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In a survey on sales of soft drinks, \(65\%\) of the surveyed customers said that they buy brand A regularly, \(60\%\) said that they buy brand B regularly, and \(32\%\) said that they do not usually buy either of the two brands. What percent of the surveyed customers said that they buy both brands A and B regularly?
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:
The problem tells us:
We need to find what percent buy both brands.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical categories
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.
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
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:
57% of the surveyed customers said that they buy both brands A and B regularly.
This matches answer choice C.
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."
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.
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.
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.
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.
When solving \(125\% - \text{overlap} = 68\%\), students might incorrectly calculate the overlap as \(125\% + 68\% = 193\%\) or make other basic arithmetic mistakes like getting \(125\% - 68\% = 43\%\).
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.
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.
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.
This set theory problem can be solved effectively using smart numbers by choosing a convenient total number of surveyed 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.
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
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
Since we chose 100 as our total, 57 customers = 57%
Answer: 57% of customers buy both brands A and B regularly.