In a group of 300 people, 150 speak Hindi and 200 can speak English. How many can speak both Hindi and English?
50
Finding the Number of Hindi and English Speakers
This question asks us to find how many people in a group can speak both Hindi and English. We are given the total number of people, the number who speak Hindi, and the number who speak English. This type of problem can be solved using the principles of set theory, specifically the inclusion-exclusion principle.
Understanding the Problem with Set Theory
Let's define sets for the people who speak each language:
- Let H be the set of people who speak Hindi.
- Let E be the set of people who speak English.
We are given the following information:
- Total number of people in the group, which represents the union of the two sets, assuming everyone speaks at least one language: $|H \cup E| = 300$.
- Number of people who speak Hindi: $|H| = 150$.
- Number of people who speak English: $|E| = 200$.
We need to find the number of people who speak both Hindi and English. This corresponds to the intersection of the two sets, $|H \cap E|$.
Applying the Inclusion-Exclusion Principle
The inclusion-exclusion principle for two sets states that the number of elements in the union of two sets is equal to the sum of the number of elements in each set minus the number of elements in their intersection. The formula is:
\(|H \cup E| = |H| + |E| - |H \cap E|\)
Step-by-Step Calculation for Hindi and English Speakers
We can plug the given values into the formula:
\(300 = 150 + 200 - |H \cap E|\)
Now, we need to solve for $|H \cap E|$, which represents the number of people who speak both Hindi and English.
\(300 = 350 - |H \cap E|\)
To isolate $|H \cap E|$, we can rearrange the equation:
\(|H \cap E| = 350 - 300\)
\(|H \cap E| = 50\)
Therefore, 50 people can speak both Hindi and English.
Summary of the Solution
Using the inclusion-exclusion principle on the given data for Hindi and English speakers:
- Total people (|H ∪ E|) = 300
- Hindi speakers (|H|) = 150
- English speakers (|E|) = 200
- People speaking both (|H ∩ E|) = ?
Formula: $|H \cup E| = |H| + |E| - |H \cap E|$
Calculation: \(300 = 150 + 200 - |H \cap E|\)
\(300 = 350 - |H \cap E|\)
\(|H \cap E| = 350 - 300\)
\(|H \cap E| = 50\)
The number of people who speak both Hindi and English is 50. This result matches one of the provided options, confirming our calculation for the Hindi and English speakers.
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