In a teachers staff room, there are three subject teachers, English, French, and German
27 teach German, and 9 teach German only and 18 teach German plus one or the other two subjects, 29 teach French , 7 teach French only and  22 teach French plus one or the other two subjects, 30 teach English, 11 teach English only and 19 teach English plus one or the other two subjects 
Find the total number of teachers teaching the languages

To solve this problem, we need to determine the total number of teachers in the staff room by accounting for those who teach different combinations of English, French, and German.

Given:
- Total teaching German (G) = 27
  - Teaching German only = 9
  - Teaching German plus one or more other languages = 18
- Total teaching French (F) = 29
  - Teaching French only = 7
  - Teaching French plus one or more other languages = 22
- Total teaching English (E) = 30
  - Teaching English only = 11
  - Teaching English plus one or more other languages = 19

### Step 1: Define Variables
- \( x \) = Number of teachers teaching all three languages (E, F, G).
- \( a \) = Number of teachers teaching both English and French but not German.
- \( b \) = Number of teachers teaching both French and German but not English.
- \( c \) = Number of teachers teaching both English and German but not French.

### Step 2: Set Up the Equations
From the given information:

1. **For German:**
   \[
   9 + b + c + x = 27
   \]
   \[
   b + c + x = 18 \quad \text{(subtracting German only)}
   \]

2. **For French:**
   \[
   7 + a + b + x = 29
   \]
   \[
   a + b + x = 22 \quad \text{(subtracting French only)}
   \]

3. **For English:**
   \[
   11 + a + c + x = 30
   \]
   \[
   a + c + x = 19 \quad \text{(subtracting English only)}
   \]

### Step 3: Solve the System of Equations
We now solve these three simultaneous equations:

1. \( b + c + x = 18 \)
2. \( a + b + x = 22 \)
3. \( a + c + x = 19 \)

#### Step 3.1: Subtract Equation 1 from Equation 2:
\[
(a + b + x) - (b + c + x) = 22 - 18
\]
\[
a - c = 4 \quad \text{(Equation 4)}
\]

#### Step 3.2: Subtract Equation 3 from Equation 2:
\[
(a + b + x) - (a + c + x) = 22 - 19
\]
\[
b - c = 3 \quad \text{(Equation 5)}
\]

#### Step 3.3: Substitute \( c = a - 4 \) (from Equation 4) into Equation 5:
\[
b - (a - 4) = 3
\]
\[
b - a + 4 = 3
\]
\[
b = a - 1 \quad \text{(Equation 6)}
\]

### Step 4: Substitute Equation 6 into the Original Equations

#### Substituting into Equation 2:
\[
a + (a - 1) + x = 22
\]
\[
2a - 1 + x = 22
\]
\[
2a + x = 23 \quad \text{(Equation 7)}
\]

#### Substituting into Equation 1:
\[
(a - 1) + (a - 4) + x = 18
\]
\[
2a - 5 + x = 18
\]
\[
2a + x = 23 \quad \text{(Same as Equation 7)}
\]

So, the equations are consistent.

### Step 5: Solve for Total Teachers
From the equations, let's sum up all the unique categories:

\[
\text{Total teachers} = (\text{only English}) + (\text{only French}) + (\text{only German}) + a + b + c + x
\]
\[
= 11 + 7 + 9 + (a + b + c + x)
\]
Using Equation 7, \(2a + x = 23\). So:
\[
a + b + c + x = 18 + x
\]
And substituting \( x = 0 \) (since \(2a + x = 23\) is consistent with the equations):
\[
\text{Total teachers} = 11 + 7 + 9 + 23 = 50
\]

So, **50 teachers** are in the staff room.

Would you like to delve into more details or explore related problems? Here are five questions for further exploration:

1. What if one more teacher taught all three languages? How would that affect the total number of teachers?
2. How would the calculation change if some teachers only taught one language?
3. Can you solve a similar problem with a different combination of subjects?
4. How would you determine the number of teachers teaching only two languages?
5. What if there were a fourth language involved?

**Tip:** Drawing a Venn diagram can be a helpful way to visualize and solve problems involving overlapping sets.

In a teachers staff room, there are three subject teachers, English, French, and German. 27 teach German, and 9 teach German only and 18 teach German plus one or the other two subjects, 29 teach French, 7 teach French only and 22 teach French plus one or the other two subjects, 30 teach English, 11 teach English only and 19 teach English plus one or the other two subjects. Find the total number of teachers teaching the languages.

Explore how to solve a set theory problem involving teachers who teach English, French, and German. This problem uses Venn diagrams and systems of equations to find the total number of teachers. Suitable for Grades 9-12.

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