 15
MP: 12
BP: 8

All three (BMP): 5

The question is: how many individuals have been surveyed?

answer options: 70, 75, 80, 110, 130

I am pretty sure I was calculating according to the right formula but I have neither of the results that are on offer. (I get: 65). Thanks for any help

[MartyMurray]

Hi AndyMichael89.

You didn't specify whether the two group overlaps are overlaps of only exactly two groups or the total two group overlaps, which overlaps also overlap the all three overlap.

Going with that they are exactly two group overlaps, we get the following.

Total = B + M + P - (Exactly Two Group Overlaps) - 2(All Three Overlap) =

110 - 35 - 10 = 65

Going with the idea that the two group overlaps are the total overlaps of two groups, we get the following.

Total = B + M + P -(Two Group Overlaps) + (Three Group Overlap) =

110 - 35 + 5 = 80

Since 65 is not among the answer choices and 80 is, the second must be the correct answer.

Answer: C

[AndyMichael89]

Dear Marty Murray,

thank you for your reply.
So far I have not been aware that, depending on the wording, I might need to use another formula than A + B + C -(AB + BC + AC) -(ABC).
How can I recognize that I have to add rather than substract (ABC)?
Here is the full information for the task:
The xxx Journal conducted a survey of wealthy German car owners. According to the survey all wealthy car owners owned one or more of the following three brands: BMW, Mercedes, or Porsche. Respondents' answers were grouped as follows: 45 owned BMW cars, 38 owned Mercedes cars and 27 owned Porsche cars. Of these, 15 owned both BMW and Mercedes cars, 12 owned both Mercedes and Porsche cars, 8 Owned both BMW and Porsche cars, and 5 persons owned all three types of cars. How many different individuals were surveyed?
a) 70, b) 75, c) 80, d) 110, e) 102

[MartyMurray]

Dear Marty Murray,

thank you for your reply.
So far I have not been aware that, depending on the wording, I might need to use another formula than A + B + C -(AB + BC + AC) -(ABC).
How can I recognize that I have to add rather than substract (ABC)?
Here is the full information for the task:
The xxx Journal conducted a survey of wealthy German car owners. According to the survey all wealthy car owners owned one or more of the following three brands: BMW, Mercedes, or Porsche. Respondents' answers were grouped as follows: 45 owned BMW cars, 38 owned Mercedes cars and 27 owned Porsche cars. Of these, 15 owned both BMW and Mercedes cars, 12 owned both Mercedes and Porsche cars, 8 Owned both BMW and Porsche cars, and 5 persons owned all three types of cars. How many different individuals were surveyed?
a) 70, b) 75, c) 80, d) 110, e) 102

That formula is only for exactly two group overlaps, except that actually you need to subtract ABC twice.

If the overlaps are not exactly two groups overlaps, then the two group overlaps overlap the all three overlap.

For example in this question 12 owned both Mercedes and Porsche cars.

Because the question does not specify that those 12 own ONLY Mercedes and Porsche cars, that 12 includes 5 who own all three.

So in subtracting the three two group overlaps, you subtract the all three overlap three times.

Actually you should subtract the three group overlap twice. So since it has been subtracted three times, you have to add it back once.

To tell which way to go, if the question says "exactly two" then use the formula that subtracts each of the "exactly two" overlaps once and the all three overlap twice.

If the question does not specify that the two group overlaps are "exactly" two group overlaps, then you can go with the assumption that they are not exactly two group overlaps and use the other formula, Total = A + B + C - (Two Group Overlaps) + (Three Group Overlap).

Also, here's another example, with some more explanations. https://www.beatthegmat.com/pizza-hoagie ... 85217.html

[AndyMichael89]

Got it,
Thank you so much for this comprehensive and enlightening answer!

[DavidG@VeritasPrep]

Now check out this official 3-set question in which employing the formula may not be the best approach...

https://www.beatthegmat.com/a-survey-was ... 70039.html