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 1095 times ]
I used a Venn diagram, too, but I got
A only = 5.
kunalsinghNS - how did you arrive at A only = 2?
GIVEN40 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
CALCULATIONSPer 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 white7. 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
Answer C*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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