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Step-by-Step Explanation
Step 1: Understand the problem
We are told that 73% of the people in an office like coffee (C), and 65% like tea (T). We denote by x the percentage of people who like both coffee and tea. We want to find which among the given options (63, 36, 54, 38) cannot be a valid value for x.
Step 2: Use the formula for the union of two sets
From set theory, for two sets C (coffee likers) and T (tea likers):
$ n(C \cup T) = n(C) + n(T) - n(C \cap T). $
The question gives the percentages of people who like coffee ($n(C) = 73$) and tea ($n(T) = 65$). Let us denote $n(C \cap T) = x$.
Step 3: Apply the fact that the total cannot exceed 100%
Because the maximum possible percentage of people is 100% (i.e., everyone in the office), we have:
$ n(C) + n(T) - n(C \cap T) \le 100. $
Substituting the known values:
$ 73 + 65 - x \le 100. $
This simplifies to:
$ 138 - x \le 100 \quad \Rightarrow \quad x \ge 38.
Step 4: Check any upper bounds
Additionally, the intersection cannot exceed the number of people liking coffee alone or tea alone. So, we must have:
$ x \le n(C) = 73, \quad x \le n(T) = 65.
Hence, combining these, we get:
$ 38 \le x \le 65.
Step 5: Identify which option is not possible
From the derived range $38 \le x \le 65$, we see that 36 is outside this range. Therefore, $x$ cannot be 36.
