# Rational Number Foundations

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-20260920-rational-number-foundations · 2026-09-20 · 100 questions

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



## 1. Valid Rational Representation

Both p=-12 and q=0 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationNo

q=0, so the denominator is zero and no ordinary rational value is defined.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0001/)

Rational number · qf-rational-valid-representation · candidate





## 2. Valid Rational Representation

Both p=-11 and q=2 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=2≠0, so -11/2 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0002/)

Rational number · qf-rational-valid-representation · candidate





## 3. Valid Rational Representation

Both p=-10 and q=3 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=3≠0, so -10/3 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0003/)

Rational number · qf-rational-valid-representation · candidate





## 4. Valid Rational Representation

Both p=-9 and q=4 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=4≠0, so -9/4 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0004/)

Rational number · qf-rational-valid-representation · candidate





## 5. Valid Rational Representation

Both p=-8 and q=5 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=5≠0, so -8/5 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0005/)

Rational number · qf-rational-valid-representation · candidate





## 6. Valid Rational Representation

Both p=-7 and q=0 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationNo

q=0, so the denominator is zero and no ordinary rational value is defined.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0006/)

Rational number · qf-rational-valid-representation · candidate





## 7. Valid Rational Representation

Both p=-6 and q=7 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=7≠0, so -6/7 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0007/)

Rational number · qf-rational-valid-representation · candidate





## 8. Valid Rational Representation

Both p=-5 and q=1 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=1≠0, so -5/1 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0008/)

Rational number · qf-rational-valid-representation · candidate





## 9. Valid Rational Representation

Both p=-4 and q=2 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=2≠0, so -4/2 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0009/)

Rational number · qf-rational-valid-representation · candidate





## 10. Valid Rational Representation

Both p=-3 and q=3 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=3≠0, so -3/3 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0010/)

Rational number · qf-rational-valid-representation · candidate





## 11. Valid Rational Representation

Both p=-2 and q=0 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationNo

q=0, so the denominator is zero and no ordinary rational value is defined.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0011/)

Rational number · qf-rational-valid-representation · candidate





## 12. Valid Rational Representation

Both p=-1 and q=5 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=5≠0, so -1/5 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0012/)

Rational number · qf-rational-valid-representation · candidate





## 13. Valid Rational Representation

Both p=0 and q=6 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=6≠0, so 0/6 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0013/)

Rational number · qf-rational-valid-representation · candidate





## 14. Valid Rational Representation

Both p=1 and q=7 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=7≠0, so 1/7 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0014/)

Rational number · qf-rational-valid-representation · candidate





## 15. Valid Rational Representation

Both p=2 and q=1 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=1≠0, so 2/1 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0015/)

Rational number · qf-rational-valid-representation · candidate





## 16. Valid Rational Representation

Both p=3 and q=0 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationNo

q=0, so the denominator is zero and no ordinary rational value is defined.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0016/)

Rational number · qf-rational-valid-representation · candidate





## 17. Valid Rational Representation

Both p=4 and q=3 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=3≠0, so 4/3 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0017/)

Rational number · qf-rational-valid-representation · candidate





## 18. Valid Rational Representation

Both p=5 and q=4 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=4≠0, so 5/4 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0018/)

Rational number · qf-rational-valid-representation · candidate





## 19. Valid Rational Representation

Both p=6 and q=5 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=5≠0, so 6/5 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0019/)

Rational number · qf-rational-valid-representation · candidate





## 20. Valid Rational Representation

Both p=7 and q=6 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=6≠0, so 7/6 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0020/)

Rational number · qf-rational-valid-representation · candidate





## 21. Valid Rational Representation

Both p=8 and q=0 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationNo

q=0, so the denominator is zero and no ordinary rational value is defined.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0021/)

Rational number · qf-rational-valid-representation · candidate





## 22. Valid Rational Representation

Both p=9 and q=1 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=1≠0, so 9/1 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0022/)

Rational number · qf-rational-valid-representation · candidate





## 23. Valid Rational Representation

Both p=10 and q=2 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=2≠0, so 10/2 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0023/)

Rational number · qf-rational-valid-representation · candidate





## 24. Valid Rational Representation

Both p=11 and q=3 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=3≠0, so 11/3 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0024/)

Rational number · qf-rational-valid-representation · candidate





## 25. Valid Rational Representation

Both p=12 and q=4 are explicitly integers. Is p/q a valid rational-number representation?

View answer and explanationYes

q=4≠0, so 12/4 is a valid representation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-valid-representation-0025/)

Rational number · qf-rational-valid-representation · candidate





## 26. Denominator-Sign Normalization

Normalize -12/-1 to an exact representation with a positive denominator.

View answer and explanation12/1

Because q=-1 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0001/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 27. Denominator-Sign Normalization

Normalize -11/-2 to an exact representation with a positive denominator.

