If 0 . Which of the following must be less than 1. (r/s rs s - r

1. Translate the problem requirements: We have constraints \(0 < \mathrm{r} < 1 < \mathrm{s} < 2\), meaning r is a positive fraction less than 1, and s is between 1 and 2. We need to determine which of the three expressions (\(\mathrm{r}/\mathrm{s}\), \(\mathrm{rs}\), \(\mathrm{s}-\mathrm{r}\)) must always be less than 1 given these constraints.
2. Analyze each expression using the constraint boundaries: For each expression, determine whether it can ever equal or exceed 1 by testing extreme values within the given ranges.
3. Apply logical reasoning to confirm which expressions must be less than 1: Use the mathematical relationships to definitively determine which expressions are always, sometimes, or never less than 1.
4. Match findings to answer choices: Identify which Roman numerals correspond to expressions that must be less than 1.

Execution of Strategic Approach

1. Translate the problem requirements

Let's first understand what we're working with in everyday language. We have two numbers: r and s.

• r is a positive number that's less than 1 (like \(0.3\), \(0.7\), or \(0.9\))
• s is a number that's bigger than 1 but smaller than 2 (like \(1.2\), \(1.5\), or \(1.8\))

We need to figure out which of these three expressions will always be less than 1, no matter what specific values of r and s we pick (as long as they follow our rules):

• Expression I: \(\mathrm{r}/\mathrm{s}\) (r divided by s)
• Expression II: \(\mathrm{rs}\) (r times s)
• Expression III: \(\mathrm{s} - \mathrm{r}\) (s minus r)

The key word here is "must" - we're looking for expressions that are guaranteed to be less than 1.

Process Skill: TRANSLATE - Converting the constraint notation into clear understanding of what values r and s can take

2. Analyze each expression using the constraint boundaries

Let's test each expression by trying some specific examples and seeing what happens at the extreme cases.

For Expression I: \(\mathrm{r}/\mathrm{s}\)

Think about this intuitively: we're dividing a number less than 1 by a number greater than 1. When you divide a small number by a bigger number, you get an even smaller result.

Let's try some examples:

• If \(\mathrm{r} = 0.9\) and \(\mathrm{s} = 1.1\), then \(\mathrm{r}/\mathrm{s} = 0.9/1.1 \approx 0.82\)
• If \(\mathrm{r} = 0.1\) and \(\mathrm{s} = 1.9\), then \(\mathrm{r}/\mathrm{s} = 0.1/1.9 \approx 0.05\)

Even in the "worst case" where r is as close to 1 as possible and s is as close to 1 as possible, we still get \(\mathrm{r}/\mathrm{s} < \mathrm{r} < 1\).

For Expression II: \(\mathrm{rs}\)

Here we're multiplying a number less than 1 by a number between 1 and 2. This could go either way.

Let's try some examples:

• If \(\mathrm{r} = 0.9\) and \(\mathrm{s} = 1.1\), then \(\mathrm{rs} = 0.9 \times 1.1 = 0.99\) (less than 1)
• If \(\mathrm{r} = 0.8\) and \(\mathrm{s} = 1.5\), then \(\mathrm{rs} = 0.8 \times 1.5 = 1.2\) (greater than 1!)

So rs can sometimes be greater than 1.

For Expression III: \(\mathrm{s} - \mathrm{r}\)

We're subtracting a number less than 1 from a number between 1 and 2.

Let's try some examples:

• If \(\mathrm{r} = 0.9\) and \(\mathrm{s} = 1.1\), then \(\mathrm{s} - \mathrm{r} = 1.1 - 0.9 = 0.2\) (less than 1)
• If \(\mathrm{r} = 0.1\) and \(\mathrm{s} = 1.9\), then \(\mathrm{s} - \mathrm{r} = 1.9 - 0.1 = 1.8\) (greater than 1!)

So \(\mathrm{s} - \mathrm{r}\) can sometimes be greater than 1.

Process Skill: CONSIDER ALL CASES - Testing boundary values to see if expressions can exceed 1

3. Apply logical reasoning to confirm which expressions must be less than 1

Now let's think more systematically about why these results make sense:

Expression I: \(\mathrm{r}/\mathrm{s}\) must always be less than 1

Since \(\mathrm{r} < 1\) and \(\mathrm{s} > 1\), we have:

\(\mathrm{r}/\mathrm{s} < \mathrm{r} < 1\)

This will always be true because dividing by a number greater than 1 makes the result smaller than the original numerator, and the numerator was already less than 1.

Expression II: \(\mathrm{rs}\) is not always less than 1

We showed with \(\mathrm{r} = 0.8\) and \(\mathrm{s} = 1.5\) that rs can equal \(1.2\), which is greater than 1. So this doesn't "must" be less than 1.

Expression III: \(\mathrm{s} - \mathrm{r}\) is not always less than 1

We showed with \(\mathrm{r} = 0.1\) and \(\mathrm{s} = 1.9\) that \(\mathrm{s} - \mathrm{r}\) can equal \(1.8\), which is greater than 1. So this doesn't "must" be less than 1.

Process Skill: INFER - Drawing the logical conclusion that only when an expression cannot possibly exceed 1 does it "must" be less than 1

4. Match findings to answer choices

From our analysis:

• Expression I (\(\mathrm{r}/\mathrm{s}\)) MUST be less than 1 ✓
• Expression II (\(\mathrm{rs}\)) does NOT always have to be less than 1 ✗
• Expression III (\(\mathrm{s} - \mathrm{r}\)) does NOT always have to be less than 1 ✗

Therefore, only Roman numeral I represents an expression that must be less than 1.

