Analyzing Student Vegetable Preferences

This problem involves analyzing data about the vegetable preferences of a group of students. We are given the number of students who like specific vegetables or combinations. Our goal is to find the percentage of students who do not like cabbage.

Understanding the Given Data

We have a total of 75 students. The preferences are broken down as follows:

• Students who like only cabbage: 12
• Students who like only cauliflower: 15
• Students who like only carrot: 21
• Students who like both carrot and cabbage: 12
• Students who like only capsicum: 13
• Students who like both capsicum and cauliflower: 2

Let's verify if the sum of these groups equals the total number of students:

\( 12 + 15 + 21 + 12 + 13 + 2 = 75 \)

Since the sum is exactly 75, this indicates that every student's preference is covered by one of these specific categories, and there are no students left out, nor are there overlaps between the 'only' groups and the 'both' groups beyond what is explicitly stated in a way that would exceed the total.

Identifying Students Who Like Cabbage

To find the students who do not like cabbage, we first need to identify all the students who *do* like cabbage. Based on the provided categories, the groups that include cabbage are:

• Students who like only cabbage: 12
• Students who like both carrot and cabbage: 12

The total number of students who like cabbage is the sum of these two groups:

Number of students who like cabbage = (Students who like only cabbage) + (Students who like both carrot and cabbage)

Number of students who like cabbage = \( 12 + 12 = 24 \)

Identifying Students Who Do Not Like Cabbage

Now, we can find the number of students who do not like cabbage. This includes all students from the groups that do not mention cabbage.

Alternatively, we can subtract the number of students who like cabbage from the total number of students.

Total students = 75

Students who like cabbage = 24

Number of students who do not like cabbage = Total students - Students who like cabbage

Number of students who do not like cabbage = \( 75 - 24 = 51 \)

Let's confirm this by summing the groups that do not like cabbage:

• Students who like only cauliflower: 15
• Students who like only carrot: 21
• Students who like only capsicum: 13
• Students who like both capsicum and cauliflower: 2

Total students who do not like cabbage = \( 15 + 21 + 13 + 2 = 51 \)

Both methods give the same result: 51 students do not like cabbage.

Calculating the Percentage

To find the percentage of students who do not like cabbage, we use the formula:

Percentage = \( \left( \frac{\text{Number of students who do not like cabbage}}{\text{Total number of students}} \right) \times 100 \)

Percentage = \( \left( \frac{51}{75} \right) \times 100 \)

We can simplify the fraction \( \frac{51}{75} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

\( \frac{51 \div 3}{75 \div 3} = \frac{17}{25} \)

Now, calculate the percentage:

Percentage = \( \frac{17}{25} \times 100 \)

Percentage = \( 17 \times \frac{100}{25} \)

Percentage = \( 17 \times 4 \)

Percentage = \( 68 \)

So, 68% of the students do not like cabbage.

Summary of Results

Students who like cabbage = \( 12 + 12 = 24 \)

Students who do not like cabbage = \( 15 + 21 + 13 + 2 = 51 \)

Percentage not liking cabbage = \( \left( \frac{51}{75} \right) \times 100 = 68\% \)

Revision Table: Key Concepts

Additional Information: Handling Set Data

When dealing with data about preferences or characteristics within a group, problems often use terms like "only," "both," "either/or," "none," etc. It's helpful to visualize this using Venn diagrams, although for this specific problem where the sum of given groups equals the total, a simple addition/subtraction approach is sufficient under the assumption that the groups partition the set of students.

For more complex problems with overlaps, Venn diagrams help ensure you don't double-count students. For instance, if a problem stated "12 like carrot and cabbage" and also gave numbers for "carrot" and "cabbage" (meaning *at least* that vegetable), you would use the principle of inclusion-exclusion: \( |A \cup B| = |A| + |B| - |A \cap B| \). However, the phrasing "only" in this problem simplifies the structure significantly, suggesting distinct, non-overlapping groups except where "both" is specified as a unique category.

Always read the question carefully to understand precisely what each number represents (e.g., "likes cabbage" vs. "likes *only* cabbage").