- Translate the problem requirements: We have 25 different integers where the middle value (13th when arranged in order) equals 50, and the difference between the largest and smallest values equals 50. We need to find the maximum possible value for the largest integer.
- Establish the constraint relationships: Set up the basic constraints using the median and range definitions, identifying what values are fixed and what can be optimized.
- Minimize the smallest value strategically: To maximize the largest value within the range constraint, determine the optimal strategy for choosing the smallest integer.
- Apply the range constraint to find maximum: Use the relationship between smallest and largest values through the range to calculate the maximum possible largest integer.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in simple terms:
- We have 25 different integers arranged in order from smallest to largest
- The median is 50, which means the middle number (the 13th number) equals 50
- The range is 50, which means the difference between the largest and smallest numbers equals 50
- We want to find the greatest possible value for the largest integer
Think of it this way: imagine lining up 25 different whole numbers from left to right, smallest to largest. The number in the middle position (position 13) must be 50, and the gap between the leftmost and rightmost numbers must be exactly 50.
Process Skill: TRANSLATE - Converting the median and range definitions into concrete positional constraints
2. Establish the constraint relationships
Now let's think about what's fixed and what we can control:
What's fixed:
- The 13th number (median) = 50
- The relationship: Largest number - Smallest number = 50
What we can optimize:
- The actual values of the smallest and largest numbers (as long as their difference is 50)
- The arrangement of the other 23 numbers
Since we want the largest number to be as big as possible, and we know that:
\(\mathrm{Largest\ number = Smallest\ number + 50}\)
This means to maximize the largest number, we need to maximize the smallest number!
3. Minimize the smallest value strategically
Wait - this seems backwards! Let me reconsider this more carefully.
Actually, let's think about this step by step. We have positions 1 through 25, and we know:
- \(\mathrm{Position\ 13 = 50}\) (the median)
- \(\mathrm{Position\ 25 - Position\ 1 = 50}\) (the range)
To maximize position 25 (the largest number), we actually want to think about what constraints we have. Since all numbers must be different integers, and 12 numbers must be smaller than 50, what's the largest that the smallest number could be?
The 12 numbers before position 13 must all be less than 50 and all different. To maximize the smallest number, we'd want these to be: 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38.
So the largest possible value for the smallest number would be 38.
Process Skill: INFER - Recognizing that maximizing the largest value requires optimizing the arrangement of all constrained values
4. Apply the range constraint to find maximum
Now we can find the maximum possible largest integer:
If the smallest number = 38, then:
\(\mathrm{Largest\ number = Smallest\ number + Range}\)
\(\mathrm{Largest\ number = 38 + 50 = 88}\)
Let's verify this works:
- We can have 12 numbers less than 50: \(\{38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49\}\)
- The median (13th number): 50
- We need 12 numbers greater than 50: \(\{51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 88\}\)
- Range check: \(\mathrm{88 - 38 = 50}\) ✓
- All 25 numbers are different ✓
5. Final Answer
The greatest possible integer that could be in this set is 88.
Looking at our answer choices:
- 62
- 68
- 75
- 88
- 100
The answer is (D) 88.
Common Faltering Points
Errors while devising the approach
- Misunderstanding the median constraint: Students often forget that in a set of 25 integers, the median is the 13th number when arranged in order, not just any middle value. They might try to place 50 anywhere in the set rather than recognizing it must be in position 13 with exactly 12 numbers below it and 12 numbers above it.
- Backwards optimization thinking: Students frequently assume that to maximize the largest number, they should minimize the smallest number without considering all constraints. They fail to realize that the range constraint (largest - smallest = 50) means they actually need to maximize the smallest possible value to maximize the largest value.
- Ignoring the 'different integers' requirement: Students may overlook that all 25 integers must be different, leading them to incorrectly assume they can have repeated values or use non-integer increments when arranging the numbers around the median.
Errors while executing the approach
- Incorrect counting of positions around the median: When trying to find the optimal arrangement, students often miscount how many integers need to be placed before and after the median (50). They might place 11 or 13 numbers instead of exactly 12 on each side of position 13.
- Non-optimal arrangement of the 12 numbers below median: Students may not arrange the 12 numbers below the median optimally. Instead of using consecutive integers like \(\{38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49\}\) to maximize the smallest value, they might spread them out more, leading to a smaller minimum value.
- Arithmetic errors in applying the range formula: When calculating the maximum value using 'largest = smallest + range', students may make simple addition errors (38 + 50) or forget to apply the range constraint correctly.
Errors while selecting the answer
- Failing to verify the solution: Students might arrive at 88 but fail to double-check that their proposed set actually satisfies all constraints (25 different integers, median of 50, range of 50), leading them to second-guess their correct answer and choose a different option.
- Confusing maximum with other calculated values: During the solution process, students calculate multiple values (like the minimum value 38, the median 50, intermediate numbers). They might accidentally select an answer choice that corresponds to one of these other values rather than the maximum value of 88.
