In a survey on sales of soft drinks, 65% of the surveyed customers said that they buy brand A regularly,...

1. Translate the problem requirements: We need to clarify what "regularly buy" means in terms of set membership, and understand that customers can belong to multiple categories (A only, B only, both A and B, or neither).
2. Identify the total customer distribution: Recognize that all surveyed customers must fall into exactly one of four mutually exclusive categories, and use the "neither" percentage to find the "at least one" percentage.
3. Apply the inclusion-exclusion principle intuitively: Understand that when we add the A buyers and B buyers, we're double-counting those who buy both brands, so we need to subtract the overlap to get the union.
4. Set up and solve the overlap equation: Use the relationship that (A buyers) + (B buyers) - (both A and B) = (at least one brand) to find the percentage buying both brands.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with in plain English. Imagine we surveyed 100 customers about their soft drink buying habits.

When someone says they "buy brand A regularly," they're in the "A group." Similarly, "buy brand B regularly" puts them in the "B group." The key insight is that a customer can be in both groups - they can buy both brands regularly.

So every customer falls into exactly one of these four categories:

• Buy only A (but not B)
• Buy only B (but not A)
• Buy both A and B
• Buy neither A nor B

The problem tells us:

• 65% buy brand A regularly (this includes people who buy A only AND people who buy both A and B)
• 60% buy brand B regularly (this includes people who buy B only AND people who buy both A and B)
• 32% don't usually buy either brand

We need to find what percent buy both brands.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical categories

2. Identify the total customer distribution

Since 32% buy neither brand, this means 68% buy at least one of the brands (either A, or B, or both).

Think of it this way: \(100\% - 32\% = 68\%\) buy at least one brand.

This 68% represents everyone who buys A, B, or both. This is a crucial number because it tells us the total size of our "at least one brand" group.

3. Apply the inclusion-exclusion principle intuitively

Here's where the magic happens. Let's think about what happens when we add up the A buyers and B buyers:

\(65\% + 60\% = 125\%\)

But wait! We only have 68% who buy at least one brand. How can we get 125%?

The answer is double-counting. When we count the 65% who buy A and the 60% who buy B separately, we're counting the people who buy BOTH brands twice - once in each group.

Imagine a customer who buys both A and B. They get counted in the "65% who buy A" AND also in the "60% who buy B." So they're counted twice in our 125% total.

Process Skill: VISUALIZE - Understanding the overlap concept through concrete reasoning

4. Set up and solve the overlap equation

Now we can set up our equation using plain English logic:

(A buyers) + (B buyers) - (People counted twice) = (People who buy at least one brand)

Substituting what we know:

\(65\% + 60\% - \text{(Both A and B buyers)} = 68\%\)

\(125\% - \text{(Both A and B buyers)} = 68\%\)

Solving for the overlap:

\text{Both A and B buyers} = 125\% - 68\% = 57\%\)

Let's verify this makes sense: If 57% buy both brands, then:

• People who buy only A = 65% - 57% = 8%
• People who buy only B = 60% - 57% = 3%
• People who buy both = 57%
• People who buy neither = 32%
• Total: \(8\% + 3\% + 57\% + 32\% = 100\%\) ✓

4. Final Answer

57% of the surveyed customers said that they buy both brands A and B regularly.

This matches answer choice C.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "32% do not usually buy either" as "32% buy neither exclusively"

Students often get confused by the language and think this 32% refers to some other category, rather than understanding it clearly means 32% buy NO brands at all. This leads them to incorrectly calculate that \(100\% - 32\% = 68\%\) represents something other than "people who buy at least one brand."

2. Not recognizing this as a classic set overlap problem

Many students see percentages that add up to more than 100% (\(65\% + 60\% + 32\% = 157\%\)) and panic, thinking there's an error in the problem. They fail to recognize that the 65% and 60% groups can overlap, which is exactly what we need to find.

3. Attempting to solve without using inclusion-exclusion principle

Some students try to guess-and-check with the answer choices or create overly complicated equations instead of recognizing this as a straightforward application of: Total = Group A + Group B - Overlap.

Errors while executing the approach

1. Incorrectly calculating the "at least one brand" percentage

Students might calculate this as \(65\% + 60\% - 32\% = 93\%\) instead of recognizing that if 32% buy neither, then \(100\% - 32\% = 68\%\) buy at least one brand.

2. Setting up the inclusion-exclusion equation incorrectly

Instead of writing \(65\% + 60\% - \text{overlap} = 68\%\), students might write equations like \(65\% + 60\% + \text{overlap} = 68\%\) or confuse which numbers go where in the formula.

3. Arithmetic errors in the final calculation

When solving \(125\% - \text{overlap} = 68\%\), students might incorrectly calculate the overlap as \(125\% + 68\% = 193\%\) or make other basic arithmetic mistakes like getting \(125\% - 68\% = 43\%\).

Errors while selecting the answer

1. Selecting the percentage who buy "at least one brand" instead of "both brands"

After calculating that 68% buy at least one brand and 57% buy both brands, students might accidentally select an answer choice close to 68% instead of 57%, confusing what the question actually asks for.

2. Choosing the complement percentage

Students might calculate \(100\% - 57\% = 43\%\) thinking they need the percentage who DON'T buy both brands, especially if they see this value among the answer choices.

No likely faltering points

The answer selection phase for this problem is relatively straightforward once the calculation is complete, as the question clearly asks for the percentage who buy both brands.

Alternate Solutions

Smart Numbers Approach

This set theory problem can be solved effectively using smart numbers by choosing a convenient total number of surveyed customers.

Step 1: Choose a Smart Number for Total Customers

Since we're dealing with percentages like 65%, 60%, and 32%, let's choose 100 customers as our total. This makes percentage calculations straightforward since each percentage point equals exactly 1 customer.

Step 2: Convert Percentages to Actual Numbers

• Customers who buy brand A regularly: \(65\% \text{ of } 100 = 65\) customers
• Customers who buy brand B regularly: \(60\% \text{ of } 100 = 60\) customers
• Customers who don't buy either brand: \(32\% \text{ of } 100 = 32\) customers

Step 3: Find Customers Who Buy At Least One Brand

If 32 customers don't buy either brand, then the remaining customers buy at least one brand:

Customers buying at least one brand = \(100 - 32 = 68\) customers

Step 4: Apply the Overlap Principle

When we add brand A buyers (65) and brand B buyers (60), we get 125 customers. But we only have 68 customers who buy at least one brand. This means we've double-counted some customers:

Double-counted customers = Those who buy both brands

Both A and B = (A buyers) + (B buyers) - (At least one brand)

Both A and B = \(65 + 60 - 68 = 57\) customers

Step 5: Convert Back to Percentage

Since we chose 100 as our total, 57 customers = 57%

Answer: 57% of customers buy both brands A and B regularly.

Why This Smart Number Works Well:

• Choosing 100 makes percentage-to-number conversion trivial
• All calculations result in whole numbers, avoiding fractions
• The logical flow mirrors the algebraic approach but with concrete values
• Easy to verify: 57 customers buying both + 8 buying only A + 3 buying only B + 32 buying neither = 100 ✓