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 with median 25 and range 25. We need to find the greatest possible value in the set. Since there are 15 integers, the median is the 8th number when arranged in order. Range is the difference between the largest and smallest values.
2. Establish the median constraint: With 15 integers in ascending order, the 8th position must be 25, meaning exactly 7 numbers are below 25 and 7 numbers are above 25.
3. Set up the range relationship: Since range = largest - smallest = 25, we can express the largest value in terms of the smallest: \(\mathrm{largest} = \mathrm{smallest} + 25\).
4. Optimize to maximize the largest value: To maximize the largest number, we need to maximize the smallest number while satisfying all constraints, particularly ensuring we can fit 7 distinct integers between the smallest and the median of 25.
5. Apply the constraint and solve: The smallest number can be at most 18 (allowing integers 19, 20, 21, 22, 23, 24 before reaching median 25). Therefore, the largest possible value is \(18 + 25 = 43\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• We have 15 different integers (all distinct numbers)
• The median is 25 (the middle value when arranged in order)
• The range is 25 (the difference between the biggest and smallest numbers)
• We want to find the greatest possible integer in this set

Since we have 15 numbers, when we arrange them from smallest to largest, the median will be the 8th number (because there are 7 numbers before it and 7 numbers after it). So the 8th position must contain the value 25.

The range tells us: \(\mathrm{largest\ number} - \mathrm{smallest\ number} = 25\)

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Establish the median constraint

Since we have 15 integers arranged in ascending order, let's think about what the median constraint means:

Position 1, 2, 3, 4, 5, 6, 7, [8], 9, 10, 11, 12, 13, 14, 15

The 8th position must be 25. This means:
• Positions 1-7 must contain 7 different integers, all less than 25
• Position 8 contains 25
• Positions 9-15 must contain 7 different integers, all greater than 25

This is a crucial constraint because all 15 numbers must be different, and we need exactly 7 numbers below 25 and exactly 7 numbers above 25.

3. Set up the range relationship

From the range constraint, we know:

\(\mathrm{Range} = \mathrm{largest\ number} - \mathrm{smallest\ number} = 25\)

This means: \(\mathrm{largest\ number} = \mathrm{smallest\ number} + 25\)

So if we can determine the largest possible value for the smallest number, we can find the largest possible value for the biggest number.

4. Optimize to maximize the largest value

To maximize the largest number, we need to maximize the smallest number. But we're constrained by the median requirement.

Let's think about this step by step. If the smallest number is S, then we need to fit exactly 7 different integers between S and 25 (not including 25).

The 7 numbers less than 25 would be: S, S+1, S+2, S+3, S+4, S+5, S+6

For this to work, the largest of these (which is S+6) must be less than 25:
\(\mathrm{S} + 6 < 25\)
\(\mathrm{S} < 19\)

Since we're dealing with integers, the largest possible value for S is 18.

Process Skill: APPLY CONSTRAINTS - Using the median requirement to limit our possibilities

5. Apply the constraint and solve

Now let's verify our solution works and calculate the answer:

If the smallest number is 18, then the 7 numbers less than the median are:
18, 19, 20, 21, 22, 23, 24

The median is 25 (8th position).

Using our range relationship:
\(\mathrm{Largest\ number} = \mathrm{smallest\ number} + 25 = 18 + 25 = 43\)

Let's verify this works: We need 7 different integers greater than 25. If the largest is 43, we could have:
26, 27, 28, 29, 30, 31, 43 (for example)

This gives us all 15 different integers with median 25 and range \(43 - 18 = 25\). ✓

4. Final Answer

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

Checking against the answer choices: ["32","37","40","43","50"]

Our answer of 43 corresponds to choice D, which matches the given correct answer.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the median position
Many students incorrectly think the median of 15 numbers is the average of the 7th and 8th positions (like it would be for an even number of values), rather than recognizing that for 15 numbers, the median is simply the 8th number when arranged in order.

Faltering Point 2: Incorrectly setting up the optimization strategy
Students often try to directly maximize the largest number without realizing that to maximize the largest number, they need to maximize the smallest number first. They fail to see the connection between the range constraint (\(\mathrm{largest} = \mathrm{smallest} + 25\)) and the optimization goal.

Faltering Point 3: Misinterpreting the "different integers" constraint
Students may overlook that all 15 numbers must be distinct, leading them to allow repeated values or not properly account for the spacing requirements between consecutive integers.

Errors while executing the approach

Faltering Point 1: Incorrect constraint setup for numbers below the median
When determining how many integers can fit below 25, students often write \(\mathrm{S} + 6 ≤ 25\) instead of \(\mathrm{S} + 6 < 25\), forgetting that the 7th number below the median cannot equal 25 (since 25 is already the median and all numbers must be different).

Faltering Point 2: Arithmetic errors in solving the inequality
Students may incorrectly solve \(\mathrm{S} + 6 < 25\), getting \(\mathrm{S} ≤ 19\) instead of \(\mathrm{S} < 19\), or forgetting that since we need integers, \(\mathrm{S} < 19\) means the maximum value of S is 18.

Errors while selecting the answer

Faltering Point 1: Applying the range formula incorrectly
After finding that the smallest number is 18, students may forget to add 25 to get the largest number, or they may add incorrectly, getting \(18 + 25 = 42\) instead of 43, or confusing which number to use in the range calculation.