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