A Survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x%

Q: Why can't $x$ be 37? If I add 63 and 76, I get 139, and 139 - 37 is 102. Is that not allowed?

Exactly! You hit the nail on the head. If the union is 102%, that exceeds our total population of 100%. Since we cannot have more than 100% of the people, $x$ must be large enough to bring that total down to 100 or less.

Q: Why is the upper bound 63? Why not 76?

Great question. The intersection $x$ represents people who read *both* A and B. You cannot have more people reading both than the total number of people reading A. Since only 63% read A, the intersection cannot exceed 63%.

Q: Is it possible for $x$ to be 0?

Mathematically, if we only had the union constraint, maybe. But here, the lower bound is 39. If $x$ were 0, the union would be 139%, which is impossible.

Question

A Survey shows that $63\%$ of the people in a city read newspaper A whereas $76\%$ read newspaper B. If $x\%$ of the people read both the newspapers, then a possible value of $x$ can be :

Options:

(A) 37

(B) 55

(C) 29

(D) 65

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in the middle of a bustling city, holding a survey about newspaper habits. You have two groups: those who read newspaper $A$ and those who read newspaper $B$.

This is the classic setup for a Venn diagram, a beautiful tool that helps us visualize how different groups interact. Let the total population be our Universal Set $U$, where $n(U) = 100\%$.

We are given that $n(A) = 63\%$ and $n(B) = 76\%$. The intersection, $A \cap B$, represents the people who read both, which we call $x\%$.

### The Principle of Inclusion-Exclusion

When we try to find the total number of people who read at least one newspaper, we use the Principle of Inclusion-Exclusion:

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

Why do we subtract the intersection? Because when we add the readers of $A$ and the readers of $B$, we count the people who read *both* twice!

Subtracting $n(A \cap B)$ corrects this double-counting. So, we have:

$$n(A \cup B) = 63 + 76 - x = 139 - x$$

### The Constraints of Reality

Now, here is the physical reality: the percentage of people who read at least one newspaper cannot exceed the total population. Therefore, $n(A \cup B) \le 100$.

Substituting our expression, we get:

$$139 - x \le 100$$

Rearranging this, we find $x \ge 139 - 100$, which means $x \ge 39$. This gives us our lower bound.

But we are not done! The intersection $x$ cannot be larger than the individual sets themselves. Since $x \le n(A)$ and $x \le n(B)$, we have $x \le 63$ and $x \le 76$.

The stricter constraint is $x \le 63$.

### The Final Range

Combining these, we find that the range for the intersection is:

$$39 \le x \le 63$$

Looking at our options, 37, 29, and 65 all fall outside this range. Only **55** fits perfectly.

It is a beautiful example of how simple constraints can define the boundaries of possibility!

Animated Solution for Mathematics - Sets and Relations: A Survey shows that $63%$ of the people in a city read newspaper A whereas $76%$ read newspaper B. If $x %$ of the people read both the newspapers, then a possible value of $x$ can be :

Select Answer:

Visualized Solution

Visualizing the Survey Data

Defining the Given Percentages

The Principle of Inclusion-Exclusion

Setting the Upper Bound for Union

Solving for the Lower Bound of $x$

Setting the Upper Bound for $x$

Determining the Possible Range

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Principle of Inclusion-Exclusion

The Constraints of Reality

The Final Range

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