- Translate the problem requirements: We have two positive fractions \(\mathrm{a/b}\) and \(\mathrm{c/d}\) where \(\mathrm{a/b < c/d}\). We need to determine which statements about \(\mathrm{(a+c)/(b+d)}\) must always be true.
- Understand fraction averaging behavior: Recognize that \(\mathrm{(a+c)/(b+d)}\) represents a weighted average of the two original fractions, which will fall between them.
- Test each statement systematically: Use the ordering relationship \(\mathrm{a/b < (a+c)/(b+d) < c/d}\) to evaluate which must always hold.
- Verify with concrete examples: Confirm conclusions using simple numerical examples to ensure the logic is sound.
Execution of Strategic Approach
1. Translate the problem requirements
Let's start by understanding what we're given and what we need to find.
We have two fractions made from positive integers: \(\mathrm{a/b}\) and \(\mathrm{c/d}\), where \(\mathrm{a/b < c/d}\).
Think of this like comparing test scores. If student A got \(\mathrm{a/b}\) correct and student C got \(\mathrm{c/d}\) correct, we know student C did better than student A.
Now we need to check three statements about a new fraction \(\mathrm{(a+c)/(b+d)}\) to see which ones MUST always be true.
Process Skill: TRANSLATE - Converting the abstract fraction comparison into concrete understanding
2. Understand fraction averaging behavior
Here's the key insight: when we form \(\mathrm{(a+c)/(b+d)}\), we're creating something like a weighted average of our two original fractions.
Let's see this with a simple example:
- Suppose \(\mathrm{a/b = 1/3}\) and \(\mathrm{c/d = 2/3}\) (so \(\mathrm{a/b < c/d}\))
- Then \(\mathrm{(a+c)/(b+d) = (1+2)/(3+3) = 3/6 = 1/2}\)
Notice that \(\mathrm{1/2}\) falls exactly between \(\mathrm{1/3}\) and \(\mathrm{2/3}\)!
This isn't a coincidence. When we combine fractions this way, the result always falls between the original fractions. In mathematical terms: \(\mathrm{a/b < (a+c)/(b+d) < c/d}\)
This happens because we're essentially taking a weighted average where the weights depend on the denominators.
Process Skill: INFER - Recognizing the non-obvious averaging property
3. Test each statement systematically
Now let's use our understanding that \(\mathrm{a/b < (a+c)/(b+d) < c/d}\) to check each statement:
Statement I: \(\mathrm{(a+c)/(b+d) < c/d}\)
From our ordering, we know \(\mathrm{(a+c)/(b+d) < c/d}\) is always true.
In our example: \(\mathrm{1/2 < 2/3}\) ✓
Statement II: \(\mathrm{(a+c)/(b+d) < a/b}\)
From our ordering, we know \(\mathrm{a/b < (a+c)/(b+d)}\), which means \(\mathrm{(a+c)/(b+d) > a/b}\).
So Statement II says the opposite of what's true!
In our example: \(\mathrm{1/2 < 1/3}\)? This is false since \(\mathrm{1/2 > 1/3}\) ✗
Statement III: \(\mathrm{(a+c)/(b+d) = a/b + c/d}\)
This would mean the combined fraction equals the sum of the original fractions.
In our example: \(\mathrm{1/2 = 1/3 + 2/3 = 1}\)? This is clearly false since \(\mathrm{1/2 ≠ 1}\) ✗
Let's verify with another example to be sure:
- Let \(\mathrm{a/b = 1/4}\) and \(\mathrm{c/d = 1/2}\)
- Then \(\mathrm{(a+c)/(b+d) = (1+1)/(4+2) = 2/6 = 1/3}\)
- Check: \(\mathrm{1/4 < 1/3 < 1/2}\) ✓
- Statement I: \(\mathrm{1/3 < 1/2}\) ✓
- Statement II: \(\mathrm{1/3 < 1/4}\)? False ✗
- Statement III: \(\mathrm{1/3 = 1/4 + 1/2 = 3/4}\)? False ✗
Process Skill: APPLY CONSTRAINTS - Using the ordering relationship to systematically evaluate each option
4. Final Answer
Only Statement I must always be true.
The answer is B. I only.
This makes intuitive sense: when we combine two quantities where one is smaller than the other, the combination will be closer to the larger quantity but still smaller than it.
