# Real Numbers through Rational Points

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-20260923-real-number-rational-points · 2026-09-23 · 100 questions

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



## 1. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 0.5 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation1/2

0.5 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 1/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0001/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 2. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 1.25 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation5/4

1.25 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 5/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0002/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 3. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string -2.4 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation-12/5

-2.4 is converted to an integer numerator over 10^1 and reduced by the gcd, giving -12/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0003/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 4. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 3.125 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation25/8

3.125 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 25/8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0004/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 5. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 0.04 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation1/25

0.04 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 1/25.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0005/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 6. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string -0.75 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation-3/4

-0.75 is converted to an integer numerator over 10^2 and reduced by the gcd, giving -3/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0006/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 7. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 12.5 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation25/2

12.5 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 25/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0007/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 8. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 0.125 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation1/8

0.125 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 1/8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0008/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 9. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 2.05 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation41/20

2.05 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 41/20.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0009/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 10. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string -1.2 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation-6/5

-1.2 is converted to an integer numerator over 10^1 and reduced by the gcd, giving -6/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0010/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 11. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 4.375 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation35/8

4.375 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 35/8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0011/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 12. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 0.625 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation5/8

0.625 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 5/8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0012/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 13. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 7.01 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation701/100

7.01 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 701/100.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0013/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 14. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string -3.5 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation-7/2

-3.5 is converted to an integer numerator over 10^1 and reduced by the gcd, giving -7/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0014/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 15. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 8.2 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation41/5

8.2 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 41/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0015/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 16. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 0.09 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation9/100

0.09 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 9/100.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0016/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 17. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string -0.01 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation-1/100

-0.01 is converted to an integer numerator over 10^2 and reduced by the gcd, giving -1/100.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0017/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 18. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 15.625 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation125/8

15.625 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 125/8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0018/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 19. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 6.4 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation32/5

6.4 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 32/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0019/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 20. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string -9.08 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation-227/25

-9.08 is converted to an integer numerator over 10^2 and reduced by the gcd, giving -227/25.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0020/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 21. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 0.2 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation1/5

0.2 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 1/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0021/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 22. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 11.25 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation45/4

11.25 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 45/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0022/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 23. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string -4.125 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation-33/8

-4.125 is converted to an integer numerator over 10^3 and reduced by the gcd, giving -33/8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0023/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 24. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 2.75 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation11/4

2.75 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 11/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0024/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 25. Exact Fraction for a Terminating Decimal

Convert the explicit decimal string 0.001 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.

View answer and explanation1/1000

0.001 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 1/1000.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-terminating-decimal-fraction-0025/)

Real number · qf-real-terminating-decimal-fraction · candidate





## 26. Exact Ordering of Rational Points

The rational points are a=-12/1 and b=-6/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-12×1 < -6×1, so -12/1 < -6/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0001/)

Real number · qf-real-rational-order · candidate





## 27. Exact Ordering of Rational Points

The rational points are a=-11/2 and b=-5/2. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-11×2 < -5×2, so -11/2 < -5/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0002/)

Real number · qf-real-rational-order · candidate





## 28. Exact Ordering of Rational Points

The rational points are a=-10/3 and b=-4/3. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-10×3 < -4×3, so -10/3 < -4/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0003/)

Real number · qf-real-rational-order · candidate





## 29. Exact Ordering of Rational Points

The rational points are a=-9/4 and b=-3/4. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-9×4 < -3×4, so -9/4 < -3/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0004/)

Real number · qf-real-rational-order · candidate





## 30. Exact Ordering of Rational Points

The rational points are a=-8/5 and b=-2/5. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-8×5 < -2×5, so -8/5 < -2/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0005/)

Real number · qf-real-rational-order · candidate





## 31. Exact Ordering of Rational Points

The rational points are a=-7/6 and b=-1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-7×1 < -1×6, so -7/6 < -1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0006/)

Real number · qf-real-rational-order · candidate





## 32. Exact Ordering of Rational Points

The rational points are a=-6/7 and b=0/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-6×1 < 0×7, so -6/7 < 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0007/)

