Overlapping Sets

timewillwait wrote:1. Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?


When I was solving this question, I did:

(4+5+9+6) + (20-(5+4+9)) + (30-(5+4+6)) + (40-(9+4+6)). I got 62 as my answer.

However, the correct answer is supposed to be 74. What am I doing wrong?

Any help is appreciated! Thank you!

Here is the short terms for the variables
M - marketing, S - sales and V - vison
MS - Marketing and sales only
MSV - Marketing, Sales and vision only


7 Groups: M + S + V + MS + MV + SV + MSV = total; MSV = 4
M + S + V + MS + MV + SV + 4 = total

5 workers are on both the Marketing and Sales teams
MS + MSV = 5; MS + 4 = 5; MS = 1

6 workers are on both the sales and Vision teams
SV + MSV = 6; SV + 4 = 6; SV = 2

9 workers are on both the Marketing and Vision teams
MV + MSV = 9; MV + 4 = 9; MV = 5

20 are on the marketing team
M + MS + MV + MSV = 20; M = 20-1-5-4 = 10

30 are on the Sales team
S + SV + MS + MSV = 30; S = 30-2-1-4 = 23

40 are on the Vision team
V + SV + MV + MSV = 40; V = 40-2-5-4 = 29

Total = M + S + V + MS + MV + SV + MSV
= 10+23+29+1+5+2+4 = 74

Let me know if this helps.

timewillwait wrote:Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

Draw a VENN DIAGRAM representing the following:
20 are on the marketing team.
30 are on the Sales team.
40 are on the Vision team.

Complete the Venn diagram by working from the INSIDE OUT.

4 workers are on all three teams.


5 workers are on the Marketing and Sales teams.
6 workers are on the Sales and Vision teams.
9 workers are on the Marketing and Vision teams.
Subtracting from these figures the 4 workers who are on all 3 teams, we get:


Subtracting the values in the diagram from M=20, S=30, and V=40, we get:


How many workers are there in total?
Adding together the values in the Venn Diagram, we get:
T = 10+1+23+5+4+2+29 = 74.