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Last year in City X, the range of daily low temperatures, in degrees Fahrenheit, was 20 for the month of June and 25 for the month of July. Which of the following is the smallest possible range of City X's daily low temperatures, in degrees Fahrenheit, for the two-month period of June and July of last year?
Let's start by understanding what we're working with in everyday terms.
Range means the difference between the highest and lowest values. Think of it like this: if the coldest day in June was 60°F and the warmest day was 80°F, then the range would be \(80 - 60 = 20\)°F.
So we know:
The key insight is that we're looking at how two separate ranges can combine to create the smallest overall range.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understandingLet's think about this with concrete numbers to make it clear. Imagine June's temperatures and July's temperatures as two separate groups:
Scenario 1 - No Overlap (Maximum Combined Range):
June: 60°F to 80°F (range = 20°F)
July: 85°F to 110°F (range = 25°F)
Combined range = \(110 - 60 = 50\)°F
Scenario 2 - Partial Overlap:
June: 60°F to 80°F (range = 20°F)
July: 75°F to 100°F (range = 25°F)
Combined range = \(100 - 60 = 40\)°F
Scenario 3 - Maximum Overlap (What we want):
This is where one month's entire range fits completely inside the other month's range.
Here's the key insight: The smallest combined range occurs when the smaller range (June's 20°F) fits completely inside the larger range (July's 25°F).
Let's see this with numbers:
Optimal arrangement:
July: 60°F to 85°F (range = 25°F)
June: 65°F to 85°F (range = 20°F)
Notice how June's entire temperature range sits within July's range. In this case:
The minimum possible range is simply the larger of the two individual ranges, which is 25°F.
Process Skill: VISUALIZE - Using concrete examples to see how ranges can overlap optimallyLet's double-check our logic:
Why can't the combined range be smaller than 25°F?
Because July alone already has a range of 25°F. No matter how we arrange the temperatures, we cannot make the overall spread smaller than the largest individual spread.
Why can't the combined range be smaller than 20°F?
Similarly, June alone has a range of 20°F, so the combined range cannot be less than this either.
Can we actually achieve 25°F?
Yes! As shown in our example, when June's range (20°F) fits entirely within July's range (25°F), the combined range equals exactly 25°F.
Therefore, the minimum possible range is 25°F.
The smallest possible range of City X's daily low temperatures for the two-month period of June and July is 25 degrees Fahrenheit.
This corresponds to answer choice C. 25.
The key insight is that when two ranges are combined, the minimum possible combined range equals the larger of the two individual ranges, achieved when the smaller range fits completely within the larger range.
Faltering Point 1: Misunderstanding what "range" means in context
Students may confuse the range of daily LOW temperatures with the range of ALL temperatures (highs and lows combined). The question specifically asks about daily low temperatures only, but students might think they need to consider both daily highs and lows, leading to an entirely wrong approach.
Faltering Point 2: Thinking the ranges must be added together
Many students instinctively think that combining two months means adding their ranges: \(20 + 25 = 45\). This leads them to choose answer E. However, the question asks for the range of the COMBINED data set, not the sum of individual ranges.
Faltering Point 3: Not recognizing that ranges can overlap
Students may assume that the temperature ranges for June and July are completely separate with no overlap possible. This leads them to think the minimum combined range would be \(20 + 25 = 45\), when in fact the ranges can overlap significantly to create a much smaller combined range.
Faltering Point 1: Setting up scenarios incorrectly
When trying to find the minimum combined range, students might set up their examples wrong. For instance, they might try to make June: 60°F to 80°F and July: 60°F to 85°F, but then incorrectly calculate the combined range as \(85 - 60 = 25\)°F without checking if June actually has a 20°F range in their setup.
Faltering Point 2: Confusing which range should contain the other
Students might try to fit July's larger range (25°F) inside June's smaller range (20°F), which is impossible. This leads to incorrect calculations and confusion about why their numbers don't work out properly.
Faltering Point 1: Choosing the difference between ranges instead of maximum overlap
Students who understand that ranges can overlap might incorrectly think the minimum combined range is the difference between the individual ranges: \(25 - 20 = 5\)°F. This leads them to choose answer A, not realizing that the combined range cannot be smaller than the largest individual range.
Faltering Point 2: Second-guessing the logical constraint
Even after correctly determining that 25°F is the minimum possible range, students might doubt their answer because it seems "too simple" that the answer equals one of the given ranges. They might switch to answer D (30°F) thinking there must be some additional constraint they're missing.