Real number · qf-real-rational-order · candidate





## 33. Exact Ordering of Rational Points

The rational points are a=-5/1 and b=1/3. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-5×3 < 1×1, so -5/1 < 1/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0008/)

Real number · qf-real-rational-order · candidate





## 34. Exact Ordering of Rational Points

The rational points are a=-2/1 and b=1/2. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-2×2 < 1×1, so -2/1 < 1/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0009/)

Real number · qf-real-rational-order · candidate





## 35. Exact Ordering of Rational Points

The rational points are a=-1/1 and b=3/5. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-1×5 < 3×1, so -1/1 < 3/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0010/)

Real number · qf-real-rational-order · candidate





## 36. Exact Ordering of Rational Points

The rational points are a=-1/2 and b=4/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-1×1 < 4×2, so -1/2 < 4/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0011/)

Real number · qf-real-rational-order · candidate





## 37. Exact Ordering of Rational Points

The rational points are a=-1/5 and b=5/2. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

-1×2 < 5×5, so -1/5 < 5/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0012/)

Real number · qf-real-rational-order · candidate





## 38. Exact Ordering of Rational Points

The rational points are a=0/1 and b=2/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation<

0×1 < 2×1, so 0/1 < 2/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0013/)

Real number · qf-real-rational-order · candidate





## 39. Exact Ordering of Rational Points

The rational points are a=1/7 and b=-3/2. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

1×2 > -3×7, so 1/7 > -3/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0014/)

Real number · qf-real-rational-order · candidate





## 40. Exact Ordering of Rational Points

The rational points are a=2/1 and b=-1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

2×1 > -1×1, so 2/1 > -1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0015/)

Real number · qf-real-rational-order · candidate





## 41. Exact Ordering of Rational Points

The rational points are a=3/2 and b=-4/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

3×1 > -4×2, so 3/2 > -4/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0016/)

Real number · qf-real-rational-order · candidate





## 42. Exact Ordering of Rational Points

The rational points are a=4/3 and b=-3/2. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

4×2 > -3×3, so 4/3 > -3/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0017/)

Real number · qf-real-rational-order · candidate





## 43. Exact Ordering of Rational Points

The rational points are a=5/4 and b=-2/3. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

5×3 > -2×4, so 5/4 > -2/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0018/)

Real number · qf-real-rational-order · candidate





## 44. Exact Ordering of Rational Points

The rational points are a=6/5 and b=-1/4. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

6×4 > -1×5, so 6/5 > -1/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0019/)

Real number · qf-real-rational-order · candidate





## 45. Exact Ordering of Rational Points

The rational points are a=7/6 and b=0/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

7×1 > 0×6, so 7/6 > 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0020/)

Real number · qf-real-rational-order · candidate





## 46. Exact Ordering of Rational Points

The rational points are a=8/7 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

8×1 > 1×7, so 8/7 > 1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0021/)

Real number · qf-real-rational-order · candidate





## 47. Exact Ordering of Rational Points

The rational points are a=9/1 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

9×1 > 1×1, so 9/1 > 1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0022/)

Real number · qf-real-rational-order · candidate





## 48. Exact Ordering of Rational Points

The rational points are a=5/1 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

5×1 > 1×1, so 5/1 > 1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0023/)

Real number · qf-real-rational-order · candidate





## 49. Exact Ordering of Rational Points

The rational points are a=11/3 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

11×1 > 1×3, so 11/3 > 1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0024/)

Real number · qf-real-rational-order · candidate





## 50. Exact Ordering of Rational Points

The rational points are a=3/1 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

View answer and explanation>

3×1 > 1×1, so 3/1 > 1/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-order-0025/)

Real number · qf-real-rational-order · candidate





## 51. A Rational Point in a Closed Interval

The closed interval [-12, -11] and rational point -12/1 are explicit. Does -12/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms -12≤-12/1≤-11; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0001/)

Real number · qf-real-rational-closed-interval · candidate





## 52. A Rational Point in a Closed Interval

The closed interval [-11, -9] and rational point -5/1 are explicit. Does -5/1 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that -5/1 lies outside [-11, -9].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0002/)

Real number · qf-real-rational-closed-interval · candidate





## 53. A Rational Point in a Closed Interval

The closed interval [-10, -7] and rational point -8/3 are explicit. Does -8/3 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that -8/3 lies outside [-10, -7].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0003/)

