Fraction Representations, Reciprocal Guards, and Raw Scaling
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261011-advance-20261015-r01-fraction-foundations · 2026-10-11 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/2, b=1/-2. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(0/1, No, undefined, undefined, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (0/1, No, undefined, undefined, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
2. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/2, b=-1/-2. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(1/1, No, undefined, undefined, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (1/1, No, undefined, undefined, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
3. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/2, b=0/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(0/1, Yes, 0/1, 0/1, Yes)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (0/1, Yes, 0/1, 0/1, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
4. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/2, b=1/4. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(3/4, Yes, 1/3, -5/12, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (3/4, Yes, 1/3, -5/12, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
5. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-8/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-11/3, Yes, -7/5, 34/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-11/3, Yes, -7/5, 34/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
6. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=7/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-19/6, Yes, 8/1, 67/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-19/6, Yes, 8/1, 67/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
7. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-6/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-8/3, Yes, -1/1, 5/3, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-8/3, Yes, -1/1, 5/3, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
8. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=5/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-13/6, Yes, 6/1, 49/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-13/6, Yes, 6/1, 49/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
9. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-4/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-5/3, Yes, -3/5, 16/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-5/3, Yes, -3/5, 16/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
10. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=3/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-7/6, Yes, 4/1, 31/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-7/6, Yes, 4/1, 31/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
11. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-2/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-2/3, Yes, -1/5, 7/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-2/3, Yes, -1/5, 7/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
12. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(-1/6, Yes, 2/1, 13/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-1/6, Yes, 2/1, 13/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
13. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(1/3, Yes, 1/5, -2/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (1/3, Yes, 1/5, -2/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
14. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-1/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(5/6, Yes, 0/1, -5/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (5/6, Yes, 0/1, -5/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
15. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=2/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(4/3, Yes, 3/5, -11/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (4/3, Yes, 3/5, -11/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
16. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-3/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(11/6, Yes, -2/1, -23/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (11/6, Yes, -2/1, -23/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
17. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=4/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(7/3, Yes, 1/1, -4/3, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (7/3, Yes, 1/1, -4/3, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
18. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-5/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(17/6, Yes, -4/1, -41/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (17/6, Yes, -4/1, -41/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
19. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=6/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(10/3, Yes, 7/5, -29/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (10/3, Yes, 7/5, -29/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
20. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-7/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(23/6, Yes, -6/1, -59/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (23/6, Yes, -6/1, -59/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
21. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=8/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(13/3, Yes, 9/5, -38/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (13/3, Yes, 9/5, -38/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
22. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-9/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(29/6, Yes, -8/1, -77/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (29/6, Yes, -8/1, -77/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
23. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=10/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(16/3, Yes, 11/5, -47/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (16/3, Yes, 11/5, -47/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
24. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-11/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(35/6, Yes, -10/1, -95/6, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (35/6, Yes, -10/1, -95/6, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
25. True Fraction Sum versus Naive Representation Addition
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=12/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).
View answer and explanation
(19/3, Yes, 13/5, -56/15, No)
Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (19/3, Yes, 13/5, -56/15, No).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-sum-naive-audit · candidate
26. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[0/1, 0/1, 0/1, 0/1]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([0/1, 0/1, 0/1], 0/1, 0/1, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([0/1, 0/1, 0/1], 0/1, 0/1, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
27. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[0/1, 1/1, 0/1, 1/1]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([1/1, -1/1, 1/1], 1/1, 1/1, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([1/1, -1/1, 1/1], 1/1, 1/1, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
28. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-10/3, 0/1, 1/2, 3/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([10/3, 1/2, 1/1], 29/6, 29/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([10/3, 1/2, 1/1], 29/6, 29/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
29. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[9/-3, 0/1, 1/2, 4/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([3/1, 1/2, 3/2], 5/1, 5/1, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([3/1, 1/2, 3/2], 5/1, 5/1, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
30. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-8/3, 0/1, 1/2, 5/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([8/3, 1/2, 2/1], 31/6, 31/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([8/3, 1/2, 2/1], 31/6, 31/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
31. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[7/-3, 0/1, 1/2, 6/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([7/3, 1/2, 5/2], 16/3, 16/3, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([7/3, 1/2, 5/2], 16/3, 16/3, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
32. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-6/3, 0/1, 1/2, 7/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([2/1, 1/2, 3/1], 11/2, 11/2, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([2/1, 1/2, 3/1], 11/2, 11/2, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
33. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[5/-3, 0/1, 1/2, 8/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([5/3, 1/2, 7/2], 17/3, 17/3, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([5/3, 1/2, 7/2], 17/3, 17/3, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
34. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-4/3, 0/1, 1/2, 9/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([4/3, 1/2, 4/1], 35/6, 35/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([4/3, 1/2, 4/1], 35/6, 35/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
35. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[3/-3, 0/1, 1/2, 10/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([1/1, 1/2, 9/2], 6/1, 6/1, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([1/1, 1/2, 9/2], 6/1, 6/1, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
36. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-2/3, 0/1, 1/2, 11/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([2/3, 1/2, 5/1], 37/6, 37/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([2/3, 1/2, 5/1], 37/6, 37/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
37. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[1/-3, 0/1, 1/2, 12/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([1/3, 1/2, 11/2], 19/3, 19/3, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([1/3, 1/2, 11/2], 19/3, 19/3, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
38. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[0/3, 0/1, 1/2, 13/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([0/1, 1/2, 6/1], 13/2, 13/2, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([0/1, 1/2, 6/1], 13/2, 13/2, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
39. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-1/-3, 0/1, 1/2, 14/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-1/3, 1/2, 13/2], 20/3, 20/3, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-1/3, 1/2, 13/2], 20/3, 20/3, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
40. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[2/3, 0/1, 1/2, 15/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-2/3, 1/2, 7/1], 41/6, 41/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-2/3, 1/2, 7/1], 41/6, 41/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
41. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-3/-3, 0/1, 1/2, 16/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-1/1, 1/2, 15/2], 7/1, 7/1, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-1/1, 1/2, 15/2], 7/1, 7/1, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
42. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[4/3, 0/1, 1/2, 17/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-4/3, 1/2, 8/1], 43/6, 43/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-4/3, 1/2, 8/1], 43/6, 43/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
43. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-5/-3, 0/1, 1/2, 18/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-5/3, 1/2, 17/2], 22/3, 22/3, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-5/3, 1/2, 17/2], 22/3, 22/3, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
44. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[6/3, 0/1, 1/2, 19/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-2/1, 1/2, 9/1], 15/2, 15/2, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-2/1, 1/2, 9/1], 15/2, 15/2, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
45. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-7/-3, 0/1, 1/2, 20/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-7/3, 1/2, 19/2], 23/3, 23/3, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-7/3, 1/2, 19/2], 23/3, 23/3, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
46. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[8/3, 0/1, 1/2, 21/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-8/3, 1/2, 10/1], 47/6, 47/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-8/3, 1/2, 10/1], 47/6, 47/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
47. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-9/-3, 0/1, 1/2, 22/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-3/1, 1/2, 21/2], 8/1, 8/1, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-3/1, 1/2, 21/2], 8/1, 8/1, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
48. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[10/3, 0/1, 1/2, 23/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-10/3, 1/2, 11/1], 49/6, 49/6, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-10/3, 1/2, 11/1], 49/6, 49/6, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
49. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-11/-3, 0/1, 1/2, 24/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-11/3, 1/2, 23/2], 25/3, 25/3, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-11/3, 1/2, 23/2], 25/3, 25/3, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
50. Ordered Fraction Differences and Endpoint Check
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[12/3, 0/1, 1/2, 25/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.
View answer and explanation
([-4/1, 1/2, 12/1], 17/2, 17/2, Yes)
Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-4/1, 1/2, 12/1], 17/2, 17/2, Yes).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-adjacent-difference-chain · candidate
51. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/-2, b=0/3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(0/1, [No, No], [undefined, undefined], undefined)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [No, No], [undefined, undefined], undefined).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
52. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/1, b=2/3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(0/1, [No, Yes], [undefined, 3/2], undefined)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [No, Yes], [undefined, 3/2], undefined).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
53. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-1/2, b=0/-5. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(0/1, [Yes, No], [-2/1, undefined], undefined)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [Yes, No], [-2/1, undefined], undefined).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
54. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-9/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(2/1, [Yes, Yes], [-1/3, -3/2], 1/2)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (2/1, [Yes, Yes], [-1/3, -3/2], 1/2).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
55. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-8/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(16/9, [Yes, Yes], [-3/8, -3/2], 9/16)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (16/9, [Yes, Yes], [-3/8, -3/2], 9/16).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
56. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-7/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(14/9, [Yes, Yes], [-3/7, -3/2], 9/14)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (14/9, [Yes, Yes], [-3/7, -3/2], 9/14).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
57. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-6/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(4/3, [Yes, Yes], [-1/2, -3/2], 3/4)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (4/3, [Yes, Yes], [-1/2, -3/2], 3/4).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
58. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-5/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(10/9, [Yes, Yes], [-3/5, -3/2], 9/10)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (10/9, [Yes, Yes], [-3/5, -3/2], 9/10).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
59. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-4/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(8/9, [Yes, Yes], [-3/4, -3/2], 9/8)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (8/9, [Yes, Yes], [-3/4, -3/2], 9/8).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
60. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-3/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(2/3, [Yes, Yes], [-1/1, -3/2], 3/2)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (2/3, [Yes, Yes], [-1/1, -3/2], 3/2).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
61. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-2/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(4/9, [Yes, Yes], [-3/2, -3/2], 9/4)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (4/9, [Yes, Yes], [-3/2, -3/2], 9/4).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
62. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-1/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(2/9, [Yes, Yes], [-3/1, -3/2], 9/2)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (2/9, [Yes, Yes], [-3/1, -3/2], 9/2).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
63. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(0/1, [No, Yes], [undefined, -3/2], undefined)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [No, Yes], [undefined, -3/2], undefined).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
64. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-2/9, [Yes, Yes], [3/1, -3/2], -9/2)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-2/9, [Yes, Yes], [3/1, -3/2], -9/2).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
65. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=2/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-4/9, [Yes, Yes], [3/2, -3/2], -9/4)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-4/9, [Yes, Yes], [3/2, -3/2], -9/4).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
66. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=3/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-2/3, [Yes, Yes], [1/1, -3/2], -3/2)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-2/3, [Yes, Yes], [1/1, -3/2], -3/2).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
67. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=4/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-8/9, [Yes, Yes], [3/4, -3/2], -9/8)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-8/9, [Yes, Yes], [3/4, -3/2], -9/8).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
68. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=5/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-10/9, [Yes, Yes], [3/5, -3/2], -9/10)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-10/9, [Yes, Yes], [3/5, -3/2], -9/10).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
69. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=6/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-4/3, [Yes, Yes], [1/2, -3/2], -3/4)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-4/3, [Yes, Yes], [1/2, -3/2], -3/4).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
70. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=7/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-14/9, [Yes, Yes], [3/7, -3/2], -9/14)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-14/9, [Yes, Yes], [3/7, -3/2], -9/14).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
71. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=8/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-16/9, [Yes, Yes], [3/8, -3/2], -9/16)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-16/9, [Yes, Yes], [3/8, -3/2], -9/16).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
72. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=9/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-2/1, [Yes, Yes], [1/3, -3/2], -1/2)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-2/1, [Yes, Yes], [1/3, -3/2], -1/2).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
73. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=10/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-20/9, [Yes, Yes], [3/10, -3/2], -9/20)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-20/9, [Yes, Yes], [3/10, -3/2], -9/20).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
74. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=11/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-22/9, [Yes, Yes], [3/11, -3/2], -9/22)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-22/9, [Yes, Yes], [3/11, -3/2], -9/22).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
75. Fraction Product and Two Reciprocal Guards
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=12/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
View answer and explanation
(-8/3, [Yes, Yes], [1/4, -3/2], -3/8)
Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-8/3, [Yes, Yes], [1/4, -3/2], -3/8).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-inverse-pair-guard · candidate
76. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=0, d=1; ordered scale factors [s,t]=[0,2]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(0/1, [[0, 0], [0, 2]], [No, Yes], [undefined, 0/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (0/1, [[0, 0], [0, 2]], [No, Yes], [undefined, 0/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
77. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=2, d=-3; ordered scale factors [s,t]=[0,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-2/3, [[0, 0], [-2, 3]], [No, Yes], [undefined, -2/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-2/3, [[0, 0], [-2, 3]], [No, Yes], [undefined, -2/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
78. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=1, d=2; ordered scale factors [s,t]=[1,0]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(1/2, [[1, 2], [0, 0]], [Yes, No], [1/2, undefined])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/2, [[1, 2], [0, 0]], [Yes, No], [1/2, undefined]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
79. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=9007199254740991, d=-7; ordered scale factors [s,t]=[9007199254740991,-2]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-9007199254740991/7, [[81129638414606663681390495662081, -63050394783186937], [-18014398509481982, 14]], [Yes, Yes], [-9007199254740991/7, -9007199254740991/7])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-9007199254740991/7, [[81129638414606663681390495662081, -63050394783186937], [-18014398509481982, 14]], [Yes, Yes], [-9007199254740991/7, -9007199254740991/7]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
80. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-8, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-4/1, [[-16, 4], [8, -2]], [Yes, Yes], [-4/1, -4/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-4/1, [[-16, 4], [8, -2]], [Yes, Yes], [-4/1, -4/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
81. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-7, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(7/3, [[-14, -6], [7, 3]], [Yes, Yes], [7/3, 7/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (7/3, [[-14, -6], [7, 3]], [Yes, Yes], [7/3, 7/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
82. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-6, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-3/1, [[-12, 4], [6, -2]], [Yes, Yes], [-3/1, -3/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-3/1, [[-12, 4], [6, -2]], [Yes, Yes], [-3/1, -3/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
83. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-5, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(5/3, [[-10, -6], [5, 3]], [Yes, Yes], [5/3, 5/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (5/3, [[-10, -6], [5, 3]], [Yes, Yes], [5/3, 5/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
84. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-4, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-2/1, [[-8, 4], [4, -2]], [Yes, Yes], [-2/1, -2/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-2/1, [[-8, 4], [4, -2]], [Yes, Yes], [-2/1, -2/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
85. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-3, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(1/1, [[-6, -6], [3, 3]], [Yes, Yes], [1/1, 1/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/1, [[-6, -6], [3, 3]], [Yes, Yes], [1/1, 1/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
86. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-2, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-1/1, [[-4, 4], [2, -2]], [Yes, Yes], [-1/1, -1/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-1/1, [[-4, 4], [2, -2]], [Yes, Yes], [-1/1, -1/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
87. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-1, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(1/3, [[-2, -6], [1, 3]], [Yes, Yes], [1/3, 1/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/3, [[-2, -6], [1, 3]], [Yes, Yes], [1/3, 1/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
88. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=0, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(0/1, [[0, 4], [0, -2]], [Yes, Yes], [0/1, 0/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (0/1, [[0, 4], [0, -2]], [Yes, Yes], [0/1, 0/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
89. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=1, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-1/3, [[2, -6], [-1, 3]], [Yes, Yes], [-1/3, -1/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-1/3, [[2, -6], [-1, 3]], [Yes, Yes], [-1/3, -1/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
90. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=2, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(1/1, [[4, 4], [-2, -2]], [Yes, Yes], [1/1, 1/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/1, [[4, 4], [-2, -2]], [Yes, Yes], [1/1, 1/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
91. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=3, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-1/1, [[6, -6], [-3, 3]], [Yes, Yes], [-1/1, -1/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-1/1, [[6, -6], [-3, 3]], [Yes, Yes], [-1/1, -1/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
92. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=4, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(2/1, [[8, 4], [-4, -2]], [Yes, Yes], [2/1, 2/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (2/1, [[8, 4], [-4, -2]], [Yes, Yes], [2/1, 2/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
93. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=5, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-5/3, [[10, -6], [-5, 3]], [Yes, Yes], [-5/3, -5/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-5/3, [[10, -6], [-5, 3]], [Yes, Yes], [-5/3, -5/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
94. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=6, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(3/1, [[12, 4], [-6, -2]], [Yes, Yes], [3/1, 3/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (3/1, [[12, 4], [-6, -2]], [Yes, Yes], [3/1, 3/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
95. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=7, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-7/3, [[14, -6], [-7, 3]], [Yes, Yes], [-7/3, -7/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-7/3, [[14, -6], [-7, 3]], [Yes, Yes], [-7/3, -7/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
96. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=8, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(4/1, [[16, 4], [-8, -2]], [Yes, Yes], [4/1, 4/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (4/1, [[16, 4], [-8, -2]], [Yes, Yes], [4/1, 4/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
97. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=9, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-3/1, [[18, -6], [-9, 3]], [Yes, Yes], [-3/1, -3/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-3/1, [[18, -6], [-9, 3]], [Yes, Yes], [-3/1, -3/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
98. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=10, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(5/1, [[20, 4], [-10, -2]], [Yes, Yes], [5/1, 5/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (5/1, [[20, 4], [-10, -2]], [Yes, Yes], [5/1, 5/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
99. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=11, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(-11/3, [[22, -6], [-11, 3]], [Yes, Yes], [-11/3, -11/3])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-11/3, [[22, -6], [-11, 3]], [Yes, Yes], [-11/3, -11/3]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
100. Two Raw Scaled Fraction Representations
Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=12, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
View answer and explanation
(6/1, [[24, 4], [-12, -2]], [Yes, Yes], [6/1, 6/1])
A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (6/1, [[24, 4], [-12, -2]], [Yes, Yes], [6/1, 6/1]).
Reviewed step-by-step derivation
Fraction · qf-density15-fraction-two-scaled-representations · candidate
Canonical question JSONL · English JSONL · Validation report