To find out how many employees eat breakfast, lunch, or both at the office, we can use the principle of inclusion-exclusion. Let's break down the information:
- Let \( A \) be the set of employees who eat breakfast at the office.
- Let \( B \) be the set of employees who eat lunch at the office.
From the survey:
- Number of employees in \( A \) (breakfast) = 20
- Number of employees in \( B \) (lunch) = 50
- Number of employees in both \( A \) and \( B \) (both breakfast and lunch) = 15
We want to find the number of employees who eat either breakfast or lunch or both, which is represented by the union of sets \( A \) and \( B \) (denoted as \( A \cup B \)).
The formula for the union of two sets using inclusion-exclusion is:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Where:
- \( |A| \) is the number of employees who eat breakfast (20)
- \( |B| \) is the number of employees who eat lunch (50)
- \( |A \cap B| \) is the number of employees who eat both (15)
Substituting the values into the formula:
\[ |A \cup B| = 20 + 50 - 15 = 55 \]
This means that 55 employees eat breakfast or lunch or both at the office.
Now, to express this as a simplified fraction of the total number of employees (80):
\[ \text{Fraction} = \frac{|A \cup B|}{\text{Total Employees}} = \frac{55}{80} \]
To simplify the fraction \( \frac{55}{80} \):
- Find the greatest common divisor (GCD) of 55 and 80. The GCD is 5.
- Divide the numerator and the denominator by their GCD:
\[ \frac{55 \div 5}{80 \div 5} = \frac{11}{16} \]
So, the final answer is that \(\frac{11}{16}\) of the employees eat breakfast or lunch at the office.