Flawed Math Problem? Veritas Prep CAT

[sunman]

Question: Of the 100 athletes in a soccer club, 40 play defense and 70 play midfield. If at least 20 of the athletes play neither midfield nor defense, the number of athletes playing both midfield and defense could be any number between:

A: 10 to 20
B: 10 to 40
C: 30 to 40
D: 30 to 70
E: 40 to 70

I got the question wrong, got caught up in the time crunch and picked "D". The answer is actually C. When I went back to review my answers, I couldn't figure out how they could possibly have 40 players playing both positions??

If you have 100 bodies, 20 play neither, so that leaves 80 bodies playing Midfield only, Defense only, or both Midfield and Defense.

Now:

30 players (Both - Mid and Def) + 10 Def only (since it must add up to 40) + 40 MF only (since it must add up to 70) = 80 bodies.

If there are 40 players who play both ways:

40 (both mid and def) + 0 def only (must add up to 40, and all are two position players) + 30 MF only (must add up to 70) = 70 bodies! It doesn't add up. It doesn't work.

Am I missing something here? Please advise.

[Bill@VeritasPrep]

Let's start with the general Venn diagram equation:

Total = Defense + Midfield - Both + Neither.

Here's what we know:

T=100.
D=40.
M=70.
B=unknown
N=unknown

Plugging in gives us:

100 = 40 + 70 - B + N

100 = 110 - B + N (let's isolate B since we're solving for possible values of B)

B = N + 10.

We know that N has to be at least 20. If N is exactly 20, then:

B = 20 + 10

B = 30

There's our minimum value. From this, we can eliminate A, B, and E. Now we just have to decide if the maximum value is 40 (C) or 70 (D). Since a total of 40 athletes play defense, we can't have 70 athletes play both defense and midfield. Thus, the correct answer must be C.

[sunman]

Let's start with the general Venn diagram equation:

Total = Defense + Midfield - Both + Neither.

Here's what we know:

T=100.
D=40.
M=70.
B=unknown
N=unknown

Plugging in gives us:

100 = 40 + 70 - B + N

100 = 110 - B + N (let's isolate B since we're solving for possible values of B)

B = N + 10.

We know that N has to be at least 20. If N is exactly 20, then:

B = 20 + 10

B = 30

There's our minimum value. From this, we can eliminate A, B, and E. Now we just have to decide if the maximum value is 40 (C) or 70 (D). Since a total of 40 athletes play defense, we can't have 70 athletes play both defense and midfield. Thus, the correct answer must be C.

I am an idiot.

The question says "at least 20", not "20".

I'm good now.

Thanks Bill.

[Bill@VeritasPrep]

No worries. We all have those moments (sometimes more often than we'd like to admit ) Venn problems are some of the easiest to misinterpret.

[LalaB]

total=100
Neither >=20

D=40
M=70

100=40+70+Neither-Both
Both=10+Neither
lets take N=20 (the min. value), then Both =10+20=30

the max value of both is 40, since there at most 40 defence players

the answ is C (30-40)

[sunman]

total=100
Neither >=20

D=40
M=70

100=40+70+Neither-Both
Both=10+Neither
lets take N=20 (the min. value), then Both =10+20=30

the max value of both is 40, since there at most 40 defence players

the answ is C (30-40)

Thanks, but I just misread the question, that's all.