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In a certain city, 80 percent of the households have cable television, and 60 percent of the households have videocassette recorders. If there are 150,000 households in the city, then the number of households that have both cable television and videocassette recorders could be any number from:
Let's start by understanding what we're really asking. We have two groups of households in this city:
- Group 1: Households with cable TV
- Group 2: Households with VCRs
Some households belong to both groups (they have BOTH cable TV AND VCRs), and we want to find the range of how many households could be in this overlap.
Think of it like this: imagine you have two circles that can overlap in different ways. We want to know the minimum and maximum size that overlapping region could be.
Process Skill: TRANSLATE - Converting the set theory language into concrete, visual understanding
Let's convert those percentages into real numbers so we can work with them more easily:
Total households in the city = 150,000
Households with cable TV = 80% of 150,000
= \(0.80 \times 150,000 = 120,000\) households
Households with VCRs = 60% of 150,000
= \(0.60 \times 150,000 = 90,000\) households
Now we have concrete numbers to work with: 120,000 households have cable TV and 90,000 households have VCRs.
To find the minimum overlap, we want to spread out the two groups as much as possible with as little overlap as we can manage.
Think about it this way: if we try to minimize overlap, we want as many households as possible to have ONLY cable TV or ONLY VCRs, but not both.
Here's the key insight: The total number of households that have AT LEAST one of these items cannot exceed the total number of households in the city (150,000).
If there were NO overlap at all, we would need:
\(120,000 + 90,000 = 210,000\) households
But we only have 150,000 households total! This means we MUST have some overlap.
The minimum overlap occurs when we use up all 150,000 households:
Minimum overlap = (Cable households + VCR households) - Total households
Minimum overlap = \(120,000 + 90,000 - 150,000 = 60,000\)
Process Skill: INFER - Recognizing that the constraint of total households forces a minimum overlap
For maximum overlap, we want as many households as possible to have BOTH items.
The maximum overlap is limited by whichever group is smaller. Think about it: you can't have more households with "both cable AND VCRs" than the total number of households that have VCRs in the first place.
Since we have:
- 120,000 households with cable TV
- 90,000 households with VCRs
The maximum possible overlap is 90,000 households. This would happen if every single household that has a VCR also has cable TV (but some cable TV households don't have VCRs).
Our analysis shows that the number of households with both cable TV and VCRs must be:
Minimum: 60,000 households
Maximum: 90,000 households
Therefore, the range is 60,000 to 90,000 inclusive.
Looking at the answer choices, this matches choice C: 60,000 to 90,000 inclusive.
The number of households that have both cable television and videocassette recorders could be any number from 60,000 to 90,000 inclusive.
Answer: C
1. Misunderstanding what "both" means in the context
Students often confuse "households that have both cable TV and VCRs" with "households that have either cable TV or VCRs." This fundamental misinterpretation leads them to calculate the union instead of the intersection of the two sets, completely changing the problem they're trying to solve.
2. Not recognizing this as a min-max range problem
Many students try to find a single exact answer rather than understanding that the question asks for a range of possible values. They might attempt to use formulas or assume equal distribution without realizing that the overlap can vary depending on how the households are distributed.
3. Failing to identify the key constraint
Students often miss that the total number of households (150,000) acts as a crucial constraint that forces a minimum overlap. They may not realize that if you try to minimize overlap completely, you would need more households than actually exist in the city.
1. Arithmetic errors in percentage calculations
Students may incorrectly calculate 80% of 150,000 or 60% of 150,000, leading to wrong base numbers (e.g., getting 12,000 instead of 120,000 for cable households). These calculation errors cascade through the entire solution.
2. Incorrect minimum overlap calculation
When finding the minimum overlap, students might incorrectly subtract in the wrong order or forget to subtract the total households. For example, they might calculate \(150,000 - 120,000 - 90,000 = -60,000\) and get confused, rather than using the correct formula: \(120,000 + 90,000 - 150,000 = 60,000\).
3. Wrong logic for maximum overlap
Students may think the maximum overlap is the larger of the two groups (120,000) or try to add the groups together, rather than understanding that the maximum overlap is limited by the smaller group (90,000 households with VCRs).
1. Mixing up minimum and maximum values
After correctly calculating that the range is 60,000 to 90,000, students might accidentally select an answer choice that has these numbers but in the wrong order, or select a choice that includes one correct boundary but extends beyond the actual range.
2. Selecting a range that includes the correct range but is too broad
Students who are unsure of their calculations might select answer choice D (60,000 to 120,000) thinking it's "safer" since it includes their calculated range of 60,000 to 90,000, not realizing that 120,000 is impossible as a maximum overlap.