Common Faltering Points
Errors while devising the approach
- Misunderstanding the constraint interpretation: Students often focus only on the given condition \(\mathrm{a/b < c/d}\) without recognizing that they need to determine what MUST be true in ALL cases. They may think they just need to find examples where the statements work, rather than proving the statements are always true.
- Missing the fraction averaging concept: Many students don't recognize that \(\mathrm{(a+c)/(b+d)}\) creates a weighted average that falls between the original fractions. Without this key insight, they resort to random algebraic manipulation or plug-in methods without strategic direction.
- Confusing "must be true" with "could be true": Students may approach this as finding statements that work in some cases, rather than identifying statements that work in ALL possible cases where \(\mathrm{a/b < c/d}\) holds.
Errors while executing the approach
- Using insufficient or biased examples: When testing with specific numbers, students often choose simple examples like \(\mathrm{1/2}\) and \(\mathrm{2/3}\), but fail to test edge cases or verify their pattern holds universally. They may conclude a statement is always true based on limited testing.
- Algebraic manipulation errors: Students attempting to prove the relationships algebraically often make sign errors when cross-multiplying inequalities or incorrectly manipulate the fraction \(\mathrm{(a+c)/(b+d)}\), especially when trying to separate it into component parts.
- Misinterpreting the direction of inequalities: Particularly for Statement II, students may correctly identify that \(\mathrm{(a+c)/(b+d)}\) and \(\mathrm{a/b}\) are related but get confused about which is larger than which, leading to incorrect conclusions about whether the statement is true or false.
Errors while selecting the answer
- Partial verification leading to wrong combinations: Students may correctly identify that Statement I is true but incorrectly conclude that other statements are also true, leading them to select "I and II" or "I and III" instead of "I only".
- Overthinking the Roman numeral format: Students sometimes get confused by the Roman numeral answer format and select based on which statements they tested rather than which statements they definitively proved must always be true.
Alternate Solutions
Smart Numbers Approach
This approach uses carefully chosen concrete values that satisfy our constraint \(\mathrm{a/b < c/d}\) to test each statement systematically.
Step 1: Choose Strategic Values
Let's select values where \(\mathrm{a/b < c/d}\) is clearly satisfied:
- Let \(\mathrm{a = 1, b = 4}\), so \(\mathrm{a/b = 1/4 = 0.25}\)
- Let \(\mathrm{c = 1, d = 2}\), so \(\mathrm{c/d = 1/2 = 0.5}\)
- Verify: \(\mathrm{1/4 < 1/2}\) ✓
Step 2: Calculate the Middle Expression
\(\mathrm{(a+c)/(b+d) = (1+1)/(4+2) = 2/6 = 1/3 ≈ 0.333}\)
Step 3: Test Each Statement
Statement I: \(\mathrm{(a+c)/(b+d) < c/d}\)
Test: \(\mathrm{1/3 < 1/2 → 0.333 < 0.5}\) ✓ TRUE
Statement II: \(\mathrm{(a+c)/(b+d) > a/b}\)
Test: \(\mathrm{1/3 > 1/4 → 0.333 > 0.25}\) ✓ FALSE (this contradicts the statement as written)
Statement III: \(\mathrm{(a+c)/(b+d) = a/b + c/d}\)
Test: \(\mathrm{1/3 = 1/4 + 1/2 → 1/3 = 3/4 → 0.333 = 0.75}\) ✗ FALSE
Step 4: Verify with Additional Values
Let's confirm with different values: \(\mathrm{a = 2, b = 6, c = 3, d = 4}\)
- \(\mathrm{a/b = 2/6 = 1/3 ≈ 0.333}\)
- \(\mathrm{c/d = 3/4 = 0.75}\)
- Constraint: \(\mathrm{1/3 < 3/4}\) ✓
- \(\mathrm{(a+c)/(b+d) = 5/10 = 1/2 = 0.5}\)
Testing statements:
- I: \(\mathrm{1/2 < 3/4 → 0.5 < 0.75}\) ✓ TRUE
- II: \(\mathrm{1/2 > 1/3 → 0.5 > 0.333}\) ✓ (Again shows II as written is false)
- III: \(\mathrm{1/2 = 1/3 + 3/4 → 0.5 = 1.083}\) ✗ FALSE
Step 5: Pattern Recognition
Both sets of values show that:
- Statement I is consistently true
- Statement II (as written) is consistently false
- Statement III is consistently false
Conclusion: Only Statement I must be true.
Answer: B (I only)
