Suppose Albers Elementary School has 39 teachers and Bothel Elementary School has 98 teachers. If the total number of teachers at Albers and Bothel combined is 122, how many teachers teach at both schools?

To solve this, we can define the problem in terms of sets and use the principle of inclusion-exclusion.

Let's define the variables:
- Let \( A \) be the number of teachers at Albers Elementary School. We know \( A = 39 \).
- Let \( B \) be the number of teachers at Bothel Elementary School. We know \( B = 98 \).
- Let \( x \) be the number of teachers who teach at both Albers and Bothel.

We are also told that the total number of teachers at Albers and Bothel combined is 122. However, since \( x \) teachers are counted in both \( A \) and \( B \), the actual number of unique teachers is given by:

\[
A + B - x = 122
\]

Substitute the known values for \( A \) and \( B \):

\[
39 + 98 - x = 122
\]

Simplify the left-hand side:

\[
137 - x = 122
\]

Now, solve for \( x \):

\[
x = 137 - 122
\]

\[
x = 15
\]

Thus, 15 teachers teach at both schools.

Would you like more details or have any questions? Here are 5 related questions:

1. How would the problem change if the total number of teachers were 130 instead of 122?
2. What is the formula for calculating the number of elements in the union of two sets?
3. How would you solve this if three schools were involved instead of two?
4. Can you represent this problem using a Venn diagram?
5. How do you adjust the solution if you know some teachers only work part-time at both schools?

**Tip**: When dealing with overlapping groups, always remember to subtract the overlap when calculating the total number of unique elements.

Learn how to calculate the number of teachers who work at both Albers and Bothel Elementary Schools using the Inclusion-Exclusion Principle. This step-by-step guide simplifies the problem with clear explanations and key formulas. Suitable for Grades 6-8.

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