Solving A Milk And Tea Preference Survey Math Problem

Hey guys! Ever wondered how to solve a math problem that involves fractions, preferences, and a little bit of logic? Well, buckle up because we're diving into a fun survey scenario about a class of 2/3 students and their love (or dislike) for milk and tea. This is a classic example of a problem that uses set theory principles, and we're going to break it down step-by-step. So, grab your thinking caps, and let's get started!

Understanding the Problem

The crux of the matter lies in figuring out how many students don't fancy either milk or tea. To unravel this, we're going to use the information we have about the students who do like milk, those who like tea, and those who don't like milk at all. We'll also use the critical piece of data about the students who don't like either beverage. These kinds of problems are super common in math because they help us understand how to organize information and find missing pieces of the puzzle. Think of it like being a detective, but with numbers and fractions!

Step-by-Step Solution

1. Defining the Sets

First, let's define our sets. A set in math is just a collection of things. In our case, these "things" are students. Let's use these notations:

• M: Set of students who like milk.
• T: Set of students who like tea.
• U: The universal set, meaning all the students in the class.

We are given that a fraction of the students like milk, specifically 2/3 of the class. We also know that 14 students don't like milk at all. This is a crucial piece of information that helps us determine the total number of students. In set notation, the number of students who don't like milk is represented as the complement of set M, often written as M' or Mc.

2. Finding the Total Number of Students

This is where we put on our detective hats! We know 14 students don't like milk, and this represents 1 - 2/3 = 1/3 of the total students. Therefore, if 1/3 of the students is 14, then the total number of students can be found by multiplying 14 by 3. Let's do the math:

Total students = 14 * 3 = 42 students

So, we've cracked our first clue! We now know there are 42 students in the class. This is a vital step because it gives us the base number we need to calculate other quantities. Remember, guys, math is like building blocks – you need a strong foundation to build something amazing!

3. Students Who Like Tea

Now, let's tackle the tea lovers. We're told that 1/4 of the total students like tea. Since we know there are 42 students in total, we can calculate the number of students who like tea:

Students who like tea = (1/4) * 42 = 10.5

Wait a minute… 10.5 students? That doesn't make sense! We can't have half a student. This tells us that there might be a slight misunderstanding or rounding issue in the problem statement. In real-world scenarios, math problems sometimes have these little quirks. For the sake of our exercise, we'll assume that the number should be a whole number. We'll round 10.5 to 10 students. It's important to acknowledge these discrepancies and make reasonable assumptions when solving problems.

4. Students Who Like Neither

We're given that 5 students like neither milk nor tea. This is another crucial piece of information. In set notation, this is the number of students who belong to the complement of both M and T, written as (M ∪ T)'. This group is outside the circles of milk and tea lovers in our imaginary Venn diagram.

5. Finding the Union of Milk and Tea Lovers

To find the number of students who don't like both tea and milk, we need to figure out how many students like either milk or tea or both. This is called the union of the sets M and T, denoted as M ∪ T. The union includes all elements that are in either set or in both.

We know the total number of students and the number of students who like neither. Therefore, the number of students who like at least one of the beverages (milk or tea) is the total number of students minus those who like neither:

Students who like milk or tea or both = Total students - Students who like neither

Students who like milk or tea or both = 42 - 5 = 37 students

6. The Number of Students Who Like Only Milk

We know 2/3 of the students like milk, which is (2/3) * 42 = 28 students. This number includes those who like only milk and those who like both milk and tea. To find the number of students who like only milk, we need to use more information later in conjunction with what we already know.

7. The Key Formula: Principle of Inclusion-Exclusion

Here's where a powerful concept comes into play: the Principle of Inclusion-Exclusion. This principle helps us find the size of the union of sets. For two sets, the formula is:

|M ∪ T| = |M| + |T| - |M ∩ T|

Where:

• |M ∪ T| is the number of students who like milk or tea or both (which we found to be 37).
• |M| is the number of students who like milk (28 students).
• |T| is the number of students who like tea (10 students).
• |M ∩ T| is the number of students who like both milk and tea.

We can rearrange the formula to solve for |M ∩ T|:

|M ∩ T| = |M| + |T| - |M ∪ T|

|M ∩ T| = 28 + 10 - 37 = 1 student

So, only 1 student likes both milk and tea. This is a small but important piece of the puzzle!

8. Finding Students Who Like Only Milk (Revisited)

Now we can refine our earlier calculation. We know 28 students like milk in total, and 1 of them likes both milk and tea. Therefore, the number of students who like only milk is:

Students who like only milk = Total students who like milk - Students who like both

Students who like only milk = 28 - 1 = 27 students

9. Finding Students Who Like Only Tea

Similarly, we know 10 students like tea in total, and 1 of them likes both milk and tea. Therefore, the number of students who like only tea is:

Students who like only tea = Total students who like tea - Students who like both

Students who like only tea = 10 - 1 = 9 students

10. The Grand Finale: Students Who Don't Like Both

Finally, we can circle back to the original question: How many students don't like both tea and milk? We already know this! It was given in the problem: 5 students like neither.

Final Answer

So, after all our calculations, the answer is:

Number of students who don't like both tea and milk: 5

Visualizing with a Venn Diagram

Guys, sometimes a picture is worth a thousand words! A Venn diagram is a fantastic tool for visualizing set relationships. Imagine two overlapping circles. One circle represents students who like milk (M), and the other represents students who like tea (T). The overlapping area represents students who like both (M ∩ T). The area outside both circles represents students who like neither (M ∪ T)'.

We can fill in the Venn diagram with the numbers we calculated:

• Only Milk: 27 students
• Only Tea: 9 students
• Both Milk and Tea: 1 student
• Neither Milk nor Tea: 5 students

This diagram makes it super clear how the different groups of students relate to each other. It's a great way to check your work and ensure your solution makes sense.

Key Takeaways

This problem might seem complicated at first, but by breaking it down into smaller steps and using key concepts like set theory and the Principle of Inclusion-Exclusion, we were able to solve it! Here are the main takeaways:

• Set Theory: Understanding sets and their notations (union, intersection, complement) is crucial for solving these types of problems.
• Principle of Inclusion-Exclusion: This formula is a powerful tool for finding the size of the union of sets.
• Venn Diagrams: Visualizing the problem with a Venn diagram can make the relationships between sets much clearer.
• Step-by-Step Approach: Breaking down a complex problem into smaller, manageable steps makes it much easier to solve.

Real-World Applications

You might be thinking,