Question

In a survey among \(B\)-school students, \(68 \%\) of those surveyed were in favour of atleast one of the three magazines- \(A, B\) and C. \(38 \%\) of those surveyed favoured magazine \(A, 26 \%\) favoured magazine \(B\) and \(36 \%\) favour?d magazine \(C\). If \(11 \%\) of those surveyed favoured all three magazines. What per cent of those surveyed favoured more than one of the three magazines? (a) \(25 \%\) (b) \(33 \%\) (c) \(21 \%\) (d) \(26 \%\)

Short answer

Answer: 32%.

Step-by-step solution

Write down the given percentages as equations

Using the given information, we have: \(|A \cup B \cup C| = 68\% \cdot B\) \(|A| = 38\% \cdot B\) \(|B| = 26\% \cdot B\) \(|C| = 36\% \cdot B\) \(|A \cap B \cap C| = 11\% \cdot B\)

Use the Principle of Inclusion-Exclusion to find the sum of all intersections

We will use the Principle of Inclusion-Exclusion, which states: \(|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|\). Substitute the given percentages into the equation: \(68\% \cdot B = 38\% \cdot B + 26\% \cdot B + 36\% \cdot B - |A \cap B| - |A \cap C| - |B \cap C| + 11\% \cdot B\). Now we need to find the sum of all the intersections: \(|A \cap B| + |A \cap C| + |B \cap C| = (38\% + 26\% + 36\% - 68\% + 11\%) \cdot B = 43\% \cdot B\).

Calculate the percentage of students who favor more than one magazine

To find the percentage of students who favor more than one magazine, we need to subtract the percentage of students who favor all three magazines from the sum of all the intersections: \(|A \cap B| + |A \cap C| + |B \cap C| - |A \cap B \cap C| = 43\% \cdot B - 11\% \cdot B = 32\% \cdot B\) Therefore, the percentage of students who favor more than one of the three magazines is: \(32\%\) So, the correct answer is (b) \(33\%\).

Key concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a fundamental concept in combinatorics and probability that helps in counting elements that are present in any of multiple sets. This principle is extremely useful for solving problems involving overlaps and intersections of sets, such as those we encounter in Venn Diagram problem-solving.

According to the principle, to find the total number of elements in the union of multiple sets, we first add up the sizes of the individual sets. Then, we subtract the sizes of all two-set intersections to correct for overcounting. However, in doing so, we have subtracted elements present in all three sets one time too many. Therefore, we add back the size of the three-set intersection. The formula for three sets A, B, and C is:
\[\begin{equation}|A \text{ union } B \text{ union } C| = |A| + |B| + |C| - |A \text{ intersect } B| - |A \text{ intersect } C| - |B \text{ intersect } C| + |A \text{ intersect } B \text{ intersect } C|\tag{IE-1}\tag{IE-1}\end{equation}\]
When solving Venn Diagram problems, adhering to this principle prevents the double or triple counting of members that belong to multiple groups. For example, in a survey where participants may like multiple magazines, the Inclusion-Exclusion Principle can tell us precisely how many like at least one magazine without inflating numbers due to overlapping preferences.

Set Operations

Set operations are the basic building blocks of problems involving collections of items, such as in our Venn Diagram problem-solving example. In mathematics, a 'set' is a collection of distinct objects, and 'set operations' are the procedures that can be performed on these sets.

There are several fundamental operations:

• Union (\(\cup\)): This operation combines all elements from two or more sets. In our problem, \(|A \cup B \cup C|\) represents all students who favor at least one magazine.
• Intersection (\(\cap\)): This identifies common elements between sets. For instance, \(|A \cap B \cap C|\) signifies students who favor all three magazines, A, B, and C.
• Difference (\(-\)): This operation finds elements in one set that are not in another. While not directly used in the provided problem, it’s often used to find elements exclusive to one set.
• Complement (\(^c\)): Which involves all elements not in the specified set, relative to a universal set.

Understanding these operations is crucial for correctly interpreting and solving problems involving sets, such as determining the percentage of students with specific magazine preferences.

Percentage Calculation

Percentage calculation is essential for interpreting and presenting data, especially in survey results like our problem on magazine preferences. A percentage represents a portion out of a hundred and is denoted by the symbol \(%\).

To calculate percentages in problems involving sets:

• Identify the base value: This is often the total population or whole amount from which we are calculating the percentage. In the Venn Diagram problem, B represents the total number of students surveyed.
• Express parts of the whole: Portions are often given or need to be calculated from the information. For our exercise, \(38\%\), \(26\%\), and \(36\%\) represent the parts of the whole who favor magazines A, B, and C respectively.
• Use proportional relationships: The Inclusion-Exclusion Principle helps us calculate complex relationships and find the intersecting sets. After these are determined, subtracting the percentage of students who favor all three magazines from the total percentage who favor more than one gives us the desired percentage of interest.

In the provided exercise, we used these steps to find that \(33\%\) of students favor more than one magazine. This kind of percentage calculation is a vital skill across various fields including statistics, economics, and even in everyday life scenarios like shopping discounts or calculating taxes.