# Fraction 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-20260930-r01-fraction-foundations · 2026-09-30 · 100 questions

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



## 1. Validity of a Fraction Representation

For integer numerator p=0 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationNo

A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0001/)

Fraction · qf-fraction-representation-validity · candidate





## 2. Validity of a Fraction Representation

For integer numerator p=0 and denominator q=5, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0002/)

Fraction · qf-fraction-representation-validity · candidate





## 3. Validity of a Fraction Representation

For integer numerator p=-10 and denominator q=3, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0003/)

Fraction · qf-fraction-representation-validity · candidate





## 4. Validity of a Fraction Representation

For integer numerator p=-9 and denominator q=-4, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0004/)

Fraction · qf-fraction-representation-validity · candidate





## 5. Validity of a Fraction Representation

For integer numerator p=-8 and denominator q=5, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0005/)

Fraction · qf-fraction-representation-validity · candidate





## 6. Validity of a Fraction Representation

For integer numerator p=-7 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationNo

A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0006/)

Fraction · qf-fraction-representation-validity · candidate





## 7. Validity of a Fraction Representation

For integer numerator p=-6 and denominator q=-7, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0007/)

Fraction · qf-fraction-representation-validity · candidate





## 8. Validity of a Fraction Representation

For integer numerator p=-5 and denominator q=1, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0008/)

Fraction · qf-fraction-representation-validity · candidate





## 9. Validity of a Fraction Representation

For integer numerator p=-4 and denominator q=2, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0009/)

Fraction · qf-fraction-representation-validity · candidate





## 10. Validity of a Fraction Representation

For integer numerator p=-3 and denominator q=-3, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0010/)

Fraction · qf-fraction-representation-validity · candidate





## 11. Validity of a Fraction Representation

For integer numerator p=-2 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationNo

A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0011/)

Fraction · qf-fraction-representation-validity · candidate





## 12. Validity of a Fraction Representation

For integer numerator p=-1 and denominator q=5, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0012/)

Fraction · qf-fraction-representation-validity · candidate





## 13. Validity of a Fraction Representation

For integer numerator p=0 and denominator q=-6, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0013/)

Fraction · qf-fraction-representation-validity · candidate





## 14. Validity of a Fraction Representation

For integer numerator p=1 and denominator q=7, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0014/)

Fraction · qf-fraction-representation-validity · candidate





## 15. Validity of a Fraction Representation

For integer numerator p=2 and denominator q=1, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0015/)

Fraction · qf-fraction-representation-validity · candidate





## 16. Validity of a Fraction Representation

For integer numerator p=3 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationNo

A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0016/)

Fraction · qf-fraction-representation-validity · candidate





## 17. Validity of a Fraction Representation

For integer numerator p=4 and denominator q=3, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0017/)

Fraction · qf-fraction-representation-validity · candidate





## 18. Validity of a Fraction Representation

For integer numerator p=5 and denominator q=4, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0018/)

Fraction · qf-fraction-representation-validity · candidate





## 19. Validity of a Fraction Representation

For integer numerator p=6 and denominator q=-5, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0019/)

Fraction · qf-fraction-representation-validity · candidate





## 20. Validity of a Fraction Representation

For integer numerator p=7 and denominator q=6, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0020/)

Fraction · qf-fraction-representation-validity · candidate





## 21. Validity of a Fraction Representation

For integer numerator p=8 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationNo

A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0021/)

Fraction · qf-fraction-representation-validity · candidate





## 22. Validity of a Fraction Representation

For integer numerator p=9 and denominator q=-1, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0022/)

Fraction · qf-fraction-representation-validity · candidate





## 23. Validity of a Fraction Representation

For integer numerator p=10 and denominator q=2, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0023/)

Fraction · qf-fraction-representation-validity · candidate





## 24. Validity of a Fraction Representation

For integer numerator p=11 and denominator q=3, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0024/)

Fraction · qf-fraction-representation-validity · candidate





## 25. Validity of a Fraction Representation

For integer numerator p=12 and denominator q=-4, is p/q a valid rational-number representation? Answer Yes or No.

