If 500 is the multiple of 100 that is closest to x and 400 is the multiple of 100 that...

Understanding the Question

We need to find which multiple of 100 is closest to x + y, given that:

• 500 is the closest multiple of 100 to x
• 400 is the closest multiple of 100 to y

What This Actually Means

If 500 is the closest multiple of 100 to x, then x must be in the range where 500 "wins" as the closest value. This happens when \(450 \leq \mathrm{x} < 550\).

Why these boundaries? At x = 450, we're exactly halfway between 400 and 500. At x = 550, we're exactly halfway between 500 and 600.

Similarly, if 400 is the closest multiple of 100 to y, then \(350 \leq \mathrm{y} < 450\).

Therefore, x + y must be in the range: \(800 \leq \mathrm{x} + \mathrm{y} < 1000\).

The Key to Sufficiency

To answer the question with certainty, we need to determine ONE specific multiple of 100 that's closest to x + y.

The possible candidates in our range are 800, 900, and 1000. The critical decision points are:

• If \(\mathrm{x} + \mathrm{y} < 850\), then 800 is closest
• If \(850 \leq \mathrm{x} + \mathrm{y} < 950\), then 900 is closest
• If \(\mathrm{x} + \mathrm{y} \geq 950\), then 1000 is closest

Notice that 850 and 950 are the "boundary points" where our answer changes. We need information that tells us which side of these boundaries x + y falls on.

Analyzing Statement 1

Statement 1: x < 500

Combined with what we already know (500 is closest to x), this narrows x to: \(450 \leq \mathrm{x} < 500\).

With y still in its full range of \(350 \leq \mathrm{y} < 450\), let's find the range of x + y:

• Minimum: 450 + 350 = 800
• Maximum: just under 500 + just under 450 = just under 950

So x + y is in the range \(800 \leq \mathrm{x} + \mathrm{y} < 950\).

Can We Get Different Answers?

This range includes the critical boundary at 850. Let's test:

• If \(\mathrm{x} = 460\) and \(\mathrm{y} = 380\), then \(\mathrm{x} + \mathrm{y} = 840\) → closest to 800
• If \(\mathrm{x} = 490\) and \(\mathrm{y} = 380\), then \(\mathrm{x} + \mathrm{y} = 870\) → closest to 900

Since we can get either 800 or 900 as our answer, Statement 1 is NOT sufficient.

This eliminates answer choices A and D.

Analyzing Statement 2

Statement 2: y < 400

Remember: We analyze each statement independently, so forget Statement 1 for now.

Combined with what we know (400 is closest to y), this narrows y to: \(350 \leq \mathrm{y} < 400\).

With x in its full range of \(450 \leq \mathrm{x} < 550\), let's find the range of x + y:

• Minimum: 450 + 350 = 800
• Maximum: just under 550 + just under 400 = just under 950

So x + y is in the range \(800 \leq \mathrm{x} + \mathrm{y} < 950\).

Same Range, Same Problem

This is exactly the same range we got with Statement 1! The critical boundary at 850 is still included. Let's verify we can get different answers:

• If \(\mathrm{x} = 470\) and \(\mathrm{y} = 360\), then \(\mathrm{x} + \mathrm{y} = 830\) → closest to 800
• If \(\mathrm{x} = 510\) and \(\mathrm{y} = 360\), then \(\mathrm{x} + \mathrm{y} = 870\) → closest to 900

Since we can get either 800 or 900 as our answer, Statement 2 is NOT sufficient.

This eliminates answer choices B and D.

Combining Both Statements

Using both statements together:

• From Statement 1: \(450 \leq \mathrm{x} < 500\)
• From Statement 2: \(350 \leq \mathrm{y} < 400\)

Therefore: \(800 \leq \mathrm{x} + \mathrm{y} < 900\)

The Final Test

Even with this narrower range, we still have that critical boundary at 850. Can x + y fall on either side of it?

Let's check with specific values that satisfy both statements:

• If \(\mathrm{x} = 460\) and \(\mathrm{y} = 380\), then \(\mathrm{x} + \mathrm{y} = 840\) → closest to 800
• If \(\mathrm{x} = 480\) and \(\mathrm{y} = 380\), then \(\mathrm{x} + \mathrm{y} = 860\) → closest to 900

Both scenarios satisfy our constraints, but they give different answers! Even with both statements combined, we cannot determine whether x + y is closest to 800 or 900.

Therefore, both statements together are NOT sufficient.

The Answer: E

The combined statements narrow x + y to the range \(800 \leq \mathrm{x} + \mathrm{y} < 900\), but this range still contains the critical boundary at 850. Since x + y can be on either side of this boundary, we cannot determine a unique answer.

Answer Choice E: "The statements together are not sufficient."