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 786 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)