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If \(0 < \mathrm{r} < 1 < \mathrm{s} < 2\). Which of the following must be less than 1.
Let's first understand what we're working with in everyday language. We have two numbers: r and s.
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):
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
Let's test each expression by trying some specific examples and seeing what happens at the extreme cases.
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:
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 \(\mathrm{r}/\mathrm{s} < \mathrm{r} < 1\).
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:
So rs can sometimes be greater than 1.
We're subtracting a number less than 1 from a number between 1 and 2.
Let's try some examples:
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
Now let's think more systematically about why these results make sense:
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.
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.
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
From our analysis:
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.
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.
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.
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.
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:
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:
Therefore, only I must be less than 1.
Answer: A