# Fraction Representations, Reciprocal Guards, and Raw Scaling

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r01-fraction-0100/)

Fraction · qf-density15-fraction-two-scaled-representations · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r01-fraction-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261011-advance-20261015-r01-fraction-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r01-fraction-foundations/validation-report.json)