Set S consists of five consecutive integers, and set T consists of seven consecutive integers. Is the median of the...

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Understanding the Question

Let's start with what we need to determine: Is the median of set S equal to the median of set T?

Given Information:

• Set S consists of five consecutive integers
• Set T consists of seven consecutive integers

What We Need to Determine:
For consecutive integers, the median is simply the middle value:

• Set S (5 numbers): The median is the 3rd number in the sequence
• Set T (7 numbers): The median is the 4th number in the sequence

Here's the key insight that will unlock this problem: For any set of consecutive integers, the sum equals \(\mathrm{(number\,of\,terms)} \times \mathrm{(median)}\). This means:

• Sum of S = \(5 \times \mathrm{median\,of\,S}\)
• Sum of T = \(7 \times \mathrm{median\,of\,T}\)

This relationship will prove crucial for our analysis.

Analyzing Statement 1

Statement 1: The median of the numbers in Set S is 0

This tells us exactly what the median of S is (which is 0), but gives us absolutely no information about set T. Without knowing anything about T's median, we cannot determine whether it also equals 0.

Let's visualize with examples:

• If T = {-3, -2, -1, 0, 1, 2, 3}, then median of T = 0 → Answer: YES, they're equal
• If T = {2, 3, 4, 5, 6, 7, 8}, then median of T = 5 → Answer: NO, they're not equal

Since we can get both YES and NO answers, Statement 1 is NOT sufficient.

[STOP - 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 sum of the numbers in set S is equal to the sum of the numbers in set T

Using our key insight from earlier:

• Sum of S = \(5 \times \mathrm{median\,of\,S}\)
• Sum of T = \(7 \times \mathrm{median\,of\,T}\)

If these sums are equal:
\(5 \times \mathrm{median\,of\,S} = 7 \times \mathrm{median\,of\,T}\)

This gives us: median of S = \(\frac{7}{5} \times \mathrm{median\,of\,T}\)

The medians can only be equal when both equal 0! Here's why:

• If median of S = median of T = m (where m ≠ 0)
• Then \(5m = 7m\), which is impossible unless \(m = 0\)

Let's test specific scenarios to confirm:

Different scenarios give different answers, so Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

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

From Statement 1: The median of S is 0
From Statement 2: Sum of S = Sum of T

Since the median of S is 0:

• Sum of S = \(5 \times 0 = 0\)

Since the sums are equal (from Statement 2):

• Sum of T = 0

Therefore:

• Sum of T = \(7 \times \mathrm{median\,of\,T} = 0\)
• This means: median of T = \(\frac{0}{7} = 0\)

Conclusion: Both medians equal 0, so we can definitively answer YES - the medians are equal.

The statements together are sufficient.

[STOP - Sufficient!] This eliminates choice E.

The Answer: C

Together, the statements tell us that both sets must have median 0, allowing us to answer the question with certainty.

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