Understanding the Student Language Preferences

The question provides us with data about the language preferences of 100 students in a class. We are given specific numbers for students who like only certain languages and students who like combinations of languages. The goal is to find out the total number of students who like French.

Let's break down the given information:

• Total number of students: 100
• 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
• Remaining students like only English.

Calculating Students Who Like French

To find the total number of students who like French, we need to consider all the groups of students where French is mentioned as a liked language. These groups are:

1. Students who like only French.
2. Students who like both Spanish and French.
3. Students who like both German and French.

Let's look at the numbers for each of these categories:

The total number of students who like French is the sum of students in these categories.

Total students who like French = (Only French) + (Spanish and French) + (German and French)

Total students who like French = $$30 + 10 + 5$$

Total students who like French = $$45$$

Therefore, 45 students like French.

Step-by-Step Solution

1. Identify all groups of students from the given data who like French. These are students who like only French, students who like Spanish and French, and students who like German and French.
2. Note the number of students in each of these identified groups.
3. Add the number of students from all these groups together to get the total number of students who like French.
4. Sum = $$30 (only\, French) + 10 (Spanish\, and\, French) + 5 (German\, and\, French) = 45$$

The total number of students who like French is 45.

Revision Table: Summary of Language Preferences

Additional Information: Understanding Survey Data

Problems like this, involving overlapping groups based on preferences or characteristics, are common in data analysis and set theory.

• Survey Data: This question uses data collected from a survey about student preferences. Analyzing survey data helps understand demographics, opinions, or, in this case, language learning interests.
• Set Theory Basics: The groups of students liking different languages can be thought of as sets. The question asks for the size of the set of students who like French, which includes elements belonging to the "only French" set, the "Spanish and French" set (intersection of Spanish and French sets), and the "German and French" set (intersection of German and French sets).
• Venn Diagrams: For problems involving 2 or 3 overlapping groups, a Venn diagram can be a very helpful visual tool. Each circle represents a language (e.g., Spanish, French, German). The overlapping regions represent students who like combinations of languages. The areas outside the circles but inside the universal set (all students) represent students who like none of these specific languages (in this case, only English).
• "Only" vs. "And": Pay close attention to wording. "Only French" means students who like French and no other language from the list. "Spanish and French" means students who like both, but they might also like German (though the data structure here implies these are distinct groups). For this specific problem structure, the numbers are given for specific non-overlapping regions (only Spanish, only French, only German, Spanish and French, German and French, only English), making the calculation straightforward addition of the relevant regions. If the question provided numbers for total students liking Spanish, total liking French, etc., along with intersections, we would use inclusion-exclusion principles or Venn diagrams more directly.

Understanding how to interpret such data is crucial for solving quantitative reasoning problems.