# Exact Rational Representations and Division Guards

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-20261009-advance-20261013-r03-rational-number-foundations · 2026-10-09 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=1/-2, b=1/3. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/6, -5/6)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-1/6, -5/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0001/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 2. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=0/-5, b=-2/-3. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(2/3, -2/3)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (2/3, -2/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0002/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 3. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-10/3, b=3/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-41/15, -59/15)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-41/15, -59/15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0003/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 4. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-9/1, b=-4/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-8/1, -10/1)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-8/1, -10/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0004/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 5. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-8/2, b=5/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/1, -5/1)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-3/1, -5/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0005/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 6. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-7/3, b=-6/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/6, -23/6)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-5/6, -23/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0006/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 7. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-6/1, b=7/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-23/5, -37/5)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-23/5, -37/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0007/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 8. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-5/2, b=-8/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/2, -9/2)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-1/2, -9/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0008/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 9. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-4/3, b=9/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(7/15, -47/15)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (7/15, -47/15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0009/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 10. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-3/1, b=-10/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/2, -11/2)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-1/2, -11/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0010/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 11. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-2/2, b=11/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(6/5, -16/5)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (6/5, -16/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0011/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 12. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=-1/3, b=-12/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(8/3, -10/3)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (8/3, -10/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0012/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 13. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=0/1, b=13/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(13/5, -13/5)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (13/5, -13/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0013/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 14. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=1/2, b=-14/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(4/1, -3/1)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (4/1, -3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0014/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 15. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=2/3, b=15/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(11/3, -7/3)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (11/3, -7/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0015/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 16. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=3/1, b=-16/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(7/1, -1/1)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (7/1, -1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0016/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 17. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=4/2, b=17/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(27/5, -7/5)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (27/5, -7/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0017/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 18. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=5/3, b=-18/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(37/6, -17/6)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (37/6, -17/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0018/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 19. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=6/1, b=19/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(49/5, 11/5)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (49/5, 11/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0019/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 20. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=7/2, b=-20/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(17/2, -3/2)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (17/2, -3/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0020/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 21. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=8/3, b=21/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(103/15, -23/15)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (103/15, -23/15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0021/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 22. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=9/1, b=-22/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(29/2, 7/2)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (29/2, 7/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0022/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 23. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=10/2, b=23/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(48/5, 2/5)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (48/5, 2/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0023/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 24. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=11/3, b=-24/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(29/3, -7/3)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (29/3, -7/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0024/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 25. Exact Ordered Rational Sum and Difference

Exact rational representations have nonzero denominators; a=12/1, b=25/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(17/1, 7/1)

Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (17/1, 7/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0025/)

Rational number · qf-density13-rational-sum-difference-pair · candidate





## 26. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-32/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -3/1)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0026/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 27. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=29/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -11/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -11/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0027/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 28. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-26/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -5/2)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -5/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0028/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 29. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=23/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -9/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0029/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 30. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-20/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -2/1)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0030/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 31. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=17/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -7/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -7/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0031/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 32. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-14/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -3/2)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -3/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0032/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 33. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=11/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -5/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -5/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0033/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 34. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-8/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -1/1)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0034/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 35. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=5/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -3/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0035/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 36. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-2/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -1/2)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0036/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 37. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-1/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(<, -1/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -1/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0037/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 38. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=4/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(=, 0/1)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (=, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0038/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 39. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-7/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 1/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 1/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0039/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 40. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=10/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 1/2)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0040/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 41. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-13/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 3/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 3/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0041/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 42. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=16/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 1/1)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0042/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 43. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-19/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 5/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 5/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0043/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 44. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=22/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 3/2)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 3/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0044/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 45. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-25/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 7/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 7/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0045/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 46. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=28/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 2/1)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0046/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 47. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-31/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 9/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0047/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 48. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=34/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 5/2)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 5/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0048/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 49. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=-37/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 11/4)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 11/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0049/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 50. Exact Rational Order and Signed Difference

Exact rational representations have nonzero denominators; a=40/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(>, 3/1)

Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0050/)

Rational number · qf-density13-rational-ordered-pair-compare · candidate





## 51. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=5/-2, b=0/-3. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(No, undefined, undefined)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (No, undefined, undefined).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0051/)

Rational number · qf-density13-rational-division-guard · candidate





## 52. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=0/-7, b=2/-3. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 0/1, 0/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 0/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0052/)

Rational number · qf-density13-rational-division-guard · candidate





## 53. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=3/-3, b=4/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -5/4, -1/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -5/4, -1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0053/)

Rational number · qf-density13-rational-division-guard · candidate





## 54. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=4/-3, b=5/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 16/15, -4/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 16/15, -4/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0054/)

Rational number · qf-density13-rational-division-guard · candidate





## 55. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=5/-3, b=6/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -25/18, -5/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -25/18, -5/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0055/)