Real number · qf-real-rational-closed-interval · candidate





## 54. A Rational Point in a Closed Interval

The closed interval [-9, -5] and rational point -3/2 are explicit. Does -3/2 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that -3/2 lies outside [-9, -5].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0004/)

Real number · qf-real-rational-closed-interval · candidate





## 55. A Rational Point in a Closed Interval

The closed interval [-8, -3] and rational point -4/1 are explicit. Does -4/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms -8≤-4/1≤-3; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0005/)

Real number · qf-real-rational-closed-interval · candidate





## 56. A Rational Point in a Closed Interval

The closed interval [-7, -1] and rational point -1/1 are explicit. Does -1/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms -7≤-1/1≤-1; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0006/)

Real number · qf-real-rational-closed-interval · candidate





## 57. A Rational Point in a Closed Interval

The closed interval [-6, -5] and rational point -2/1 are explicit. Does -2/1 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that -2/1 lies outside [-6, -5].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0007/)

Real number · qf-real-rational-closed-interval · candidate





## 58. A Rational Point in a Closed Interval

The closed interval [-5, -3] and rational point -1/1 are explicit. Does -1/1 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that -1/1 lies outside [-5, -3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0008/)

Real number · qf-real-rational-closed-interval · candidate





## 59. A Rational Point in a Closed Interval

The closed interval [-4, -1] and rational point -4/1 are explicit. Does -4/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms -4≤-4/1≤-1; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0009/)

Real number · qf-real-rational-closed-interval · candidate





## 60. A Rational Point in a Closed Interval

The closed interval [-3, 1] and rational point 1/2 are explicit. Does 1/2 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms -3≤1/2≤1; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0010/)

Real number · qf-real-rational-closed-interval · candidate





## 61. A Rational Point in a Closed Interval

The closed interval [-2, 3] and rational point 2/3 are explicit. Does 2/3 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms -2≤2/3≤3; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0011/)

Real number · qf-real-rational-closed-interval · candidate





## 62. A Rational Point in a Closed Interval

The closed interval [-1, 5] and rational point 3/4 are explicit. Does 3/4 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms -1≤3/4≤5; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0012/)

Real number · qf-real-rational-closed-interval · candidate





## 63. A Rational Point in a Closed Interval

The closed interval [0, 1] and rational point 0/1 are explicit. Does 0/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms 0≤0/1≤1; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0013/)

Real number · qf-real-rational-closed-interval · candidate





## 64. A Rational Point in a Closed Interval

The closed interval [1, 3] and rational point 1/1 are explicit. Does 1/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms 1≤1/1≤3; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0014/)

Real number · qf-real-rational-closed-interval · candidate





## 65. A Rational Point in a Closed Interval

The closed interval [2, 5] and rational point 4/3 are explicit. Does 4/3 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 4/3 lies outside [2, 5].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0015/)

Real number · qf-real-rational-closed-interval · candidate





## 66. A Rational Point in a Closed Interval

The closed interval [3, 7] and rational point 3/4 are explicit. Does 3/4 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 3/4 lies outside [3, 7].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0016/)

Real number · qf-real-rational-closed-interval · candidate





## 67. A Rational Point in a Closed Interval

The closed interval [4, 9] and rational point 8/1 are explicit. Does 8/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms 4≤8/1≤9; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0017/)

Real number · qf-real-rational-closed-interval · candidate





## 68. A Rational Point in a Closed Interval

The closed interval [5, 11] and rational point 4/1 are explicit. Does 4/1 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 4/1 lies outside [5, 11].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0018/)

Real number · qf-real-rational-closed-interval · candidate





## 69. A Rational Point in a Closed Interval

The closed interval [6, 7] and rational point 2/1 are explicit. Does 2/1 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 2/1 lies outside [6, 7].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0019/)

Real number · qf-real-rational-closed-interval · candidate





## 70. A Rational Point in a Closed Interval

The closed interval [7, 9] and rational point 2/1 are explicit. Does 2/1 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 2/1 lies outside [7, 9].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0020/)

