How many integers n are there such that r ? (s - r = 5 r and s are not...

Understanding the Question

We need to find the exact count of integers n where \(\mathrm{r} < \mathrm{n} < \mathrm{s}\).

This is a value question - we need a specific number as our answer. For this question to be sufficient, we must be able to determine exactly one count of integers between r and s.

Given Information

• Two values r and s (we don't know if r < s or s < r initially)
• We're counting integers strictly between them (not including r or s)
• No other constraints are given

Key Insight

The number of integers between two real numbers depends on:

1. The distance between them
2. Their positions relative to integers on the number line

For example, between 1.5 and 4.5 (distance = 3), there are 2 integers (2 and 3). But between 1.9 and 4.9 (also distance = 3), there are 3 integers (2, 3, and 4). Position matters!

Analyzing Statement 1

Statement 1: \(\mathrm{s} - \mathrm{r} = 5\)

This tells us that s is 5 units greater than r (so r < s), and the exact distance between them is 5. However, it doesn't tell us where they are positioned on the number line.

Let's test different positions to see if we always get the same count:

Test Case 1: \(\mathrm{r} = 1, \mathrm{s} = 6\)
- Integers between 1 and 6: 2, 3, 4, 5
- Count = 4

Test Case 2: \(\mathrm{r} = 1.5, \mathrm{s} = 6.5\)
- Integers between 1.5 and 6.5: 2, 3, 4, 5, 6
- Count = 5

Since different positions give different counts (4 vs 5), we cannot determine a unique answer.

[STOP - Not Sufficient!] Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: r and s are not integers

This eliminates boundary cases where r or s might equal an integer, but tells us nothing about:

• Which value is larger (r or s)
• The distance between them
• Their specific positions

Test Case 1: \(\mathrm{r} = 1.5, \mathrm{s} = 2.5\) (both non-integers)
- Integers between them: just 2
- Count = 1

Test Case 2: \(\mathrm{r} = 1.5, \mathrm{s} = 10.5\) (both non-integers)
- Integers between them: 2, 3, 4, 5, 6, 7, 8, 9, 10
- Count = 9

We get completely different counts (1 vs 9), so we cannot determine a unique answer.

[STOP - Not Sufficient!] Statement 2 alone is NOT sufficient.

This eliminates choice B.

Since neither statement alone is sufficient, our answer must be C or E.

Combining Both Statements

With both statements together:

• \(\mathrm{s} - \mathrm{r} = 5\) (exact distance, with s > r)
• Neither r nor s is an integer

Here's the key insight: When two non-integers are exactly 5 units apart, there are always exactly 5 integers between them!

To understand why, imagine r falls between consecutive integers k and k+1. Since r is not an integer, we can write \(\mathrm{r} = \mathrm{k} + \text{(some fraction between 0 and 1)}\). Then \(\mathrm{s} = \mathrm{r} + 5 = \mathrm{k} + \text{(same fraction)} + 5\).

The integers between r and s must be: k+1, k+2, k+3, k+4, and k+5. That's exactly 5 integers, regardless of where r starts!

Let's verify with concrete examples:

• \(\mathrm{r} = 1.3, \mathrm{s} = 6.3\): integers between are 2, 3, 4, 5, 6 → count = 5
• \(\mathrm{r} = 7.8, \mathrm{s} = 12.8\): integers between are 8, 9, 10, 11, 12 → count = 5
• \(\mathrm{r} = -2.1, \mathrm{s} = 2.9\): integers between are -2, -1, 0, 1, 2 → count = 5

The count is always 5!

[STOP - Sufficient!] Both statements together are sufficient.

This eliminates choice E.

The Answer: C

Both statements together give us exactly one answer (5 integers), but neither statement alone is sufficient.

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