If q, s, and t are all different numbers, is q ? (t - q = |t - s| +...

Understanding the Question

We need to determine whether three different numbers q, s, and t are ordered specifically as \(\mathrm{q} < \mathrm{s} < \mathrm{t}\).

Let's think about what this means: we're asking if these three distinct numbers appear in increasing order with q being the smallest, s in the middle, and t being the largest.

What constitutes sufficiency: Since this is a yes/no question, we need to definitively answer either:

• YES: The order is exactly \(\mathrm{q} < \mathrm{s} < \mathrm{t}\)
• NO: The numbers are in any other order (like \(\mathrm{s} < \mathrm{q} < \mathrm{t}\), or \(\mathrm{t} < \mathrm{s} < \mathrm{q}\), etc.)

Given Information:

• q, s, and t are all different numbers
• We need to establish their exact order

Key Insight: Statement 1 involves an elegant geometric interpretation. The equation \(\mathrm{t} - \mathrm{q} = |\mathrm{t} - \mathrm{s}| + |\mathrm{s} - \mathrm{q}|\) represents distances on a number line, where the direct distance from q to t equals the sum of distances when traveling through s. This is only possible when s lies between the endpoints q and t.

Analyzing Statement 1

Statement 1: \(\mathrm{t} - \mathrm{q} = |\mathrm{t} - \mathrm{s}| + |\mathrm{s} - \mathrm{q}|\)

This equation relates distances between our three numbers. The left side \((\mathrm{t} - \mathrm{q})\) represents the direct distance from q to t, while the right side represents the sum of distances from q to s and from s to t.

Here's where the geometric insight becomes powerful: imagine these numbers positioned on a number line. For the equation to hold true, we first need \(\mathrm{t} > \mathrm{q}\) (otherwise \(\mathrm{t} - \mathrm{q}\) would be negative while the right side, being a sum of absolute values, is always non-negative).

Given that \(\mathrm{t} > \mathrm{q}\), when would the direct distance equal the sum of two intermediate distances? Only when the intermediate point s lies on the straight path between q and t. Think of it like this: if you're traveling from city q to city t, going through city s only covers the same total distance if s is on the direct route between q and t.

Let's verify this insight with concrete scenarios:

• If \(\mathrm{q} < \mathrm{s} < \mathrm{t}\): The path \(\mathrm{q} \rightarrow \mathrm{s} \rightarrow \mathrm{t}\) covers the same ground as \(\mathrm{q} \rightarrow \mathrm{t}\) directly ✓
• If \(\mathrm{s} < \mathrm{q} < \mathrm{t}\): Going from q to s means backtracking, then forward to t = longer path ✗
• If \(\mathrm{q} < \mathrm{t} < \mathrm{s}\): Going from q past t to reach s, then back to t = longer path ✗

Therefore, Statement 1 tells us that \(\mathrm{q} < \mathrm{s} < \mathrm{t}\) must be true.

[STOP - Statement 1 is SUFFICIENT!]

Statement 1 is sufficient. This eliminates choices B, C, and E.

Analyzing Statement 2

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

Statement 2: \(\mathrm{t} > \mathrm{q}\)

This simply tells us that t is greater than q, but provides no information about where s fits in the ordering.

With only \(\mathrm{t} > \mathrm{q}\), we could have:

• \(\mathrm{q} < \mathrm{s} < \mathrm{t}\) (which would answer YES to our question)
• \(\mathrm{q} < \mathrm{t} < \mathrm{s}\) (which would answer NO to our question)
• \(\mathrm{s} < \mathrm{q} < \mathrm{t}\) (which would answer NO to our question)

Since we can construct scenarios that lead to different answers to our question, we cannot determine whether \(\mathrm{q} < \mathrm{s} < \mathrm{t}\).

Statement 2 alone is NOT sufficient. This eliminates choices B and D.

The Answer: A

Statement 1 alone is sufficient because the distance equation can only be satisfied when s lies between q and t, giving us exactly the order we need to confirm. Statement 2 alone only tells us that \(\mathrm{t} > \mathrm{q}\), leaving s's position completely undetermined.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."