If m, p, s and v are positive, and m/p , which of the following must be between (m/p and...

1. Translate the problem requirements: We have positive variables m, p, s, v with \(\mathrm{m/p} < \mathrm{s/v}\). We need to determine which of three expressions must always fall between these two fractions.
2. Test each expression with strategic examples: Use simple values that satisfy \(\mathrm{m/p} < \mathrm{s/v}\) to check if each expression consistently falls between the bounds.
3. Apply weighted average reasoning for expression I: Recognize that \(\mathrm{(m+s)/(p+v)}\) represents a weighted average of the two fractions, which must lie between them.
4. Verify counterexamples for expressions II and III: Show that expressions II and III can fall outside the required range using concrete examples.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're working with in plain English. We have four positive numbers: m, p, s, and v. We're told that the fraction \(\mathrm{m/p}\) is smaller than the fraction \(\mathrm{s/v}\).

Think of this like comparing test scores. If \(\mathrm{m/p}\) represents one student's score and \(\mathrm{s/v}\) represents another student's score, we know the second student scored higher.

Now we need to figure out which of three different expressions will always fall between these two scores - no matter what positive numbers we choose for m, p, s, and v (as long as \(\mathrm{m/p} < \mathrm{s/v}\)).

Process Skill: TRANSLATE - Converting the inequality constraint into a clear understanding of what we need to find

2. Test each expression with strategic examples

Let's use simple numbers to test our intuition. Say \(\mathrm{m = 1, p = 2, s = 3, v = 4}\).

This gives us \(\mathrm{m/p = 1/2 = 0.5}\) and \(\mathrm{s/v = 3/4 = 0.75}\).

So we need to check if each expression falls between \(\mathrm{0.5}\) and \(\mathrm{0.75}\).

Testing Expression I: \(\mathrm{(m+s)/(p+v)}\)
\(\mathrm{(1+3)/(2+4) = 4/6 = 2/3 ≈ 0.67}\)
This falls between \(\mathrm{0.5}\) and \(\mathrm{0.75}\) ✓

Testing Expression II: \(\mathrm{ms/(pv)}\)
\(\mathrm{(1×3)/(2×4) = 3/8 = 0.375}\)
This is less than \(\mathrm{0.5}\), so it's not between our bounds ✗

Testing Expression III: \(\mathrm{s/v - m/p}\)
\(\mathrm{3/4 - 1/2 = 0.75 - 0.5 = 0.25}\)
This is less than \(\mathrm{0.5}\), so it's not between our bounds ✗

But we need to check if this pattern holds for ALL possible values, not just our example.

3. Apply weighted average reasoning for expression I

Here's the key insight for Expression I: \(\mathrm{(m+s)/(p+v)}\) is actually a weighted average!

Think of it this way: imagine you're mixing two different concentrations of a solution. If you have m parts of substance in p parts of solution, that's concentration \(\mathrm{m/p}\). If you have s parts of substance in v parts of solution, that's concentration \(\mathrm{s/v}\).

When you combine them, you get \(\mathrm{(m+s)}\) total parts of substance in \(\mathrm{(p+v)}\) total parts of solution. The resulting concentration \(\mathrm{(m+s)/(p+v)}\) must be between the two original concentrations.

Mathematically, this weighted average property means:
If \(\mathrm{m/p < s/v}\), then \(\mathrm{m/p < (m+s)/(p+v) < s/v}\)

This will ALWAYS be true, regardless of what positive values we choose for m, p, s, and v.

Process Skill: INFER - Recognizing the weighted average property that guarantees Expression I always falls between the bounds

4. Verify counterexamples for expressions II and III

For Expression II and III, we need to show they don't always fall between \(\mathrm{m/p}\) and \(\mathrm{s/v}\).

Expression II: \(\mathrm{ms/(pv)}\)
This is the product of the two fractions: \(\mathrm{(m/p) × (s/v)}\)
Let's try \(\mathrm{m = 1, p = 3, s = 2, v = 4}\):

• \(\mathrm{m/p = 1/3 ≈ 0.33}\)
• \(\mathrm{s/v = 2/4 = 0.5}\)
• \(\mathrm{ms/(pv) = (1×2)/(3×4) = 2/12 = 1/6 ≈ 0.17}\)

Since \(\mathrm{0.17 < 0.33}\), the product falls below our lower bound. Expression II doesn't always work.

Expression III: \(\mathrm{s/v - m/p}\)
This represents the difference between the two fractions.
Using the same values: \(\mathrm{s/v - m/p = 0.5 - 0.33 = 0.17}\)

Since \(\mathrm{0.17 < 0.33}\), the difference also falls below our lower bound. Expression III doesn't always work either.

Process Skill: CONSIDER ALL CASES - Testing different examples to show that Expressions II and III can fall outside the required range

4. Final Answer

Only Expression I, \(\mathrm{(m+s)/(p+v)}\), must always fall between \(\mathrm{m/p}\) and \(\mathrm{s/v}\) due to the weighted average property. Expressions II and III can fall outside this range as we demonstrated with counterexamples.

