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24 Jun 2023, 05:45
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C
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Difficulty:
45%
(medium)
Question Stats:
62% (00:59) correct
38%
(01:29)
wrong
based on 263
sessions
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The range of the monthly salary in 1999 is 240, the range of the monthly salary in 2000 is 320, what is the minimal range of the monthly salary in both 1999 and 2000?
A. 80
B. 240
C. 320
D. 400
E. 480
A. 80
B. 240
C. 320
D. 400
E. 480
25 Jun 2023, 08:41
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Bunuel
1999
- Lowest salary = x
- Highest salary = x+240
2000
- Lowest salary = y
- Highest salary = y + 320
Combined
Case 1: x < y
If x < y → x + 240 < y + 320
- Lowest Value = x
- Highest Value = y + 320
If this was the case, the range will be greater than 320.
Reason: To obtain the range, we are now subtracting a smaller value, x, from y + 320.
Case 2: x = y
- Lowest Value = x = y
- Highest Value = y + 320 = x + 320
Range = 320.
Case 3: x > y
- Lowest Value = y
- The highest value can be either y + 320 (denoted by position B), or x+240 (denoted by position C)
Range =
- If the highest value = y + 320 (i.e. x + 240 \(\leq\) y + 320), range = 320
- If the highest value = x + 240 (i.e. x + 240 > y + 320), range > 320.
Reason: As x + 240 is farther right on the number line than y +320 is, the distance of y from x + 240 will be greater than 320.
Therefore the lowest value of range from all three cases = 320
Option C
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General Discussion
25 Jun 2023, 18:57
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Bunuel
The range of the combined data set cannot be smaller than either of the ranges of the original data sets. So, we have:
range_combined ≥ max(range_1999,range_2000)
range_combined ≥ max(240,320) = 320
The combined range takes its smallest possible value if the original data set with the greater range “contains” the original data set with the smaller range. In other words, the original data set with the smaller range has no element that is outside the range of values of the original data set with the greater range.
Therefore, the smallest possible value of the combined range is 320.
Answer: C
