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# There are 40 people, 28 invested in A, 16 invested in B, and 18 invest

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Math Revolution GMAT Instructor
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There are 40 people, 28 invested in A, 16 invested in B, and 18 invest  [#permalink]

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17 Aug 2017, 00:55
00:00

Difficulty:

55% (hard)

Question Stats:

63% (02:29) correct 37% (02:55) wrong based on 73 sessions

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There are 40 people, 28 invested in A, 16 invested in B, and 18 invested in C. Among 10 of the people who invested in both A and C, 3 also invested in B. All people who invested in B also invested in A. How many people did not invest in any of the three investments?

A. 2 B. 3 C. 4 D. 5 E. 6

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There are 40 people, 28 invested in A, 16 invested in B, and 18 invest  [#permalink]

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17 Aug 2017, 07:02
1
MathRevolution wrote:
There are 40 people, 28 invested in A, 16 invested in B, and 18 invested in C. Among 10 of the people who invested in both A and C, 3 also invested in B. All people who invested in B also invested in A. How many people did not invest in any of the three investments?

A. 2 B. 3 C. 4 D. 5 E. 6

Attachment:

ABCvenn.png [ 7.7 KiB | Viewed 1180 times ]

I used a Venn diagram, too, but I got A only = 5. kunalsinghNS - how did you arrive at A only = 2?

GIVEN
40 people total
A = 28. B = 16. C = 18
A + C = 10
-------> and A + C + B = 3
All people who invested in B also invested in A
---If B, then A
---B only = 0

CALCULATIONS

Per diagram, I entered numbers into color spaces in this order: blue, pink, green, yellow, gray, purple, white

1. Take most restrictive condition first. Blue. Three circles overlap.

"Among 10 of the people who invested in both A and C, 3 also invested in B." That means

A + C + B = 3. Put 3 in blue*

2. Next pink, which is A + C, which follows from #1.

A + C = 10.
Careful - don't put 10 in pink. Blue already has 3 of those who are also in A + C.
A + B + C = 3, where "B also" = 3
10 - 3 = 7. Put 7 in pink

3. Green. Another restrictive condition: All B are A

B = 16
So B + A total must = 16
But blue has 3 of those 16. 16 - 3 = 13 for the rest of A + B (green). Put 13 in green

4. Yellow and gray. Finish B. B = 16. Yellow is B only. Gray is B + C.

All 16 B are accounted for.
Put ZERO in yellow.
Put ZERO in gray

5. Purple, C only. Finish C. C = 18

Areas surrounding C are pink, blue, gray. All have numbers now. Add their numbers: 7 + 3 + 0 = 10
Total 18 - 10 = 8. C only = 8. Put 8 in purple

6. White, A only. Finish A. A = 28

Areas surrounding A are pink, blue, green. All have numbers now. Add their numbers: 7 + 3 + 13 = 23
A only: 28 - 23 = 5. A only = 5. Put 5 in white

7. Now add all numbers, from every color, to get total that have "fallen" into the Venn diagram. Working outside to inside, clockwise from A only

5 + 0 + 8 + 13 + 0 + 7 + 3 = 36 invested the three investments A, B, C

8. There are 40 people total. (40 - 36) = 4 who did not invest in any of the three investments

*Blue will affect everything. Check its numbers and contents (A + B + C) for overlap with surrounding intersecting spaces (green, pink, gray: A + B, B + C, and C + A, respectively)
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There are 40 people, 28 invested in A, 16 invested in B, and 18 invest  [#permalink]

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17 Aug 2017, 08:53
1
Given data:
P(Total) = 40
P(A) = 28
P(B) = 16
P(C) = 18

From the question stem, we can get the following information
P(all 3) = 3
P(A U C) = 7(10 - 3)
P(A U B) = 13(16 - 3)
P(B U C) = 0

P(Total) = P(A) + P(B) + P(C) - P(Only 2) - 2*P(All 3) + P(Neither)
40 = 28 + 16 + 18 -(7 + 13) - 2(3) + P(Neither)
40 = 28 + 34 - 26 + P(Neither)
40 = 36 + P(Neither)
P(Neither) = 4

Therefore, the number of people who didn't invest in any of the three investments is 4(Option C)
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Re: There are 40 people, 28 invested in A, 16 invested in B, and 18 invest  [#permalink]

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17 Aug 2017, 04:39
Used ven diagram to solve this one :

Only C = 8
Intersection of A,B and C = 3
Intersection of A and C = 7
" of B and A = 16

Only A = 2

hence total combined it comes down to 36

Subtracting it from 40 we get "4" hence, answer = "C"
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Re: There are 40 people, 28 invested in A, 16 invested in B, and 18 invest  [#permalink]

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17 Aug 2017, 07:18
From Venn diagram Only C= 8
Since complete B is inside A, we need not it, as we count A

So 28+8 = 36
40-36=4
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GMAT 1: 760 Q51 V42
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Re: There are 40 people, 28 invested in A, 16 invested in B, and 18 invest  [#permalink]

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20 Aug 2017, 19:12
Attachment:

8.21.png [ 15.32 KiB | Viewed 1006 times ]

Ans: C
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Re: There are 40 people, 28 invested in A, 16 invested in B, and 18 invest   [#permalink] 20 Aug 2017, 19:12
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