The answer is B. I only.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "must be between" means
Students often confuse "must be between" with "could be between." The question asks which expressions will ALWAYS fall between \(\mathrm{m/p}\) and \(\mathrm{s/v}\) for ANY positive values satisfying \(\mathrm{m/p < s/v}\). Many students test just one example, see that an expression falls between the bounds, and incorrectly conclude it "must be between" without realizing they need to prove it works for ALL possible values.

2. Failing to recognize the need for both algebraic proof AND counterexamples
Students may rely solely on testing specific numbers without understanding that: (a) For expressions that work, they need to understand WHY they always work (like the weighted average property), and (b) For expressions that don't work, they need to find counterexamples where the expression falls outside the bounds.

3. Not connecting to the weighted average concept for Expression I
Many students miss that \(\mathrm{(m+s)/(p+v)}\) represents a weighted average of \(\mathrm{m/p}\) and \(\mathrm{s/v}\). Without this insight, they can't understand why Expression I will ALWAYS fall between the two fractions, leading them to treat it the same as the other expressions.

Errors while executing the approach

1. Using the same numerical example for all expressions
Students often test all three expressions using identical values like \(\mathrm{m=1, p=2, s=3, v=4}\). While this might work to verify Expression I, it won't necessarily reveal that Expressions II and III can fall outside the bounds. They need to try different sets of values to find counterexamples.

2. Arithmetic errors when testing expressions
When calculating fractions like \(\mathrm{ms/(pv)}\) or \(\mathrm{s/v - m/p}\), students frequently make computational mistakes. For example, when computing \(\mathrm{3/8}\) as a decimal or subtracting fractions with different denominators, these errors can lead to wrong conclusions about whether expressions fall within the required range.

3. Incorrectly assuming that products and differences behave like weighted averages
Students may intuitively think that since \(\mathrm{m/p < s/v}\), the product \(\mathrm{ms/(pv)}\) or difference \(\mathrm{s/v - m/p}\) should also fall between them. They fail to recognize that multiplication of fractions typically yields a smaller result, and differences can be much smaller than either original fraction.

Errors while selecting the answer

1. Selecting based on limited testing rather than comprehensive analysis
After testing expressions with one or two examples, students may jump to conclusions. If their examples show Expression I works but II and III don't, they might correctly choose "I only." However, if their limited examples happen to show multiple expressions working, they could incorrectly select "I and II both" without finding the necessary counterexamples.

2. Confusing the Roman numeral system in answer choices
Students sometimes mix up which Roman numerals correspond to which expressions, especially when they've determined that only the first expression works. They might know Expression I is correct but accidentally select an answer choice referring to a different Roman numeral.

Alternate Solutions

Smart Numbers Approach

Since we need to determine which expressions must always fall between \(\mathrm{m/p}\) and \(\mathrm{s/v}\), we can test each expression using carefully chosen values that satisfy our constraint \(\mathrm{m/p < s/v}\).

Step 1: Choose our first set of smart numbers

Let's use simple values: \(\mathrm{m = 1, p = 2, s = 3, v = 4}\)
This gives us: \(\mathrm{m/p = 1/2 = 0.5}\) and \(\mathrm{s/v = 3/4 = 0.75}\)
✓ Constraint satisfied: \(\mathrm{0.5 < 0.75}\)

Step 2: Test each expression

Expression I: \(\mathrm{(m+s)/(p+v) = (1+3)/(2+4) = 4/6 = 2/3 ≈ 0.667}\)
Since \(\mathrm{0.5 < 0.667 < 0.75}\), Expression I falls between our bounds ✓

Expression II: \(\mathrm{(ms)/(pv) = (1×3)/(2×4) = 3/8 = 0.375}\)
Since \(\mathrm{0.375 < 0.5}\), Expression II is below our lower bound ✗

Expression III: \(\mathrm{s/v - m/p = 3/4 - 1/2 = 3/4 - 2/4 = 1/4 = 0.25}\)
Since \(\mathrm{0.25 < 0.5}\), Expression III is below our lower bound ✗

Step 3: Verify with a second set of numbers

Let's try: \(\mathrm{m = 2, p = 5, s = 4, v = 5}\)
This gives us: \(\mathrm{m/p = 2/5 = 0.4}\) and \(\mathrm{s/v = 4/5 = 0.8}\)
✓ Constraint satisfied: \(\mathrm{0.4 < 0.8}\)

Expression I: \(\mathrm{(m+s)/(p+v) = (2+4)/(5+5) = 6/10 = 0.6}\)
Since \(\mathrm{0.4 < 0.6 < 0.8}\), Expression I again falls between our bounds ✓

Expression II: \(\mathrm{(ms)/(pv) = (2×4)/(5×5) = 8/25 = 0.32}\)
Since \(\mathrm{0.32 < 0.4}\), Expression II is again below our lower bound ✗

Expression III: \(\mathrm{s/v - m/p = 4/5 - 2/5 = 2/5 = 0.4}\)
Since \(\mathrm{0.4}\) is equal to our lower bound (not strictly between), Expression III fails ✗

Step 4: Conclusion

Through our smart number testing:

• Expression I consistently falls between \(\mathrm{m/p}\) and \(\mathrm{s/v}\)
• Expression II consistently falls below the range
• Expression III consistently falls below or at the boundary of the range

Only Expression I must be between \(\mathrm{m/p}\) and \(\mathrm{s/v}\).

Answer: B (I only)