Workers are grouped by then area of experise and are placed on at least one team. There are 20 workders on the Marketing team, 30 on the Sales team, and 40 on the Vision team. Five workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision terms, 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?

Answer

To solve the problem of finding the total number of workers across the Marketing, Sales, and Vision teams, we can use the principle of inclusion-exclusion. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Sets
Let:
- \( P_M \) = Number of workers in the Marketing team = 20
- \( P_S \) = Number of workers in the Sales team = 30
- \( P_V \) = Number of workers in the Vision team = 40

### Step 2: Define the Intersections
From the problem, we know:
- \( P_{M \cap S} \) = Number of workers in both Marketing and Sales = 5
- \( P_{S \cap V} \) = Number of workers in both Sales and Vision = 6
- \( P_{M \cap V} \) = Number of workers in both Marketing and Vision = 9
- \( P_{M \cap S \cap V} \) = Number of workers in all three teams = 4

### Step 3: Apply the Inclusion-Exclusion Principle
The formula for the total number of workers in at least one of the teams is given by:

\[ P_{M \cup S \cup V} = P_M + P_S + P_V - P_{M \cap S} - P_{S \cap V} - P_{M \cap V} + P_{M \cap S \cap V} \]

### Step 4: Substitute the Values
Now, substituting the known values into the formula:

\[ P_{M \cup S \cup V} = 20 + 30 + 40 - 5 - 6 - 9 + 4 \]

### Step 5: Calculate
Calculating step-by-step:

1. Sum of individual teams:
\[ 20 + 30 + 40 = 90 \]

2. Sum of intersections (subtract):
\[ 90 - 5 - 6 - 9 = 90 - 20 = 70 \]

3. Add back the intersection of all three teams:
\[ 70 + 4 = 74 \]

### Final Answer
Thus, the total number of workers is:
\[ \boxed{74} \]