Final Answer: A. I only

This matches our given correct answer, confirming our solution is correct.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "must be less than 1"
Students often confuse "must be less than 1" with "can be less than 1." They may look for expressions that are sometimes less than 1, rather than expressions that are ALWAYS less than 1 for any valid values of r and s. This leads them to incorrectly include expressions II and III in their answer.

Faltering Point 2: Not fully utilizing the constraint boundaries
Students may not recognize that they need to test extreme cases within the given constraints \((0 < \mathrm{r} < 1 < \mathrm{s} < 2)\). They might only test one or two convenient values instead of checking boundary cases like r close to 1 or s close to 2, missing scenarios where expressions can exceed 1.

Faltering Point 3: Overlooking the need for systematic analysis
Students may rely on intuition or quick mental math instead of systematically analyzing each expression. They might assume that since \(\mathrm{r} < 1\), any operation involving r will result in a value less than 1, without considering how the operations and the value of s affect the final result.

Errors while executing the approach

Faltering Point 1: Choosing non-representative test values
When testing expressions with specific values, students often pick "nice" numbers that are easy to calculate (like \(\mathrm{r} = 0.5\), \(\mathrm{s} = 1.5\)) but may not reveal cases where expressions exceed 1. They might miss testing values like \(\mathrm{r} = 0.8\) and \(\mathrm{s} = 1.4\), which would show that \(\mathrm{rs} = 1.12 > 1\).

Faltering Point 2: Arithmetic errors in calculations
Students may make simple computational mistakes when evaluating expressions like rs or \(\mathrm{s} - \mathrm{r}\), leading to incorrect conclusions about whether these expressions can exceed 1. For example, miscalculating \(0.8 \times 1.5\) or \(1.9 - 0.1\).

Faltering Point 3: Insufficient testing of boundary cases
Students might test only one example for each expression and conclude based on that single result, rather than testing multiple cases including extreme values within the constraints. This incomplete testing can lead to missing counterexamples that disprove their initial conclusions.

Errors while selecting the answer

Faltering Point 1: Misreading Roman numeral combinations
After correctly identifying that only Expression I must be less than 1, students may misread the answer choices and select "D. I and II" or "E. I and III" thinking these represent Expression I alone, rather than carefully noting that "A. I only" is the correct choice.

Faltering Point 2: Second-guessing correct analysis
Students who correctly determine that only Expression I must always be less than 1 might second-guess themselves, thinking "this seems too simple" or "there must be more than one correct expression," leading them to change their answer to include additional expressions.

Alternate Solutions

Smart Numbers Approach

For this problem, we can strategically choose specific values for r and s that satisfy our constraints \(0 < \mathrm{r} < 1 < \mathrm{s} < 2\), then test each expression to see which must be less than 1.

Step 1: Choose smart values within the constraints

Let's select values that make calculations easy while representing different scenarios:

• First test: \(\mathrm{r} = 0.5\), \(\mathrm{s} = 1.5\)
• Second test: \(\mathrm{r} = 0.9\), \(\mathrm{s} = 1.1\) (values close to the boundaries)

Step 2: Test each expression with our first set of values (\(\mathrm{r} = 0.5\), \(\mathrm{s} = 1.5\))

Expression I: \(\mathrm{r}/\mathrm{s}\)

\(\mathrm{r}/\mathrm{s} = 0.5/1.5 = 1/3 \approx 0.33 < 1\) ✓

Expression II: \(\mathrm{rs}\)

\(\mathrm{rs} = 0.5 \times 1.5 = 0.75 < 1\) ✓

Expression III: \(\mathrm{s} - \mathrm{r}\)

\(\mathrm{s} - \mathrm{r} = 1.5 - 0.5 = 1.0 = 1\) (not less than 1) ✗

Step 3: Test with boundary values (\(\mathrm{r} = 0.9\), \(\mathrm{s} = 1.1\))

Expression I: \(\mathrm{r}/\mathrm{s}\)

\(\mathrm{r}/\mathrm{s} = 0.9/1.1 \approx 0.82 < 1\) ✓

Expression II: \(\mathrm{rs}\)

\(\mathrm{rs} = 0.9 \times 1.1 = 0.99 < 1\) ✓

Expression III: \(\mathrm{s} - \mathrm{r}\)

\(\mathrm{s} - \mathrm{r} = 1.1 - 0.9 = 0.2 < 1\) ✓

Step 4: Test extreme case to check Expression II

Let's try \(\mathrm{r} = 0.9\) and \(\mathrm{s} = 1.9\):

\(\mathrm{rs} = 0.9 \times 1.9 = 1.71 > 1\) ✗

Step 5: Test extreme case to check Expression III

Let's try \(\mathrm{r} = 0.1\) and \(\mathrm{s} = 1.9\):

\(\mathrm{s} - \mathrm{r} = 1.9 - 0.1 = 1.8 > 1\) ✗

Step 6: Verify Expression I always works

Since \(\mathrm{r} < 1\) and \(\mathrm{s} > 1\), we have \(\mathrm{r}/\mathrm{s} < 1/1 = 1\). Let's confirm with extreme values:

\(\mathrm{r} = 0.99\), \(\mathrm{s} = 1.01\): \(\mathrm{r}/\mathrm{s} = 0.99/1.01 \approx 0.98 < 1\) ✓

Conclusion:

Our smart number testing shows that:

• Expression I (\(\mathrm{r}/\mathrm{s}\)) is always less than 1
• Expression II (\(\mathrm{rs}\)) can be greater than 1
• Expression III (\(\mathrm{s} - \mathrm{r}\)) can be greater than 1

Therefore, only I must be less than 1.

Answer: A