NIMCET 2007 — Reasoning PYQ

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Question

NIMCET 2007 — Reasoning PYQ

NIMCET | Reasoning | 2007

In a group of $15$ people, $7$ read French, $8$ read English while $3$ of them read none of these two. How many of them read French and English both:

Choose the correct answer:

Explanation

To solve this problem clearly, we can use basic set theory principles or a conceptual breakdown of the groups.

1. Identifying the Universal Set Component

Let the total number of people in the entire group be represented as the Universal set, $n (U)$ .

$n (U) = 15$

The problem states that $3$ people do not read either of the two languages. Therefore, we find the number of people who read at least one language (the union of the two sets, $n (F \cup E)$ ) by subtracting the non-readers from the total group size:

$n (F \cup E) = Total People - People who read none$

$n (F \cup E) = 15 - 3 = 12$

This means exactly $12$ people read either French, English, or both languages.

2. Setting Up the Set Theory Formula

Let us define our individual language counts:

Number of French readers, $n (F) = 7$
Number of English readers, $n (E) = 8$

We need to find the number of people who read both languages, which is represented as the intersection of the two sets, $n (F \cap E)$ .

According to the fundamental formula of set theory:

$n (F \cup E) = n (F) + n (E) - n (F \cap E)$

3. Calculating the Intersection Value

Substitute the known values into our formula:

$12 = 7 + 8 - n (F \cap E)$

$12 = 15 - n (F \cap E)$

Now, isolate $n (F \cap E)$ by rearranging the equation:

$n (F \cap E) = 15 - 12$

$n (F \cap E) = 3$