View answer and explanationYes

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



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-representation-validity-0025/)

Fraction · qf-fraction-representation-validity · candidate





## 26. Equivalence of Two Fractions

The valid fractions are -12/(1) and 24/(-2), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are 24 and 24; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0001/)

Fraction · qf-fraction-equivalence · candidate





## 27. Equivalence of Two Fractions

The valid fractions are -11/(-2) and -32/(-6), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are 66 and 64; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0002/)

Fraction · qf-fraction-equivalence · candidate





## 28. Equivalence of Two Fractions

The valid fractions are -10/(3) and -40/(12), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are -120 and -120; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0003/)

Fraction · qf-fraction-equivalence · candidate





## 29. Equivalence of Two Fractions

The valid fractions are -9/(-4) and 46/(20), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are -180 and -184; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0004/)

Fraction · qf-fraction-equivalence · candidate





## 30. Equivalence of Two Fractions

The valid fractions are -8/(5) and -16/(10), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are -80 and -80; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0005/)

Fraction · qf-fraction-equivalence · candidate





## 31. Equivalence of Two Fractions

The valid fractions are -7/(-6) and -20/(-18), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are 126 and 120; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0006/)

Fraction · qf-fraction-equivalence · candidate





## 32. Equivalence of Two Fractions

The valid fractions are -6/(7) and 24/(-28), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are 168 and 168; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0007/)

Fraction · qf-fraction-equivalence · candidate





## 33. Equivalence of Two Fractions

The valid fractions are -5/(-1) and -24/(-5), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are 25 and 24; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0008/)

Fraction · qf-fraction-equivalence · candidate





## 34. Equivalence of Two Fractions

The valid fractions are -4/(2) and -8/(4), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are -16 and -16; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0009/)

Fraction · qf-fraction-equivalence · candidate





## 35. Equivalence of Two Fractions

The valid fractions are -3/(-3) and 10/(9), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are -27 and -30; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0010/)

Fraction · qf-fraction-equivalence · candidate





## 36. Equivalence of Two Fractions

The valid fractions are -2/(4) and -8/(16), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are -32 and -32; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0011/)

Fraction · qf-fraction-equivalence · candidate





## 37. Equivalence of Two Fractions

The valid fractions are -1/(-5) and -4/(-25), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are 25 and 20; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0012/)

Fraction · qf-fraction-equivalence · candidate





## 38. Equivalence of Two Fractions

The valid fractions are 0/(6) and 0/(-12), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are 0 and 0; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0013/)

Fraction · qf-fraction-equivalence · candidate





## 39. Equivalence of Two Fractions

The valid fractions are 1/(-7) and 4/(-21), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are -21 and -28; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0014/)

Fraction · qf-fraction-equivalence · candidate





## 40. Equivalence of Two Fractions

The valid fractions are 2/(1) and 8/(4), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are 8 and 8; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0015/)

Fraction · qf-fraction-equivalence · candidate





## 41. Equivalence of Two Fractions

The valid fractions are 3/(-2) and -14/(10), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are 30 and 28; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0016/)

Fraction · qf-fraction-equivalence · candidate





## 42. Equivalence of Two Fractions

The valid fractions are 4/(3) and 8/(6), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are 24 and 24; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0017/)

Fraction · qf-fraction-equivalence · candidate





## 43. Equivalence of Two Fractions

The valid fractions are 5/(-4) and 16/(-12), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are -60 and -64; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0018/)

Fraction · qf-fraction-equivalence · candidate





## 44. Equivalence of Two Fractions

The valid fractions are 6/(5) and -24/(-20), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are -120 and -120; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0019/)

Fraction · qf-fraction-equivalence · candidate





## 45. Equivalence of Two Fractions

The valid fractions are 7/(-6) and 36/(-30), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are -210 and -216; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0020/)

Fraction · qf-fraction-equivalence · candidate





## 46. Equivalence of Two Fractions

The valid fractions are 8/(7) and 16/(14), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are 112 and 112; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0021/)

Fraction · qf-fraction-equivalence · candidate





## 47. Equivalence of Two Fractions

The valid fractions are 9/(-1) and -26/(3), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are 27 and 26; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0022/)

Fraction · qf-fraction-equivalence · candidate





## 48. Equivalence of Two Fractions

The valid fractions are 10/(2) and 40/(8), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are 80 and 80; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0023/)

Fraction · qf-fraction-equivalence · candidate





## 49. Equivalence of Two Fractions

The valid fractions are 11/(-3) and 56/(-15), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationNo

The two cross products are -165 and -168; they are different, so the answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0024/)

Fraction · qf-fraction-equivalence · candidate





## 50. Equivalence of Two Fractions

The valid fractions are 12/(4) and -24/(-8), with both denominators nonzero. Do they represent the same rational number?