View answer and explanation11/2

Because q=-2 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0002/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 28. Denominator-Sign Normalization

Normalize -10/-3 to an exact representation with a positive denominator.

View answer and explanation10/3

Because q=-3 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0003/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 29. Denominator-Sign Normalization

Normalize -9/-4 to an exact representation with a positive denominator.

View answer and explanation9/4

Because q=-4 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0004/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 30. Denominator-Sign Normalization

Normalize -8/-5 to an exact representation with a positive denominator.

View answer and explanation8/5

Because q=-5 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0005/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 31. Denominator-Sign Normalization

Normalize -7/-6 to an exact representation with a positive denominator.

View answer and explanation7/6

Because q=-6 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0006/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 32. Denominator-Sign Normalization

Normalize -6/-7 to an exact representation with a positive denominator.

View answer and explanation6/7

Because q=-7 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0007/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 33. Denominator-Sign Normalization

Normalize -5/-8 to an exact representation with a positive denominator.

View answer and explanation5/8

Because q=-8 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0008/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 34. Denominator-Sign Normalization

Normalize -4/-9 to an exact representation with a positive denominator.

View answer and explanation4/9

Because q=-9 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0009/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 35. Denominator-Sign Normalization

Normalize -3/-1 to an exact representation with a positive denominator.

View answer and explanation3/1

Because q=-1 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0010/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 36. Denominator-Sign Normalization

Normalize -2/-2 to an exact representation with a positive denominator.

View answer and explanation2/2

Because q=-2 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0011/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 37. Denominator-Sign Normalization

Normalize -1/-3 to an exact representation with a positive denominator.

View answer and explanation1/3

Because q=-3 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0012/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 38. Denominator-Sign Normalization

Normalize 0/-4 to an exact representation with a positive denominator.

View answer and explanation0/4

Because q=-4 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0013/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 39. Denominator-Sign Normalization

Normalize 1/-5 to an exact representation with a positive denominator.

View answer and explanation-1/5

Because q=-5 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0014/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 40. Denominator-Sign Normalization

Normalize 2/-6 to an exact representation with a positive denominator.

View answer and explanation-2/6

Because q=-6 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0015/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 41. Denominator-Sign Normalization

Normalize 3/-7 to an exact representation with a positive denominator.

View answer and explanation-3/7

Because q=-7 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0016/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 42. Denominator-Sign Normalization

Normalize 4/-8 to an exact representation with a positive denominator.

View answer and explanation-4/8

Because q=-8 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0017/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 43. Denominator-Sign Normalization

Normalize 5/-9 to an exact representation with a positive denominator.

View answer and explanation-5/9

Because q=-9 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0018/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 44. Denominator-Sign Normalization

Normalize 6/-1 to an exact representation with a positive denominator.

View answer and explanation-6/1

Because q=-1 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0019/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 45. Denominator-Sign Normalization

Normalize 7/-2 to an exact representation with a positive denominator.

View answer and explanation-7/2

Because q=-2 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0020/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 46. Denominator-Sign Normalization

Normalize 8/-3 to an exact representation with a positive denominator.

View answer and explanation-8/3

Because q=-3 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0021/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 47. Denominator-Sign Normalization

Normalize 9/-4 to an exact representation with a positive denominator.

View answer and explanation-9/4

Because q=-4 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0022/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 48. Denominator-Sign Normalization

Normalize 10/-5 to an exact representation with a positive denominator.

View answer and explanation-10/5

Because q=-5 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0023/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 49. Denominator-Sign Normalization

Normalize 11/-6 to an exact representation with a positive denominator.

View answer and explanation-11/6

Because q=-6 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0024/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 50. Denominator-Sign Normalization

Normalize 12/-7 to an exact representation with a positive denominator.

View answer and explanation-12/7

Because q=-7 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-denominator-sign-normalization-0025/)

Rational number · qf-rational-denominator-sign-normalization · candidate





## 51. Equivalent Fractions

The denominators q=1 and s=2 are nonzero. Are -11/1 and -22/2 equivalent?

View answer and explanationYes

-11×2 = -22×1; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0001/)

Rational number · qf-rational-equivalent-fractions · candidate





## 52. Equivalent Fractions

The denominators q=2 and s=4 are nonzero. Are -10/2 and -19/4 equivalent?

View answer and explanationNo

-10×4 ≠ -19×2; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0002/)

Rational number · qf-rational-equivalent-fractions · candidate





## 53. Equivalent Fractions

The denominators q=3 and s=6 are nonzero. Are -9/3 and -18/6 equivalent?

View answer and explanationYes

-9×6 = -18×3; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0003/)

Rational number · qf-rational-equivalent-fractions · candidate





## 54. Equivalent Fractions

The denominators q=4 and s=8 are nonzero. Are -8/4 and -15/8 equivalent?

