Analyzing Student Language Preferences

This question asks us to determine the number of students who like only one language based on the provided data about a class of 100 students.

Understanding the Given Information

We are given the following breakdown of language preferences among the 100 students:

• Students who like only Spanish: 20
• Students who like only French: 30
• Students who like only German: 15
• Students who like both Spanish and French: 10
• Students who like both German and French: 5
• The remaining students like only English.

Calculating Students Who Like Only English

The total number of students in the class is 100. We know the counts for all groups except those who like only English. To find the number of students who like only English, we need to subtract the sum of students in all other categories from the total number of students.

Sum of students in known categories (excluding only English):

$$ \text{Only Spanish} + \text{Only French} + \text{Only German} + \text{Spanish and French} + \text{German and French} $$

$$ 20 + 30 + 15 + 10 + 5 = 80 $$

The number of students who like only English is the total number of students minus the sum calculated above:

$$ \text{Total Students} - \text{Students in other categories} $$

$$ 100 - 80 = 20 $$

So, 20 students like only English.

Identifying Students Who Like Only One Language

The question asks for the number of students who like only one language. Based on the given information, the groups of students who like only one language are:

• Students who like only Spanish
• Students who like only French
• Students who like only German
• Students who like only English (as calculated above)

Calculating Total Students Who Like Only One Language

To find the total number of students who like only one language, we sum the counts of these four groups:

$$ \text{Only Spanish} + \text{Only French} + \text{Only German} + \text{Only English} $$

$$ 20 + 30 + 15 + 20 $$

Let's add these numbers:

• $20 + 30 = 50$
• $50 + 15 = 65$
• $65 + 20 = 85$

The total number of students who like only one language is 85.

Based on our calculation, 85 students like only one language.

Revision Table: Language Preference Analysis

Additional Information: Set Theory Concepts

Problems like this can be visualized using Venn diagrams, which are helpful in set theory. Each circle in a Venn diagram represents a set (in this case, students who like a particular language). The overlapping areas represent students who like more than one language.

• The region unique to one circle represents students who like only that language.
• The overlapping region between two circles represents students who like both those languages.
• If there were students who liked three languages, there would be a region where all three circles overlap.
• Students who like none of the listed languages (but perhaps another) or who like languages not mentioned might be represented outside the circles, but within the universal set (the total class). In this problem, 'only English' acts somewhat like this for the Spanish, French, German sets.

This problem is relatively simple as it gives counts for 'only' categories and specific 'both' categories without complex overlaps (like three languages). The key is careful calculation of the 'remaining' group.