Each of the five divisions of a certain company sent representatives to a conference. If the numbers of representatives sent...

Understanding the Question

We need to determine whether the range of representatives sent by five divisions is greater than 2.

Given Information

• Five divisions sent representatives to a conference
• Four divisions sent: 3, 4, 5, and 5 representatives
• One division sent an unknown number of representatives (let's call it x)

What We Need to Find

The range = (maximum value - minimum value)

Here's the key insight: The range depends entirely on where x falls:

• If x is between 3 and 5 (inclusive): \(\mathrm{range} = 5 - 3 = 2\)
• If \(x < 3\): then x becomes the minimum, so \(\mathrm{range} = 5 - x > 2\)
• If \(x > 5\): then x becomes the maximum, so \(\mathrm{range} = x - 3 > 2\)

So we're really asking: Is x outside the interval \([3, 5]\)?

For this question to be answerable with certainty, we need information that forces x to be either less than 3 or greater than 5.

Analyzing Statement 1

Statement 1: The median of the five numbers was greater than the average.

This gives us a relationship between two key statistics. Let's think through what this means:

• The median is the middle value when sorted
• The average includes all values equally
• When median > average, it typically indicates the data is left-skewed (some low values pulling the average down)

Let's test whether this forces x outside \([3, 5]\):

Test Case 1: Let \(x = 4\) (right in the middle of our known values)

• Sorted values: 3, 4, 4, 5, 5
• Median = 4
• Average = \(21/5 = 4.2\)
• But here median < average! ❌

This tells us x can't be in the middle range. Good start! Let's check the extremes:

Test Case 2: Let \(x = 2\) (less than 3)

• Sorted values: 2, 3, 4, 5, 5
• Median = 4
• Average = \(19/5 = 3.8\)
• Median > average ✓ and \(\mathrm{range} = 5 - 2 = 3 > 2\) ✓

Test Case 3: Let \(x = 5\) (at the upper boundary)

• Sorted values: 3, 4, 5, 5, 5
• Median = 5
• Average = \(22/5 = 4.4\)
• Median > average ✓ but \(\mathrm{range} = 5 - 3 = 2\) ❌

Different values of x that satisfy the condition lead to different answers about whether range > 2.

Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The median equals 4.

For the median to be 4, when we sort all five numbers, the middle one must be 4. Since we already have a 4 among our known values, this happens when x doesn't push the 4 out of the middle position.

This occurs when \(x \leq 4\).

Let's test whether this constraint determines if range > 2:

Test Case 1: Let \(x = 2\) (less than 3)

• Sorted: 2, 3, 4, 5, 5
• Median = 4 ✓
• Range = \(5 - 2 = 3 > 2\) ✓

Test Case 2: Let \(x = 3.5\) (between 3 and 4)

• Sorted: 3, 3.5, 4, 5, 5
• Median = 4 ✓
• Range = \(5 - 3 = 2\) ❌

Again, different values of x that make the median 4 lead to different conclusions about whether range > 2.

Statement 2 is NOT sufficient.

This eliminates choice B.

Combining Statements

Now let's see what happens when we use both statements together.

From Statement 1: Median > Average
From Statement 2: Median = 4

This means: \(4 > \mathrm{Average}\)

Since the sum of our known values is \(3 + 4 + 5 + 5 = 17\), the average = \((17 + x)/5\).

So we need:
\(4 > (17 + x)/5\)

Solving this inequality:

• \(20 > 17 + x\)
• \(x < 3\)

Here's the crucial insight:

• Statement 2 tells us \(x \leq 4\) (for median to be 4)
• Statement 1 further restricts this to \(x < 3\) (for median to exceed average)

When \(x < 3\), it becomes the minimum value in our set, making the \(\mathrm{range} = 5 - x\).
Since \(x < 3\), we have:
\(\mathrm{range} = 5 - x > 5 - 3 = 2\) ✓

[STOP - Sufficient!]

Therefore, when both statements are true, the range must be greater than 2.

The statements together are sufficient.

This eliminates choice E.

The Answer: C

Both statements together tell us that \(x < 3\), which guarantees the range is greater than 2. Neither statement alone provides this certainty.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."