40% persons read newspaper X and 50% persons read newspaper Y. 10% persons read both newspapers. How many persons do not read any of the newspapers ?
20%
Newspaper Readership Problem Explained
This problem involves calculating the percentage of people who do not read any newspapers, given the percentages of people who read specific newspapers and those who read both.
Understanding the Given Information
We are provided with the following details:
- Percentage of persons who read newspaper X: 40%
- Percentage of persons who read newspaper Y: 50%
- Percentage of persons who read both newspaper X and newspaper Y: 10%
Applying the Principle of Inclusion-Exclusion
To find the percentage of people who read at least one newspaper (either X or Y or both), we use the principle of inclusion-exclusion from set theory. The formula is:
$P(X \cup Y) = P(X) + P(Y) - P(X \cap Y)$
Where:
- $P(X \cup Y)$ represents the percentage of people who read newspaper X OR newspaper Y (or both).
- $P(X)$ represents the percentage of people who read newspaper X.
- $P(Y)$ represents the percentage of people who read newspaper Y.
- $P(X \cap Y)$ represents the percentage of people who read both newspaper X AND newspaper Y.
Step-by-Step Calculation
First, let's calculate the percentage of people who read at least one newspaper:
- Substitute the given values into the formula:
$P(X \cup Y) = 40\% + 50\% - 10\%$
- Perform the addition:
$P(X \cup Y) = 90\% - 10\%$
- Perform the subtraction to find the total percentage reading at least one newspaper:
$P(X \cup Y) = 80\%$
Calculating Persons Who Read No Newspapers
The total percentage of people surveyed is 100%. The percentage of people who do not read any newspaper is the total percentage minus the percentage of people who read at least one newspaper.
Percentage of persons who do not read any newspaper = $100\% - P(X \cup Y)$
Now, substitute the value we found for $P(X \cup Y)$:
Percentage of persons who do not read any newspaper = $100\% - 80\%$
Percentage of persons who do not read any newspaper = $20\%$
Final Answer Derivation
Therefore, 20% of the persons surveyed do not read either newspaper X or newspaper Y.
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