View answer and explanationYes

The two cross products are -96 and -96; they are equal, so the answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-equivalence-0025/)

Fraction · qf-fraction-equivalence · candidate





## 51. Fraction Reduction and Normalization

Reduce the valid integer fraction -24/(2) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-12/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -12/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0001/)

Fraction · qf-fraction-reduction · candidate





## 52. Fraction Reduction and Normalization

Reduce the valid integer fraction -33/(-6) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation11/2

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 11/2; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0002/)

Fraction · qf-fraction-reduction · candidate





## 53. Fraction Reduction and Normalization

Reduce the valid integer fraction -40/(12) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-10/3

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -10/3; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0003/)

Fraction · qf-fraction-reduction · candidate





## 54. Fraction Reduction and Normalization

Reduce the valid integer fraction -45/(-20) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation9/4

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 9/4; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0004/)

Fraction · qf-fraction-reduction · candidate





## 55. Fraction Reduction and Normalization

Reduce the valid integer fraction -16/(10) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-8/5

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -8/5; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0005/)

Fraction · qf-fraction-reduction · candidate





## 56. Fraction Reduction and Normalization

Reduce the valid integer fraction -21/(-18) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation7/6

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 7/6; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0006/)

Fraction · qf-fraction-reduction · candidate





## 57. Fraction Reduction and Normalization

Reduce the valid integer fraction -24/(28) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-6/7

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -6/7; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0007/)

Fraction · qf-fraction-reduction · candidate





## 58. Fraction Reduction and Normalization

Reduce the valid integer fraction -25/(-5) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation5/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 5/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0008/)

Fraction · qf-fraction-reduction · candidate





## 59. Fraction Reduction and Normalization

Reduce the valid integer fraction -8/(4) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-2/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -2/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0009/)

Fraction · qf-fraction-reduction · candidate





## 60. Fraction Reduction and Normalization

Reduce the valid integer fraction -9/(-9) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation1/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 1/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0010/)

Fraction · qf-fraction-reduction · candidate





## 61. Fraction Reduction and Normalization

Reduce the valid integer fraction -8/(16) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-1/2

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -1/2; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0011/)

Fraction · qf-fraction-reduction · candidate





## 62. Fraction Reduction and Normalization

Reduce the valid integer fraction -5/(-25) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation1/5

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 1/5; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0012/)

Fraction · qf-fraction-reduction · candidate





## 63. Fraction Reduction and Normalization

Reduce the valid integer fraction 0/(12) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation0/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 0/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0013/)

Fraction · qf-fraction-reduction · candidate





## 64. Fraction Reduction and Normalization

Reduce the valid integer fraction 3/(-21) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-1/7

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -1/7; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0014/)

Fraction · qf-fraction-reduction · candidate





## 65. Fraction Reduction and Normalization

Reduce the valid integer fraction 8/(4) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation2/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 2/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0015/)

Fraction · qf-fraction-reduction · candidate





## 66. Fraction Reduction and Normalization

Reduce the valid integer fraction 15/(-10) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-3/2

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -3/2; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0016/)

Fraction · qf-fraction-reduction · candidate





## 67. Fraction Reduction and Normalization

Reduce the valid integer fraction 8/(6) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation4/3

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 4/3; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0017/)

Fraction · qf-fraction-reduction · candidate





## 68. Fraction Reduction and Normalization

Reduce the valid integer fraction 15/(-12) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-5/4

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -5/4; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0018/)

Fraction · qf-fraction-reduction · candidate





## 69. Fraction Reduction and Normalization

Reduce the valid integer fraction 24/(20) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation6/5

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 6/5; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0019/)

Fraction · qf-fraction-reduction · candidate





## 70. Fraction Reduction and Normalization

Reduce the valid integer fraction 35/(-30) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-7/6