Real number · qf-real-rational-closed-interval · candidate





## 71. A Rational Point in a Closed Interval

The closed interval [8, 11] and rational point 8/1 are explicit. Does 8/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms 8≤8/1≤11; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0021/)

Real number · qf-real-rational-closed-interval · candidate





## 72. A Rational Point in a Closed Interval

The closed interval [9, 13] and rational point 5/1 are explicit. Does 5/1 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 5/1 lies outside [9, 13].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0022/)

Real number · qf-real-rational-closed-interval · candidate





## 73. A Rational Point in a Closed Interval

The closed interval [10, 15] and rational point 14/3 are explicit. Does 14/3 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 14/3 lies outside [10, 15].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0023/)

Real number · qf-real-rational-closed-interval · candidate





## 74. A Rational Point in a Closed Interval

The closed interval [11, 17] and rational point 13/4 are explicit. Does 13/4 belong to this closed interval?

View answer and explanationNo

Exact cross multiplication shows that 13/4 lies outside [11, 17].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0024/)

Real number · qf-real-rational-closed-interval · candidate





## 75. A Rational Point in a Closed Interval

The closed interval [12, 13] and rational point 12/1 are explicit. Does 12/1 belong to this closed interval?

View answer and explanationYes

Exact cross multiplication confirms 12≤12/1≤13; equality with an endpoint is included in a closed interval.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-closed-interval-0025/)

Real number · qf-real-rational-closed-interval · candidate





## 76. Exact Distance Between Rational Points

Two rational real points are -10/1 and 12/1. Find the exact distance |-10/1−12/1| as a lowest-terms fraction.

View answer and explanation22/1

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 22/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0001/)

Real number · qf-real-rational-point-distance · candidate





## 77. Exact Distance Between Rational Points

Two rational real points are -9/2 and 11/2. Find the exact distance |-9/2−11/2| as a lowest-terms fraction.

View answer and explanation10/1

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 10/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0002/)

Real number · qf-real-rational-point-distance · candidate





## 78. Exact Distance Between Rational Points

Two rational real points are -8/3 and 10/3. Find the exact distance |-8/3−10/3| as a lowest-terms fraction.

View answer and explanation6/1

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 6/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0003/)

Real number · qf-real-rational-point-distance · candidate





## 79. Exact Distance Between Rational Points

Two rational real points are -7/4 and 9/4. Find the exact distance |-7/4−9/4| as a lowest-terms fraction.

View answer and explanation4/1

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 4/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0004/)

Real number · qf-real-rational-point-distance · candidate





## 80. Exact Distance Between Rational Points

Two rational real points are -6/5 and 8/5. Find the exact distance |-6/5−8/5| as a lowest-terms fraction.

View answer and explanation14/5

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 14/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0005/)

Real number · qf-real-rational-point-distance · candidate





## 81. Exact Distance Between Rational Points

Two rational real points are -5/1 and 7/6. Find the exact distance |-5/1−7/6| as a lowest-terms fraction.

View answer and explanation37/6

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 37/6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0006/)

Real number · qf-real-rational-point-distance · candidate





## 82. Exact Distance Between Rational Points

Two rational real points are -2/1 and 6/7. Find the exact distance |-2/1−6/7| as a lowest-terms fraction.

View answer and explanation20/7

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 20/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0007/)

Real number · qf-real-rational-point-distance · candidate





## 83. Exact Distance Between Rational Points

Two rational real points are -1/1 and 5/1. Find the exact distance |-1/1−5/1| as a lowest-terms fraction.

View answer and explanation6/1

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 6/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0008/)

Real number · qf-real-rational-point-distance · candidate





## 84. Exact Distance Between Rational Points

Two rational real points are -1/2 and 2/1. Find the exact distance |-1/2−2/1| as a lowest-terms fraction.

View answer and explanation5/2

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 5/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0009/)

Real number · qf-real-rational-point-distance · candidate





## 85. Exact Distance Between Rational Points

Two rational real points are -1/5 and 1/1. Find the exact distance |-1/5−1/1| as a lowest-terms fraction.

View answer and explanation6/5

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 6/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0010/)

Real number · qf-real-rational-point-distance · candidate





## 86. Exact Distance Between Rational Points

Two rational real points are 0/1 and 1/2. Find the exact distance |0/1−1/2| as a lowest-terms fraction.

