Question

Three hundred students at College \(Q\) study a foreign language. Of these, 110 of those students study French and 170 study Spanish. If at least 90 students who study a foreign language at College \(Q\) study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from A 10 to 40 B 40 to 100 C 60 to 100 D 60 to 110 E 70 to 110

Short answer

The number of students who study Spanish but not French could be any number from 60 to 100. Answer: C

Step-by-step solution

- Identify Total Students Studying a Foreign Language

We know there are 300 students at College Q studying a foreign language.

- Calculate Students Studying both French and Spanish

Let x represent the number of students studying both French and Spanish. We know the following from the problem:110 (students studying French) + 170 (students studying Spanish) - x (students studying both) + 90 (students studying neither) = 300. Therefore:110 + 170 - x + 90 = 300.

- Simplify the Equation

Simplify the provided equation:370 - x = 300 This means:x = 70.

- Determine Students Studying Only Spanish

Now, to find the students studying only Spanish, use the formula:170 (students studying Spanish) - x (students studying both French and Spanish) Substitute in the value of x:170 - 70 = 100.

- Calculate the Range

The maximum number of students studying only Spanish is 100. The minimum given by the problem is 60. Therefore, the range is from 60 to 100.

Key concepts

overlapping sets

In problems involving overlapping sets, it's crucial to understand the concept of set theory. These problems often involve calculating the numbers of groups that have some common elements. In this exercise, students studying both French and Spanish are the overlapping set.
To solve overlapping set problems, one effective approach is to use the principle of inclusion-exclusion. This principle helps ensure that we do not count any student more than once.
Here are the steps to solve such problems:

• Identify the total number of elements.
• Identify the elements in each subset.
• Subtract the overlap from these subsets to avoid counting elements twice.

In mathematical notation, if we have two sets A and B, the principle of inclusion-exclusion can be written as:
\( |A \cup B| = |A| + |B| - |A \cap B| \).
This formula allows us to find the number of elements in either one set or the other, while accounting for the overlap. Understanding this principle is fundamental when dealing with overlapping sets problems.

foreign language education

Learning a foreign language is a valuable skill that opens doors to new cultures and opportunities. At College Q, students are learning multiple languages, which is a tangible example of the benefits of a diverse education.
There's often an overlap where students study multiple languages, such as French and Spanish in this exercise. This overlap might indicate the interdisciplinary nature of foreign language education, where understanding one language can help in learning another.
Here are some benefits of studying foreign languages:

• Enhanced Mental Abilities: Improves cognitive skills and problem-solving.
• Better Career Opportunities: Multilingual individuals have access to more job prospects.
• Cultural Appreciation: Enables a deeper understanding of different cultures.

Encouraging students to study foreign languages, especially more than one, equips them with the tools they need to thrive in a globalized world.

mathematical reasoning

Mathematical reasoning is the process of thinking logically about numbers and equations to solve problems. In our exercise, students use mathematical reasoning to solve an overlapping set problem.
Here’s how mathematical reasoning is applied step-by-step:

• Identify Known Values: Start by noting the total number of students and the number studying each language.
• Set Up Equations: Create equations based on the information provided and the relationships between sets.
• Simplify: Use algebraic methods to simplify the equations and find the unknown value, such as the number of students studying both languages.
• Interpret Results: Translate the mathematical findings back to the context of the problem - in this case, determining the range of students studying only Spanish.

By applying these steps, students develop a robust skill set that includes critical thinking and problem-solving, which are essential in both academic and real-world contexts.