A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer...

1. Translate the problem requirements: We have 15 different integers arranged in order. The median (8th number) is 25, and the range (largest - smallest) is 25. We need to find the maximum possible value for the largest integer.
2. Establish the median constraint: Since we have 15 integers, the 8th integer when arranged in ascending order must be 25. This means 7 integers are below 25 and 7 integers are above 25.
3. Set up the range relationship: \(\mathrm{Range} = \mathrm{Largest} - \mathrm{Smallest} = 25\). To maximize the largest number, we need to minimize the smallest number while maintaining the median constraint.
4. Determine the minimum possible smallest value: The smallest 7 integers must all be less than 25. To minimize the smallest value while keeping all integers different, we work backwards from the median constraint.
5. Calculate the maximum largest value: Once we find the minimum possible smallest integer, we can use the range formula to find the maximum possible largest integer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in everyday terms. Imagine we have 15 different test scores lined up from lowest to highest. The middle score (which is the 8th score since there are 7 scores below it and 7 above it) is 25. This middle score is called the median.

The range tells us the difference between the highest and lowest scores. Since the range is 25, this means: Highest Score - Lowest Score = 25.

Our goal is to find the largest possible value for the highest score in this lineup.

Process Skill: TRANSLATE - Converting the median and range definitions into concrete understanding

2. Establish the median constraint

Since we have 15 different integers, when we arrange them from smallest to largest, the 8th position will be our median of 25.

This gives us a clear picture:

• Positions 1, 2, 3, 4, 5, 6, 7: All must be less than 25
• Position 8: Must be exactly 25 (our median)
• Positions 9, 10, 11, 12, 13, 14, 15: All must be greater than 25

Since all integers must be different, none of the other 14 integers can equal 25.

3. Set up the range relationship

Now we use the range constraint. We know that:
\(\mathrm{Range} = \mathrm{Largest\,integer} - \mathrm{Smallest\,integer} = 25\)

To find the maximum possible largest integer, we need to think strategically. Since:
\(\mathrm{Largest\,integer} = \mathrm{Smallest\,integer} + 25\)

To maximize the largest integer, we need to maximize the smallest integer while still satisfying all our constraints.

Process Skill: INFER - Recognizing that maximizing the largest requires maximizing the smallest

4. Determine the minimum possible smallest value

Wait - let me reconsider this. To maximize the largest integer, we actually want to minimize the smallest integer, not maximize it. Let me think through this correctly.

Since \(\mathrm{Largest} = \mathrm{Smallest} + 25\), to maximize the Largest, we want to maximize the Smallest.

But we have a constraint: the smallest integer must be small enough that we can fit 7 different integers between it and 25 (our median).

If our smallest integer is S, then we need 7 different integers between S and 25.
The largest these 7 integers can be is: 24, 23, 22, 21, 20, 19, 18

So our smallest integer S must be at most 17 (since 18, 19, 20, 21, 22, 23, 24 gives us exactly 7 integers less than 25).

Therefore, the maximum value for our smallest integer is 18 (making the 7 integers: 18, 19, 20, 21, 22, 23, 24, and then 25 as the median).

Process Skill: APPLY CONSTRAINTS - Ensuring we can fit exactly 7 different integers between smallest and median

5. Calculate the maximum largest value

Now we can find our answer:
If the smallest integer is 18, then:
\(\mathrm{Largest\,integer} = \mathrm{Smallest\,integer} + \mathrm{Range}\)
\(\mathrm{Largest\,integer} = 18 + 25 = 43\)

Let's verify this works:

• Our 15 integers could be: 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 43
• Median (8th position): 25 ✓
• Range: \(43 - 18 = 25\) ✓
• All different integers ✓

Final Answer

The greatest possible integer that could be in this set is 43.

This matches answer choice D. 43.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding median position: Students may incorrectly think the median of 15 numbers is the 7th or 9th number instead of the 8th. With 15 numbers arranged in order, there are exactly 7 numbers below and 7 numbers above the median, making the 8th position the median.

2. Confusing the optimization strategy: Students may think they need to minimize the smallest number to maximize the largest number. Since \(\mathrm{Range} = \mathrm{Largest} - \mathrm{Smallest} = 25\), many students incorrectly reason that making the smallest number as small as possible will make the largest number as large as possible. However, the constraint that exactly 7 different integers must fit between the smallest and the median (25) actually requires maximizing the smallest number.

3. Overlooking the "different integers" constraint: Students may forget that all 15 integers must be different, leading them to allow repeated values or not properly account for the spacing needed between consecutive integers.

Errors while executing the approach

1. Incorrect counting of integers between smallest and median: When trying to fit 7 different integers between the smallest number and 25, students may miscount. For example, if the smallest is 18, they need to verify that 19, 20, 21, 22, 23, 24 gives exactly 6 more integers (total of 7 including 18) before reaching 25.

2. Arithmetic errors in range calculation: Students may make simple calculation mistakes when computing \(\mathrm{Largest} = \mathrm{Smallest} + \mathrm{Range}\), especially when working backwards from the constraint that the smallest number can be at most 18.

Errors while selecting the answer

1. Failing to verify the complete solution: Students may arrive at 43 but not double-check that their proposed set of 15 integers actually satisfies both the median = 25 and range = 25 constraints simultaneously, potentially selecting an answer that doesn't work when fully tested.