Animated Solution for Mathematics - Sets and Relations: A survey shows that $73\%$ of the persons working in an office like coffee, whereas $65\%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be :

Welcome, fellow traveler of the mathematical arts. Today, we are not just solving a problem about coffee and tea; we are exploring the boundaries of possibility.

Imagine you are standing in the middle of a bustling office. You have a clipboard, and you are conducting a survey. You find that $73\%$ of your colleagues are coffee enthusiasts, and $65\%$ are tea lovers.

At first glance, this seems straightforward. But as we dive deeper, we encounter a beautiful, logical puzzle: how many people like both? And more importantly, what are the limits of this overlap?

Before we touch a single variable, we must establish our stage. In any survey involving percentages, the entire population is our universal set, which we define as $n(U)=100$.

This is our absolute ceiling. No matter how you slice the data, the percentage of people who like at least one beverage—the union of our sets—cannot exceed this $100\%$ boundary. This is the first law of our reality in this problem.

Now, let us define our players. Let $C$ be the set of coffee lovers, where $n(C)=73$, and $T$ be the set of tea lovers, where $n(T)=65$.

We are looking for $x$, the percentage of people who like both, which is mathematically represented as the intersection, $n(C\cap T)=x$. To connect these, we invoke the Principle of Inclusion-Exclusion:

This formula is the heartbeat of set theory. It tells us that if we simply add the coffee lovers and the tea lovers, we are double-counting the people who like both. By subtracting $x$, we correct this double-counting and find the true union.

Substituting our values, we get $n(C\cup T)=73+65-x$, which simplifies to:

Now, we hit the exciting part: the constraints. We have two 'walls' that define the range of $x$.

The first wall is the total population. Since $n(C\cup T)\le 100$, we substitute our expression:

Rearranging this, we find $x\ge 38$. This is our lower bound. It tells us that at least $38\%$ of the office must like both, or else the total percentage of people who like at least one drink would exceed $100\%$.

But there is a second wall. The intersection $n(C\cap T)$ cannot be larger than the individual sets themselves. Specifically, it cannot exceed the smaller of the two sets.

Since $n(C)=73$ and $n(T)=65$, the intersection $x$ must satisfy $x\le \min (73,65)$, which means $x\le 65$. This is our upper bound.

We have arrived at our destination. The valid range for $x$ is the interval $[38,65]$. Any value outside this range is a logical impossibility.

Looking at our options, we see $63$, $36$, $38$, and $54$. We are looking for the value that $x$ cannot be.

Since $36$ is strictly less than $38$, it falls outside our valid range. It is the impossible value. By understanding the constraints of the system, we have not just solved for $x$; we have mapped the entire landscape of possibility.