View answer and explanationNo

-8×8 ≠ -15×4; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0004/)

Rational number · qf-rational-equivalent-fractions · candidate





## 55. Equivalent Fractions

The denominators q=5 and s=10 are nonzero. Are -7/5 and -14/10 equivalent?

View answer and explanationYes

-7×10 = -14×5; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0005/)

Rational number · qf-rational-equivalent-fractions · candidate





## 56. Equivalent Fractions

The denominators q=6 and s=12 are nonzero. Are -6/6 and -11/12 equivalent?

View answer and explanationNo

-6×12 ≠ -11×6; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0006/)

Rational number · qf-rational-equivalent-fractions · candidate





## 57. Equivalent Fractions

The denominators q=7 and s=14 are nonzero. Are -5/7 and -10/14 equivalent?

View answer and explanationYes

-5×14 = -10×7; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0007/)

Rational number · qf-rational-equivalent-fractions · candidate





## 58. Equivalent Fractions

The denominators q=1 and s=2 are nonzero. Are -4/1 and -7/2 equivalent?

View answer and explanationNo

-4×2 ≠ -7×1; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0008/)

Rational number · qf-rational-equivalent-fractions · candidate





## 59. Equivalent Fractions

The denominators q=2 and s=4 are nonzero. Are -3/2 and -6/4 equivalent?

View answer and explanationYes

-3×4 = -6×2; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0009/)

Rational number · qf-rational-equivalent-fractions · candidate





## 60. Equivalent Fractions

The denominators q=3 and s=6 are nonzero. Are -2/3 and -3/6 equivalent?

View answer and explanationNo

-2×6 ≠ -3×3; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0010/)

Rational number · qf-rational-equivalent-fractions · candidate





## 61. Equivalent Fractions

The denominators q=4 and s=8 are nonzero. Are -1/4 and -2/8 equivalent?

View answer and explanationYes

-1×8 = -2×4; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0011/)

Rational number · qf-rational-equivalent-fractions · candidate





## 62. Equivalent Fractions

The denominators q=5 and s=10 are nonzero. Are 0/5 and 1/10 equivalent?

View answer and explanationNo

0×10 ≠ 1×5; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0012/)

Rational number · qf-rational-equivalent-fractions · candidate





## 63. Equivalent Fractions

The denominators q=6 and s=12 are nonzero. Are 1/6 and 2/12 equivalent?

View answer and explanationYes

1×12 = 2×6; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0013/)

Rational number · qf-rational-equivalent-fractions · candidate





## 64. Equivalent Fractions

The denominators q=7 and s=14 are nonzero. Are 2/7 and 5/14 equivalent?

View answer and explanationNo

2×14 ≠ 5×7; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0014/)

Rational number · qf-rational-equivalent-fractions · candidate





## 65. Equivalent Fractions

The denominators q=1 and s=2 are nonzero. Are 3/1 and 6/2 equivalent?

View answer and explanationYes

3×2 = 6×1; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0015/)

Rational number · qf-rational-equivalent-fractions · candidate





## 66. Equivalent Fractions

The denominators q=2 and s=4 are nonzero. Are 4/2 and 9/4 equivalent?

View answer and explanationNo

4×4 ≠ 9×2; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0016/)

Rational number · qf-rational-equivalent-fractions · candidate





## 67. Equivalent Fractions

The denominators q=3 and s=6 are nonzero. Are 5/3 and 10/6 equivalent?

View answer and explanationYes

5×6 = 10×3; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0017/)

Rational number · qf-rational-equivalent-fractions · candidate





## 68. Equivalent Fractions

The denominators q=4 and s=8 are nonzero. Are 6/4 and 13/8 equivalent?

View answer and explanationNo

6×8 ≠ 13×4; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0018/)

Rational number · qf-rational-equivalent-fractions · candidate





## 69. Equivalent Fractions

The denominators q=5 and s=10 are nonzero. Are 7/5 and 14/10 equivalent?

View answer and explanationYes

7×10 = 14×5; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0019/)

Rational number · qf-rational-equivalent-fractions · candidate





## 70. Equivalent Fractions

The denominators q=6 and s=12 are nonzero. Are 8/6 and 17/12 equivalent?

View answer and explanationNo

8×12 ≠ 17×6; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0020/)

Rational number · qf-rational-equivalent-fractions · candidate





## 71. Equivalent Fractions

The denominators q=7 and s=14 are nonzero. Are 9/7 and 18/14 equivalent?

View answer and explanationYes

9×14 = 18×7; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0021/)

Rational number · qf-rational-equivalent-fractions · candidate





## 72. Equivalent Fractions

The denominators q=1 and s=2 are nonzero. Are 10/1 and 21/2 equivalent?