Rational number · qf-density13-rational-division-guard · candidate





## 56. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=6/-3, b=7/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 8/7, -2/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 8/7, -2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0056/)

Rational number · qf-density13-rational-division-guard · candidate





## 57. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=7/-3, b=8/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -35/24, -7/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -35/24, -7/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0057/)

Rational number · qf-density13-rational-division-guard · candidate





## 58. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=8/-3, b=9/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 32/27, -8/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 32/27, -8/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0058/)

Rational number · qf-density13-rational-division-guard · candidate





## 59. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=9/-3, b=10/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -3/2, -3/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -3/2, -3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0059/)

Rational number · qf-density13-rational-division-guard · candidate





## 60. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=10/-3, b=11/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 40/33, -10/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 40/33, -10/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0060/)

Rational number · qf-density13-rational-division-guard · candidate





## 61. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=11/-3, b=12/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -55/36, -11/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -55/36, -11/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0061/)

Rational number · qf-density13-rational-division-guard · candidate





## 62. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=12/-3, b=13/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 16/13, -4/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 16/13, -4/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0062/)

Rational number · qf-density13-rational-division-guard · candidate





## 63. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=13/-3, b=14/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -65/42, -13/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -65/42, -13/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0063/)

Rational number · qf-density13-rational-division-guard · candidate





## 64. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=14/-3, b=15/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 56/45, -14/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 56/45, -14/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0064/)

Rational number · qf-density13-rational-division-guard · candidate





## 65. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=15/-3, b=16/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -25/16, -5/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -25/16, -5/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0065/)

Rational number · qf-density13-rational-division-guard · candidate





## 66. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=16/-3, b=17/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 64/51, -16/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 64/51, -16/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0066/)

Rational number · qf-density13-rational-division-guard · candidate





## 67. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=17/-3, b=18/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -85/54, -17/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -85/54, -17/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0067/)

Rational number · qf-density13-rational-division-guard · candidate





## 68. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=18/-3, b=19/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 24/19, -6/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 24/19, -6/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0068/)

Rational number · qf-density13-rational-division-guard · candidate





## 69. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=19/-3, b=20/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -19/12, -19/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -19/12, -19/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0069/)

Rational number · qf-density13-rational-division-guard · candidate





## 70. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=20/-3, b=21/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 80/63, -20/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 80/63, -20/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0070/)

Rational number · qf-density13-rational-division-guard · candidate





## 71. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=21/-3, b=22/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -35/22, -7/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -35/22, -7/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0071/)

Rational number · qf-density13-rational-division-guard · candidate





## 72. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=22/-3, b=23/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 88/69, -22/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 88/69, -22/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0072/)

Rational number · qf-density13-rational-division-guard · candidate





## 73. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=23/-3, b=24/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -115/72, -23/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -115/72, -23/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0073/)

Rational number · qf-density13-rational-division-guard · candidate





## 74. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=24/-3, b=25/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, 32/25, -8/1)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 32/25, -8/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0074/)

Rational number · qf-density13-rational-division-guard · candidate





## 75. Rational Division Eligibility and Multiplication Back

Exact rational representations have nonzero denominators; a=25/-3, b=26/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, -125/78, -25/3)

Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -125/78, -25/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0075/)

Rational number · qf-density13-rational-division-guard · candidate





## 76. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -23/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-4/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-4/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0076/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 77. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 21/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-11/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-11/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0077/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 78. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -19/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-10/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-10/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0078/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 79. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 17/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-3/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0079/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 80. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -15/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-8/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-8/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0080/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 81. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 13/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-7/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-7/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0081/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 82. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -11/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-2/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0082/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 83. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 9/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-5/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0083/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 84. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -7/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-4/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-4/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0084/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 85. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 5/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-1/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0085/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 86. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -3/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-2/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0086/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 87. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 1/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-1/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0087/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 88. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 1/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(0/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (0/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0088/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 89. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -3/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(1/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (1/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0089/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 90. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 5/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(2/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (2/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0090/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 91. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -7/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(1/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (1/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0091/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 92. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 9/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(4/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (4/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0092/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 93. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -11/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(5/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (5/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0093/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 94. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 13/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(2/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (2/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0094/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 95. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -15/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(7/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (7/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0095/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 96. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 17/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(8/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (8/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0096/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 97. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -19/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(3/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (3/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0097/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 98. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 21/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(10/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (10/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0098/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 99. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -23/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(11/3, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (11/3, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0099/)

Rational number · qf-density13-rational-target-missing-addend · candidate





## 100. Missing Rational Addend and Substitution Defect

Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 25/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(4/1, 0/1)

Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (4/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r03-rational-number-0100/)

Rational number · qf-density13-rational-target-missing-addend · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r03-rational-number-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261009-advance-20261013-r03-rational-number-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r03-rational-number-foundations/validation-report.json)