View answer and explanation1/2

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 1/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0011/)

Real number · qf-real-rational-point-distance · candidate





## 87. Exact Distance Between Rational Points

Two rational real points are 1/2 and 1/5. Find the exact distance |1/2−1/5| as a lowest-terms fraction.

View answer and explanation3/10

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 3/10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0012/)

Real number · qf-real-rational-point-distance · candidate





## 88. Exact Distance Between Rational Points

Two rational real points are 2/3 and 0/1. Find the exact distance |2/3−0/1| as a lowest-terms fraction.

View answer and explanation2/3

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 2/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0013/)

Real number · qf-real-rational-point-distance · candidate





## 89. Exact Distance Between Rational Points

Two rational real points are 3/4 and -1/7. Find the exact distance |3/4−-1/7| as a lowest-terms fraction.

View answer and explanation25/28

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 25/28.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0014/)

Real number · qf-real-rational-point-distance · candidate





## 90. Exact Distance Between Rational Points

Two rational real points are 4/5 and -2/1. Find the exact distance |4/5−-2/1| as a lowest-terms fraction.

View answer and explanation14/5

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 14/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0015/)

Real number · qf-real-rational-point-distance · candidate





## 91. Exact Distance Between Rational Points

Two rational real points are 5/1 and -3/2. Find the exact distance |5/1−-3/2| as a lowest-terms fraction.

View answer and explanation13/2

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 13/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0016/)

Real number · qf-real-rational-point-distance · candidate





## 92. Exact Distance Between Rational Points

Two rational real points are 3/1 and -4/3. Find the exact distance |3/1−-4/3| as a lowest-terms fraction.

View answer and explanation13/3

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 13/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0017/)

Real number · qf-real-rational-point-distance · candidate





## 93. Exact Distance Between Rational Points

Two rational real points are 7/3 and -5/4. Find the exact distance |7/3−-5/4| as a lowest-terms fraction.

View answer and explanation43/12

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 43/12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0018/)

Real number · qf-real-rational-point-distance · candidate





## 94. Exact Distance Between Rational Points

Two rational real points are 2/1 and -6/5. Find the exact distance |2/1−-6/5| as a lowest-terms fraction.

View answer and explanation16/5

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 16/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0019/)

Real number · qf-real-rational-point-distance · candidate





## 95. Exact Distance Between Rational Points

Two rational real points are 9/5 and -7/6. Find the exact distance |9/5−-7/6| as a lowest-terms fraction.

View answer and explanation89/30

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 89/30.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0020/)

Real number · qf-real-rational-point-distance · candidate





## 96. Exact Distance Between Rational Points

Two rational real points are 10/1 and -8/7. Find the exact distance |10/1−-8/7| as a lowest-terms fraction.

View answer and explanation78/7

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 78/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0021/)

Real number · qf-real-rational-point-distance · candidate





## 97. Exact Distance Between Rational Points

Two rational real points are 11/2 and -9/1. Find the exact distance |11/2−-9/1| as a lowest-terms fraction.

View answer and explanation29/2

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 29/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0022/)

Real number · qf-real-rational-point-distance · candidate





## 98. Exact Distance Between Rational Points

Two rational real points are 4/1 and -5/1. Find the exact distance |4/1−-5/1| as a lowest-terms fraction.

View answer and explanation9/1

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 9/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0023/)

Real number · qf-real-rational-point-distance · candidate





## 99. Exact Distance Between Rational Points

Two rational real points are 13/4 and -11/3. Find the exact distance |13/4−-11/3| as a lowest-terms fraction.

View answer and explanation83/12

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 83/12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0024/)

Real number · qf-real-rational-point-distance · candidate





## 100. Exact Distance Between Rational Points

Two rational real points are 14/5 and -3/1. Find the exact distance |14/5−-3/1| as a lowest-terms fraction.

View answer and explanation29/5

Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 29/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260923-real-rational-point-distance-0025/)

Real number · qf-real-rational-point-distance · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260923-real-number-rational-points/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260923-real-number-rational-points.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260923-real-number-rational-points/validation-report.json)