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Of the \(\mathrm{190}\) bicycle stores in a certain region, \(\mathrm{90\%}\) repair bicycles and \(\mathrm{40\%}\) rent bicycles. If the integer \(\mathrm{n}\) denotes the number of bicycle stores in the region that both repair and rent bicycles, which of the following indicates all possible values of \(\mathrm{n}\)?
Let's start by understanding what we have in plain English. We're dealing with 190 bicycle stores in a region, and we know two things about what these stores do:
Now let's convert these percentages into actual numbers of stores:
We need to find the range of possible values for n, which represents stores that do BOTH activities - repair AND rent bicycles.
Process Skill: TRANSLATE - Converting percentage language into concrete numbers
To find all possible values of n, we need to think about what limits exist. There are two key constraints:
Upper Limit Constraint: The number of stores doing both activities cannot exceed the smaller group. Since we have 171 repair stores and 76 rental stores, the overlap cannot be more than 76 stores (you can't have more stores doing both than the total doing just one of the activities).
Lower Limit Constraint: We need to ensure all 190 stores are accounted for. If some stores do both, some do only repair, some do only rent, and some do neither, the total must equal 190.
Process Skill: APPLY CONSTRAINTS - Recognizing the logical limits on our answer
To find the minimum value of n, we want to minimize the overlap. This happens when we try to fit as many stores as possible without them doing both activities.
Think of it this way: We have 171 repair stores and 76 rental stores. If we tried to make these completely separate groups, we'd need \(\mathrm{171 + 76 = 247}\) stores. But we only have 190 stores total!
This means we're "short" by \(\mathrm{247 - 190 = 57}\) stores. These 57 stores MUST be doing both activities - there's no way around it.
So the minimum value of \(\mathrm{n = 57}\).
Process Skill: INFER - Drawing the non-obvious conclusion about forced overlap
For the maximum overlap, we already identified this in step 2. The overlap cannot exceed the smaller of our two groups.
Since we have 76 stores that rent (which is smaller than 171 stores that repair), the maximum possible overlap is when ALL rental stores also repair bicycles.
So the maximum value of \(\mathrm{n = 76}\).
Let's verify this works: If 76 stores do both, then we have:
Total: \(\mathrm{76 + 95 + 0 + 19 = 190}\) ✓
Our calculated range is: \(\mathrm{57 \leq n \leq 76}\)
Looking at the answer choices:
The answer is B. \(\mathrm{57 \leq n \leq 76}\)
This range represents all possible values for the number of stores that both repair and rent bicycles, where 57 is the minimum forced overlap due to the constraint of having only 190 total stores, and 76 is the maximum possible overlap limited by the smaller rental group.
1. Misunderstanding what "both" means in set overlap problems
Students often confuse "stores that both repair AND rent" with "stores that do either repair OR rent." This fundamental misinterpretation leads them to approach the problem as finding the union of two sets rather than the intersection, completely derailing their solution from the start.
2. Failing to recognize the constraint that forces minimum overlap
Many students don't realize that when the sum of individual percentages (\(\mathrm{90\% + 40\% = 130\%}\)) exceeds \(\mathrm{100\%}\), there MUST be overlap. They might attempt to find scenarios where n could be 0 or very small, not understanding that the total number of stores (190) creates a mathematical constraint that forces a minimum overlap.
3. Not establishing both upper and lower bounds systematically
Students often focus on finding just one boundary (usually the maximum) without recognizing that this is a range problem requiring both minimum and maximum values. They might calculate one limit correctly but fail to consider what constraints determine the other boundary.
1. Arithmetic errors when converting percentages to actual numbers
Students frequently make calculation mistakes when computing \(\mathrm{90\% \text{ of } 190 = 171}\) or \(\mathrm{40\% \text{ of } 190 = 76}\). These seemingly simple calculations are often rushed, leading to incorrect base numbers that cascade through the entire solution.
2. Incorrect application of the overlap formula
When calculating minimum overlap, students might incorrectly apply the principle. Instead of recognizing that minimum \(\mathrm{n = (171 + 76) - 190 = 57}\), they might subtract in the wrong order or misapply the constraint, leading to incorrect boundary calculations.
3. Failing to verify their calculated range with concrete examples
Students often don't check their work by testing whether their calculated minimum and maximum values actually satisfy all the given conditions (totaling 190 stores with correct repair and rental percentages), missing opportunities to catch calculation errors.
1. Choosing an answer choice that matches only one boundary correctly
Students might calculate one boundary correctly (say, maximum = 76) and select an answer choice that contains this correct value without verifying that the other boundary is also correct. For instance, they might choose option A (\(\mathrm{38 \leq n \leq 76}\)) because the upper limit matches their calculation while ignoring that the lower limit is wrong.
2. Selecting the range with the largest span assuming it's "safer"
When uncertain, some students gravitate toward answer choices with wider ranges (like option C or E) thinking these are more likely to be correct since they include more possibilities, rather than focusing on the precise mathematical constraints that determine the exact range.