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -7/6; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0020/)

Fraction · qf-fraction-reduction · candidate





## 71. Fraction Reduction and Normalization

Reduce the valid integer fraction 16/(14) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation8/7

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 8/7; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0021/)

Fraction · qf-fraction-reduction · candidate





## 72. Fraction Reduction and Normalization

Reduce the valid integer fraction 27/(-3) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-9/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -9/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0022/)

Fraction · qf-fraction-reduction · candidate





## 73. Fraction Reduction and Normalization

Reduce the valid integer fraction 40/(8) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation5/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 5/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0023/)

Fraction · qf-fraction-reduction · candidate





## 74. Fraction Reduction and Normalization

Reduce the valid integer fraction 55/(-15) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation-11/3

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -11/3; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0024/)

Fraction · qf-fraction-reduction · candidate





## 75. Fraction Reduction and Normalization

Reduce the valid integer fraction 24/(8) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

View answer and explanation3/1

Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 3/1; zero is written 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-reduction-0025/)

Fraction · qf-fraction-reduction · candidate





## 76. Exact Fraction Addition

Compute 0/(1)+(0)/(-1) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation0/1

Use the common-denominator numerator 0×-1+(0)×1 and denominator 1×-1, then normalize and reduce to 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0001/)

Fraction · qf-fraction-exact-addition · candidate





## 77. Exact Fraction Addition

Compute -11/(-2)+(9)/(2) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation10/1

Use the common-denominator numerator -11×2+(9)×-2 and denominator -2×2, then normalize and reduce to 10/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0002/)

Fraction · qf-fraction-exact-addition · candidate





## 78. Exact Fraction Addition

Compute -10/(3)+(6)/(3) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-4/3

Use the common-denominator numerator -10×3+(6)×3 and denominator 3×3, then normalize and reduce to -4/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0003/)

Fraction · qf-fraction-exact-addition · candidate





## 79. Exact Fraction Addition

Compute -9/(-4)+(3)/(-4) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation3/2

Use the common-denominator numerator -9×-4+(3)×-4 and denominator -4×-4, then normalize and reduce to 3/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0004/)

Fraction · qf-fraction-exact-addition · candidate





## 80. Exact Fraction Addition

Compute -8/(5)+(0)/(5) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-8/5

Use the common-denominator numerator -8×5+(0)×5 and denominator 5×5, then normalize and reduce to -8/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0005/)

Fraction · qf-fraction-exact-addition · candidate





## 81. Exact Fraction Addition

Compute -7/(-6)+(-3)/(1) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-11/6

Use the common-denominator numerator -7×1+(-3)×-6 and denominator -6×1, then normalize and reduce to -11/6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0006/)

Fraction · qf-fraction-exact-addition · candidate





## 82. Exact Fraction Addition

Compute -6/(7)+(-6)/(-2) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation15/7

Use the common-denominator numerator -6×-2+(-6)×7 and denominator 7×-2, then normalize and reduce to 15/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0007/)

Fraction · qf-fraction-exact-addition · candidate





## 83. Exact Fraction Addition

Compute -5/(-1)+(-9)/(3) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation2/1

Use the common-denominator numerator -5×3+(-9)×-1 and denominator -1×3, then normalize and reduce to 2/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0008/)

Fraction · qf-fraction-exact-addition · candidate





## 84. Exact Fraction Addition

Compute -4/(2)+(-12)/(4) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-5/1

Use the common-denominator numerator -4×4+(-12)×2 and denominator 2×4, then normalize and reduce to -5/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0009/)

Fraction · qf-fraction-exact-addition · candidate





## 85. Exact Fraction Addition

Compute -3/(-3)+(10)/(-5) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-1/1

Use the common-denominator numerator -3×-5+(10)×-3 and denominator -3×-5, then normalize and reduce to -1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0010/)

Fraction · qf-fraction-exact-addition · candidate





## 86. Exact Fraction Addition

Compute -2/(4)+(7)/(1) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation13/2

Use the common-denominator numerator -2×1+(7)×4 and denominator 4×1, then normalize and reduce to 13/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0011/)

