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All the apartments in a certain building are either three-bedroom apartments or four-bedroom apartments. The range of the monthly rents for all the apartments is \(\$600\), and the range of the monthly rents for the three-bedroom apartments is \(\$200\). What is the range of the monthly rents for the four-bedroom apartments?
We need to find: What is the range of monthly rents for the four-bedroom apartments?
Given Information:
- Building has only 3-bedroom and 4-bedroom apartments
- Range of ALL apartments = \(\$600\)
- Range of 3-bedroom apartments = \(\$200\)
What We Need to Determine:
To find the range of 4-bedroom apartments, we need to know both:
- The lowest 4-bedroom rent
- The highest 4-bedroom rent
Key Insight: The crucial insight here is understanding how the two apartment types contribute to the overall \(\$600\) range. Think of it this way: we have two groups of apartments with their own internal ranges, and together they create a combined range of \(\$600\). The key question is: do these ranges overlap, or are they separate?
Statement 1: The lowest monthly rent for the four-bedroom apartments is \(\$800\).
What Statement 1 Tells Us:
We now know the starting point for 4-bedroom rents (\(\$800\)), but we still don't know where this sits relative to the 3-bedroom range.
Testing Different Scenarios:
Let's test whether the 3-bedroom range could be in different positions:
Scenario 1: 3-bedrooms range from \(\$500\) to \(\$700\)
- Overall lowest = \(\$500\) (from 3-bedrooms)
- Overall range = \(\$600\), so overall highest = \(\$500 + \$600 = \$1,100\)
- Since 3-bedrooms only go up to \(\$700\), the 4-bedrooms must reach \(\$1,100\)
- Therefore: 4-bedroom range = \(\$1,100 - \$800 = \$300\) ✓
Scenario 2: 3-bedrooms range from \(\$900\) to \(\$1,100\)
- Overall lowest = \(\$800\) (from 4-bedrooms, since \(\$800 < \$900\))
- Overall range = \(\$600\), so overall highest = \(\$800 + \$600 = \$1,400\)
- Since 3-bedrooms only go up to \(\$1,100\), the 4-bedrooms must reach \(\$1,400\)
- Therefore: 4-bedroom range = \(\$1,400 - \$800 = \$600\) ✓
Conclusion: Different positions of the 3-bedroom range lead to different 4-bedroom ranges (\(\$300\) vs \(\$600\)). Statement 1 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The lowest monthly rent for the four-bedroom apartments is \(\$100\) greater than the highest monthly rent for the three-bedroom apartments.
What Statement 2 Provides:
This creates a fascinating situation - there's a \(\$100\) gap between the ranges! The 3-bedroom and 4-bedroom ranges don't overlap at all.
Visual Understanding:
Picture this arrangement:
3-bedroom apartments: [-----$200 range-----]
← $100 gap →
4-bedroom apartments: [-----? range-----]
Overall range: [------------------------$600------------------------]
Since there's no overlap:
- The overall lowest rent = the 3-bedroom lowest
- The overall highest rent = the 4-bedroom highest
- These two points must be exactly \(\$600\) apart
The Elegant Solution:
The total \(\$600\) range is made up of exactly three components:
- The 3-bedroom range: \(\$200\)
- The gap between ranges: \(\$100\)
- The 4-bedroom range: ?
So: \(\$200 + \$100 + ? = \$600\)
Therefore: The 4-bedroom range = \(\$300\)
Verification: Let's verify with a concrete example. If 3-bedrooms range from \(\$1,000\) to \(\$1,200\):
- Lowest 4-bedroom = \(\$1,200 + \$100 = \$1,300\)
- Overall lowest = \(\$1,000\), overall highest = \(\$1,000 + \$600 = \$1,600\)
- Therefore, highest 4-bedroom = \(\$1,600\)
- 4-bedroom range = \(\$1,600 - \$1,300 = \$300\) ✓
Conclusion: Statement 2 gives us a unique answer for the 4-bedroom range. Statement 2 is sufficient.
[STOP - Sufficient!] This eliminates choices C and E.
Statement 2 alone tells us exactly how the ranges relate to each other (with a \(\$100\) gap), which allows us to determine that the 4-bedroom range must be \(\$300\).
Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."