Of the 190 bicycle stores in a certain region, 90% repair bicycles and 40% rent bicycles. If the integer n...

1. Translate the problem requirements: We have 190 stores total, with \(\mathrm{90\%}\) (171 stores) that repair and \(\mathrm{40\%}\) (76 stores) that rent. We need to find all possible values for stores that do both activities.
2. Establish the constraint boundaries: Determine the minimum and maximum possible overlap between the repair and rent groups using logical reasoning about set intersections.
3. Calculate the minimum overlap scenario: Find the smallest possible value of n by maximizing the number of stores that do only one activity.
4. Calculate the maximum overlap scenario: Find the largest possible value of n by recognizing that overlap cannot exceed the smaller of the two groups.
5. Verify the range against answer choices: Confirm our calculated range matches one of the provided options.

Execution of Strategic Approach

1. Translate the problem requirements

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:

• \(\mathrm{90\%}\) of them repair bicycles
• \(\mathrm{40\%}\) of them rent bicycles

Now let's convert these percentages into actual numbers of stores:

• Stores that repair: \(\mathrm{90\% \text{ of } 190 = 0.90 \times 190 = 171}\) stores
• Stores that rent: \(\mathrm{40\% \text{ of } 190 = 0.40 \times 190 = 76}\) 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

2. Establish the constraint boundaries

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

3. Calculate the minimum overlap scenario

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

4. Calculate the maximum overlap scenario

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:

• 76 stores doing both repair and rent
• \(\mathrm{171 - 76 = 95}\) stores doing only repair
• \(\mathrm{76 - 76 = 0}\) stores doing only rent
• \(\mathrm{190 - 76 - 95 - 0 = 19}\) stores doing neither

Total: \(\mathrm{76 + 95 + 0 + 19 = 190}\) ✓

5. Verify the range against answer choices

Our calculated range is: \(\mathrm{57 \leq n \leq 76}\)

Looking at the answer choices:

1. \(\mathrm{38 \leq n \leq 76}\) (minimum too low)
2. \(\mathrm{57 \leq n \leq 76}\) (matches our calculation exactly)
3. \(\mathrm{57 \leq n \leq 171}\) (maximum too high)
4. \(\mathrm{76 \leq n \leq 95}\) (minimum too high)
5. \(\mathrm{76 \leq n \leq 171}\) (minimum too high, maximum too high)

Final Answer

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.

Common Faltering Points

Errors while devising the approach

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.

Errors while executing the approach

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.

Errors while selecting the answer

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.