In a group of 500 people, 350 people can speak English, and 400 people can speak French. Find how many people can speak both languages.

Analysis

This is a classic set theory problem involving two sets. We are given the total number of people (the universal set) and the number of people in two overlapping categories (English speakers and French speakers). We need to find the intersection of these two sets (those who speak both).

Concept

We use the principle of inclusion-exclusion for two sets:

Where:

• $n(A \cup B)$ is the total number of people (Universal set).
• $n(A)$ is the number of people who speak English.
• $n(B)$ is the number of people who speak French.
• $n(A \cap B)$ is the number of people who speak both languages.

Solution

Step 1: Identify the given values.

• Total number of people, $n(A \cup B) = 500$
• Number of English speakers, $n(E) = 350$
• Number of French speakers, $n(F) = 400$

Step 2: Apply the formula.

Step 3: Simplify the equation.

Step 4: Solve for $n(E \cap F)$.

The number of people who speak both languages is 250.

Exam Strategy

When dealing with set theory questions in JAMB, always check if the total given is the sum of the individual sets. If the sum of the individual sets ($350 + 400 = 750$) is greater than the total number of people ($500$), the difference ($750 - 500 = 250$) represents the people who were counted twice because they belong to both groups. This is a quick way to verify your answer!