Question

List A has seven integers; whose range is 80 and median is 240. The median for the three smallest integers in List A is 180. What is the possible range for the largest three integers in the List A?
Possible Values:
I. 75
II. 24
III. 0

• A. I only
• B. II only
• C. I and III only
• D. II and III only
• E. III only

Hint:

When asked for a possible range or value for a quantity based on statistical properties, try to construct extreme cases. To find the minimum and maximum possible values, push the unknown variables to their limits while still satisfying all given conditions.

Answer 1

Answer: (E)

Note: The question format is unusual. It asks for a "possible range" but provides single numbers as options. It should be interpreted as "Which of the following is a possible value for the range of the three largest integers?".
Step 1: Understanding the Concept:
We are given properties of a set of seven integers and need to find the possible values for the range of the three largest integers in the set.
Let the seven integers be \(a_1, a_2, a_3, a_4, a_5, a_6, a_7\) in non-decreasing order.
Step 2: Detailed Explanation:
From the problem statement, we have:

Median of List A is 240: The median of seven integers is the 4th term. So, \(a_4 = 240\).
Range of List A is 80: \(a_7 - a_1 = 80\).
Median of the three smallest integers is 180: The three smallest are \(a_1, a_2, a_3\). Their median is the middle term, \(a_2\). So, \(a_2 = 180\).
We can now establish the relationships and constraints between the integers: \[ a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5 \leq a_6 \leq a_7 \] Substituting the known values: \[ a_1 \leq 180 \leq a_3 \leq 240 \leq a_5 \leq a_6 \leq a_7 \] The question asks for the range of the largest three integers, which is \(a_7 - a_5\). We need to find the possible values for this expression.
Finding the bounds for \(a_7\) and \(a_5\):

From \(a_2=180\), we know \(a_1 \leq 180\).
From the range, \(a_7 = a_1 + 80\). Since \(a_1 \leq 180\), then \(a_7 \leq 180 + 80 = 260\).
We also know \(a_4=240 \leq a_5 \leq a_6 \leq a_7\). This implies \(a_7 \geq 240\).
So, the possible values for \(a_7\) are in the range [240, 260].
The possible values for \(a_5\) are constrained by \(240 \leq a_5 \leq a_7\).
Finding the bounds for the range \(a_7 - a_5\):

Minimum value: To minimize the range \(a_7 - a_5\), we want \(a_7\) and \(a_5\) to be as close as possible. This occurs when \(a_5 = a_7\). Is this possible? Let's try to construct a set. Let \(a_7 = 240\). Then \(a_1 = 240 - 80 = 160\). A possible set is \{160, 180, 200, 240, 240, 240, 240\}. All conditions are met. In this case, the range of the largest three integers \(\{240, 240, 240\}\) is \(240 - 240 = 0\). So, a range of 0 is possible.
Maximum value: To maximize the range \(a_7 - a_5\), we want \(a_7\) to be as large as possible and \(a_5\) to be as small as possible. Max value for \(a_7\) is 260. This occurs when \(a_1=180\). Min value for \(a_5\) is 240 (since \(a_4 \leq a_5\)). A possible set is \{180, 180, 180, 240, 240, 250, 260\}. All conditions are met. In this case, the range of the largest three integers \(\{240, 250, 260\}\) is \(260 - 240 = 20\).
The possible values for the range \(a_7 - a_5\) are in the interval [0, 20].
Evaluate the options:

I. 75: Not possible, as it is outside the interval [0, 20].
II. 24: Not possible, as it is outside the interval [0, 20].
III. 0: Possible, as it is within the interval [0, 20].
Step 3: Final Answer:
Only 0 is a possible value for the range of the three largest integers. This corresponds to option (E).