Fraction · qf-fraction-exact-addition · candidate





## 87. Exact Fraction Addition

Compute -1/(-5)+(4)/(2) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation11/5

Use the common-denominator numerator -1×2+(4)×-5 and denominator -5×2, then normalize and reduce to 11/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0012/)

Fraction · qf-fraction-exact-addition · candidate





## 88. Exact Fraction Addition

Compute 0/(6)+(1)/(-3) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-1/3

Use the common-denominator numerator 0×-3+(1)×6 and denominator 6×-3, then normalize and reduce to -1/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0013/)

Fraction · qf-fraction-exact-addition · candidate





## 89. Exact Fraction Addition

Compute 1/(-7)+(-2)/(4) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-9/14

Use the common-denominator numerator 1×4+(-2)×-7 and denominator -7×4, then normalize and reduce to -9/14.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0014/)

Fraction · qf-fraction-exact-addition · candidate





## 90. Exact Fraction Addition

Compute 2/(1)+(-5)/(5) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation1/1

Use the common-denominator numerator 2×5+(-5)×1 and denominator 1×5, then normalize and reduce to 1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0015/)

Fraction · qf-fraction-exact-addition · candidate





## 91. Exact Fraction Addition

Compute 3/(-2)+(-8)/(-1) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation13/2

Use the common-denominator numerator 3×-1+(-8)×-2 and denominator -2×-1, then normalize and reduce to 13/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0016/)

Fraction · qf-fraction-exact-addition · candidate





## 92. Exact Fraction Addition

Compute 4/(3)+(-11)/(2) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-25/6

Use the common-denominator numerator 4×2+(-11)×3 and denominator 3×2, then normalize and reduce to -25/6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0017/)

Fraction · qf-fraction-exact-addition · candidate





## 93. Exact Fraction Addition

Compute 5/(-4)+(11)/(3) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation29/12

Use the common-denominator numerator 5×3+(11)×-4 and denominator -4×3, then normalize and reduce to 29/12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0018/)

Fraction · qf-fraction-exact-addition · candidate





## 94. Exact Fraction Addition

Compute 6/(5)+(8)/(-4) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-4/5

Use the common-denominator numerator 6×-4+(8)×5 and denominator 5×-4, then normalize and reduce to -4/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0019/)

Fraction · qf-fraction-exact-addition · candidate





## 95. Exact Fraction Addition

Compute 7/(-6)+(5)/(5) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-1/6

Use the common-denominator numerator 7×5+(5)×-6 and denominator -6×5, then normalize and reduce to -1/6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0020/)

Fraction · qf-fraction-exact-addition · candidate





## 96. Exact Fraction Addition

Compute 8/(7)+(2)/(1) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation22/7

Use the common-denominator numerator 8×1+(2)×7 and denominator 7×1, then normalize and reduce to 22/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0021/)

Fraction · qf-fraction-exact-addition · candidate





## 97. Exact Fraction Addition

Compute 9/(-1)+(-1)/(-2) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-17/2

Use the common-denominator numerator 9×-2+(-1)×-1 and denominator -1×-2, then normalize and reduce to -17/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0022/)

Fraction · qf-fraction-exact-addition · candidate





## 98. Exact Fraction Addition

Compute 10/(2)+(-4)/(3) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation11/3

Use the common-denominator numerator 10×3+(-4)×2 and denominator 2×3, then normalize and reduce to 11/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0023/)

Fraction · qf-fraction-exact-addition · candidate





## 99. Exact Fraction Addition

Compute 11/(-3)+(-7)/(4) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation-65/12

Use the common-denominator numerator 11×4+(-7)×-3 and denominator -3×4, then normalize and reduce to -65/12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0024/)

Fraction · qf-fraction-exact-addition · candidate





## 100. Exact Fraction Addition

Compute 12/(4)+(-10)/(-5) exactly and write the result in lowest terms with a positive denominator.

View answer and explanation5/1

Use the common-denominator numerator 12×-5+(-10)×4 and denominator 4×-5, then normalize and reduce to 5/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r01-fraction-exact-addition-0025/)

Fraction · qf-fraction-exact-addition · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260930-r01-fraction-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260930-r01-fraction-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260930-r01-fraction-foundations/validation-report.json)