Problem Solving

Out of 72 track stars, 20 run hurdles, 22 long jump, and 30 sprint. If 5 people sprint and also run hurdles, 4 people run hurdles and also long jump, 3 people sprint and also long jump, and 2 people do all 3, how many track stars do not do any of the events?

A. 10
B. 12
C. 14
D. 16
E. 18

I used the formula: Total = Group 1 + Group 2 + Group 3 - Both - 2*All3 + None

72 = 20 + 22 + 30 - 12 - 2(2) + N

N = 16

Please help me understand what I'm doing wrong because that is the wrong answer. Thanks!!!

GMATFrantic wrote:Out of 72 track stars, 20 run hurdles, 22 long jump, and 30 sprint. If 5 people sprint and also run hurdles, 4 people run hurdles and also long jump, 3 people sprint and also long jump, and 2 people do all 3, how many track stars do not do any of the events?

A. 10
B. 12
C. 14
D. 16
E. 18

Draw a VENN DIAGRAM representing the following:
Total = 72
Hurdles = 20
Long jump = 22
Sprint = 30
None = ?


Complete the Venn diagram by working from the INSIDE OUT.

2 people do all 3:


5 people sprint and also run hurdles.
4 people run hurdles and also long jump.
3 people sprint and also long jump.
Subtracting from these figures the 3 people who do all 3 events, we get:


Subtracting the values in the diagram from H=20, L=22, and S=30, we get:


Subtracting the values in the diagram from Total = 72, we get:


Thus:
N=10.

The correct answer is A.

GMATFrantic wrote:Out of 72 track stars, 20 run hurdles, 22 long jump, and 30 sprint. If 5 people sprint and also run hurdles, 4 people run hurdles and also long jump, 3 people sprint and also long jump, and 2 people do all 3, how many track stars do not do any of the events?

A. 10
B. 12
C. 14
D. 16
E. 18

I used the formula: Total = Group 1 + Group 2 + Group 3 - Both - 2*All3 + None

In the formula above, both represents the number of people in EXACTLY 2 GROUPS.
But the prompt here does not give information about the number of people in exactly 2 groups.
Instead, it gives information about the number of people in AT LEAST 2 GROUPS:

5 people sprint and also run hurdles = people who do AT LEAST these 2 events = (sprint + hurdles) + (all 3 events).
4 people run hurdles and also long jump = people who do AT LEAST these 2 events = (hurdles + long jump) + (all 3 events).
3 people sprint and also long jump = people who do AT LEAST these 2 events = (sprint + long jump) + (all 3 events).

When given information about the number of people in AT LEAST 2 GROUPS, we can use the following formula:
T = G� + G₂ + G₃ - (at least 2) + (all 3) + N.

In the problem above, we get:
72 = 20 + 22 + 30 - (5 + 4 + 3) + 2 + N
72 = 62 + N
N = 10.

The correct answer is A.

GMATFrantic wrote:Out of 72 track stars, 20 run hurdles, 22 long jump, and 30 sprint. If 5 people sprint and also run hurdles, 4 people run hurdles and also long jump, 3 people sprint and also long jump, and 2 people do all 3, how many track stars do not do any of the events?

A. 10
B. 12
C. 14
D. 16
E. 18

I used the formula: Total = Group 1 + Group 2 + Group 3 - Both - 2*All3 + None

72 = 20 + 22 + 30 - 12 - 2(2) + N

N = 16

Please help me understand what I'm doing wrong because that is the wrong answer. Thanks!!!

Hi
I guess the mistake you made is all these three numbers 5, 4 and 3 include 2 as ONLY is not mentioned in the question. So infact you deducted 2 5 times instead of 2. So your answer is 6 more. In my opinion the answer should be 10.