View answer and explanationNo

10×2 ≠ 21×1; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0022/)

Rational number · qf-rational-equivalent-fractions · candidate





## 73. Equivalent Fractions

The denominators q=2 and s=4 are nonzero. Are 11/2 and 22/4 equivalent?

View answer and explanationYes

11×4 = 22×2; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0023/)

Rational number · qf-rational-equivalent-fractions · candidate





## 74. Equivalent Fractions

The denominators q=3 and s=6 are nonzero. Are 12/3 and 25/6 equivalent?

View answer and explanationNo

12×6 ≠ 25×3; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0024/)

Rational number · qf-rational-equivalent-fractions · candidate





## 75. Equivalent Fractions

The denominators q=4 and s=8 are nonzero. Are 13/4 and 26/8 equivalent?

View answer and explanationYes

13×8 = 26×4; the fractions are equivalent. This uses exact cross products rather than decimal appearance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-equivalent-fractions-0025/)

Rational number · qf-rational-equivalent-fractions · candidate





## 76. Reduced Rational Form

Reduce 0/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation0/1

The canonical reduced form of zero is 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0001/)

Rational number · qf-rational-reduced-form · candidate





## 77. Reduced Rational Form

Reduce -12/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-3/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -3/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0002/)

Rational number · qf-rational-reduced-form · candidate





## 78. Reduced Rational Form

Reduce 27/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation9/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 9/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0003/)

Rational number · qf-rational-reduced-form · candidate





## 79. Reduced Rational Form

Reduce -24/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-3/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -3/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0004/)

Rational number · qf-rational-reduced-form · candidate





## 80. Reduced Rational Form

Reduce 45/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation15/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 15/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0005/)

Rational number · qf-rational-reduced-form · candidate





## 81. Reduced Rational Form

Reduce -36/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-9/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -9/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0006/)

Rational number · qf-rational-reduced-form · candidate





## 82. Reduced Rational Form

Reduce 63/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation21/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 21/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0007/)

Rational number · qf-rational-reduced-form · candidate





## 83. Reduced Rational Form

Reduce -48/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-6/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -6/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0008/)

Rational number · qf-rational-reduced-form · candidate





## 84. Reduced Rational Form

Reduce 81/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation27/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 27/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0009/)

Rational number · qf-rational-reduced-form · candidate





## 85. Reduced Rational Form

Reduce -60/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-15/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -15/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0010/)

Rational number · qf-rational-reduced-form · candidate





## 86. Reduced Rational Form

Reduce 99/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation33/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 33/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0011/)

Rational number · qf-rational-reduced-form · candidate





## 87. Reduced Rational Form

Reduce -72/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-9/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -9/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0012/)

Rational number · qf-rational-reduced-form · candidate





## 88. Reduced Rational Form

Reduce 117/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation39/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 39/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0013/)

Rational number · qf-rational-reduced-form · candidate





## 89. Reduced Rational Form

Reduce -84/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-21/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -21/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0014/)

Rational number · qf-rational-reduced-form · candidate





## 90. Reduced Rational Form

Reduce 135/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation45/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 45/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0015/)

Rational number · qf-rational-reduced-form · candidate





## 91. Reduced Rational Form

Reduce -96/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-12/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -12/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0016/)

Rational number · qf-rational-reduced-form · candidate





## 92. Reduced Rational Form

Reduce 153/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation51/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 51/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0017/)

Rational number · qf-rational-reduced-form · candidate





## 93. Reduced Rational Form

Reduce -108/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-27/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -27/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0018/)

Rational number · qf-rational-reduced-form · candidate





## 94. Reduced Rational Form

Reduce 171/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation57/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 57/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0019/)

Rational number · qf-rational-reduced-form · candidate





## 95. Reduced Rational Form

Reduce -120/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-15/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -15/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0020/)

Rational number · qf-rational-reduced-form · candidate





## 96. Reduced Rational Form

Reduce 189/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation63/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 63/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0021/)

Rational number · qf-rational-reduced-form · candidate





## 97. Reduced Rational Form

Reduce -132/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-33/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -33/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0022/)

Rational number · qf-rational-reduced-form · candidate





## 98. Reduced Rational Form

Reduce 207/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation69/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 69/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0023/)

Rational number · qf-rational-reduced-form · candidate





## 99. Reduced Rational Form

Reduce -144/8 to an exact lowest-terms form with a positive denominator.

View answer and explanation-18/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -18/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0024/)

Rational number · qf-rational-reduced-form · candidate





## 100. Reduced Rational Form

Reduce 225/12 to an exact lowest-terms form with a positive denominator.

View answer and explanation75/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 75/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260920-rational-reduced-form-0025/)

Rational number · qf-rational-reduced-form · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260920-rational-number-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260920-rational-number-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260920-rational-number-foundations/validation-report.json)