Exact Rational Errors and Inclusive Tolerance Bounds
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batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=3/1, reference=1/1. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(3, 1, 2)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 1, 2).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
2. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=5/2, reference=2/2. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(5/2, 1, 3/2)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 1, 3/2).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
3. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=7/3, reference=3/3. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(7/3, 1, 4/3)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7/3, 1, 4/3).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
4. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=9/4, reference=4/4. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(9/4, 1, 5/4)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9/4, 1, 5/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
5. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=11/5, reference=5/1. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(11/5, 5, 14/5)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 5, 14/5).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
6. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=13/1, reference=6/2. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(13, 3, 10)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13, 3, 10).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
7. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=15/2, reference=7/3. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(15/2, 7/3, 31/6)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/2, 7/3, 31/6).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
8. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=17/3, reference=8/4. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(17/3, 2, 11/3)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (17/3, 2, 11/3).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
9. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=19/4, reference=9/1. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(19/4, 9, 17/4)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (19/4, 9, 17/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
10. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=21/5, reference=1/2. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(21/5, 1/2, 37/10)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (21/5, 1/2, 37/10).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
11. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=23/1, reference=2/3. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(23, 2/3, 67/3)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (23, 2/3, 67/3).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
12. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=25/2, reference=3/4. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(25/2, 3/4, 47/4)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (25/2, 3/4, 47/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
13. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=27/3, reference=4/1. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(9, 4, 5)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 4, 5).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
14. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=29/4, reference=5/2. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(29/4, 5/2, 19/4)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (29/4, 5/2, 19/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
15. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=31/5, reference=6/3. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(31/5, 2, 21/5)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (31/5, 2, 21/5).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
16. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=33/1, reference=7/4. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(33, 7/4, 125/4)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (33, 7/4, 125/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
17. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=35/2, reference=8/1. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(35/2, 8, 19/2)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (35/2, 8, 19/2).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
18. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=37/3, reference=9/2. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(37/3, 9/2, 47/6)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (37/3, 9/2, 47/6).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
19. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=39/4, reference=1/3. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(39/4, 1/3, 113/12)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (39/4, 1/3, 113/12).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
20. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=41/5, reference=2/4. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(41/5, 1/2, 77/10)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (41/5, 1/2, 77/10).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
21. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=43/1, reference=3/1. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(43, 3, 40)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (43, 3, 40).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
22. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=45/2, reference=4/2. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(45/2, 2, 41/2)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (45/2, 2, 41/2).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
23. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=47/3, reference=5/3. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(47/3, 5/3, 14)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (47/3, 5/3, 14).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
24. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=49/4, reference=6/4. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(49/4, 3/2, 43/4)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (49/4, 3/2, 43/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
25. Exact Absolute Difference from a Specified Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=51/5, reference=7/1. Give (normalized approximation, normalized reference, absolute error).
View answer and explanation
(51/5, 7, 16/5)
Take the absolute value of the exact difference, retaining denominator-sign normalization. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (51/5, 7, 16/5).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-absolute-error · candidate
26. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=3/1, reference=1/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(3, 1, 2, 2)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 1, 2, 2).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
27. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=5/2, reference=2/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(5/2, 1, 3/2, 3/2)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 1, 3/2, 3/2).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
28. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=7/3, reference=3/3 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(7/3, 1, 4/3, 4/3)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7/3, 1, 4/3, 4/3).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
29. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=9/4, reference=4/4 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(9/4, 1, 5/4, 5/4)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9/4, 1, 5/4, 5/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
30. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=11/5, reference=5/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(11/5, 5, 14/5, 14/25)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 5, 14/5, 14/25).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
31. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=13/1, reference=6/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(13, 3, 10, 10/3)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13, 3, 10, 10/3).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
32. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=15/2, reference=7/3 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(15/2, 7/3, 31/6, 31/14)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/2, 7/3, 31/6, 31/14).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
33. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=17/3, reference=8/4 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(17/3, 2, 11/3, 11/6)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (17/3, 2, 11/3, 11/6).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
34. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=19/4, reference=9/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(19/4, 9, 17/4, 17/36)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (19/4, 9, 17/4, 17/36).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
35. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=21/5, reference=1/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(21/5, 1/2, 37/10, 37/5)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (21/5, 1/2, 37/10, 37/5).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
36. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=23/1, reference=2/3 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(23, 2/3, 67/3, 67/2)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (23, 2/3, 67/3, 67/2).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
37. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=25/2, reference=3/4 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(25/2, 3/4, 47/4, 47/3)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (25/2, 3/4, 47/4, 47/3).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
38. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=27/3, reference=4/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(9, 4, 5, 5/4)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 4, 5, 5/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
39. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=29/4, reference=5/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(29/4, 5/2, 19/4, 19/10)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (29/4, 5/2, 19/4, 19/10).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
40. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=31/5, reference=6/3 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(31/5, 2, 21/5, 21/10)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (31/5, 2, 21/5, 21/10).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
41. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=33/1, reference=7/4 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(33, 7/4, 125/4, 125/7)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (33, 7/4, 125/4, 125/7).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
42. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=35/2, reference=8/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(35/2, 8, 19/2, 19/16)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (35/2, 8, 19/2, 19/16).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
43. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=37/3, reference=9/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(37/3, 9/2, 47/6, 47/27)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (37/3, 9/2, 47/6, 47/27).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
44. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=39/4, reference=1/3 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(39/4, 1/3, 113/12, 113/4)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (39/4, 1/3, 113/12, 113/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
45. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=41/5, reference=2/4 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(41/5, 1/2, 77/10, 77/5)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (41/5, 1/2, 77/10, 77/5).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
46. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=43/1, reference=3/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(43, 3, 40, 40/3)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (43, 3, 40, 40/3).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
47. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=45/2, reference=4/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(45/2, 2, 41/2, 41/4)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (45/2, 2, 41/2, 41/4).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
48. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=47/3, reference=5/3 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(47/3, 5/3, 14, 42/5)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (47/3, 5/3, 14, 42/5).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
49. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=49/4, reference=6/4 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(49/4, 3/2, 43/4, 43/6)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (49/4, 3/2, 43/4, 43/6).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
50. Relative Error with an Explicit Nonzero Reference
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=51/5, reference=7/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.
View answer and explanation
(51/5, 7, 16/5, 16/35)
Check the nonzero-reference premise and use its absolute value, including for a negative reference. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (51/5, 7, 16/5, 16/35).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-relative-error · candidate
51. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=3/1, reference=1/1, tolerance=1/2≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(3, 1, 2, 1/2, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 1, 2, 1/2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
52. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=5/2, reference=2/2, tolerance=2/3≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(5/2, 1, 3/2, 2/3, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 1, 3/2, 2/3, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
53. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=7/3, reference=3/3, tolerance=3/4≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(7/3, 1, 4/3, 3/4, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7/3, 1, 4/3, 3/4, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
54. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=9/4, reference=4/4, tolerance=4/5≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(9/4, 1, 5/4, 4/5, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9/4, 1, 5/4, 4/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
55. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=11/5, reference=5/1, tolerance=5/6≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(11/5, 5, 14/5, 5/6, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 5, 14/5, 5/6, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
56. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=13/1, reference=6/2, tolerance=6/7≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(13, 3, 10, 6/7, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13, 3, 10, 6/7, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
57. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=15/2, reference=7/3, tolerance=1/8≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(15/2, 7/3, 31/6, 1/8, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/2, 7/3, 31/6, 1/8, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
58. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=17/3, reference=8/4, tolerance=2/2≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(17/3, 2, 11/3, 1, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (17/3, 2, 11/3, 1, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
59. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=19/4, reference=9/1, tolerance=3/3≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(19/4, 9, 17/4, 1, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (19/4, 9, 17/4, 1, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
60. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=21/5, reference=1/2, tolerance=4/4≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(21/5, 1/2, 37/10, 1, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (21/5, 1/2, 37/10, 1, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
61. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=23/1, reference=2/3, tolerance=5/5≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(23, 2/3, 67/3, 1, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (23, 2/3, 67/3, 1, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
62. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=25/2, reference=3/4, tolerance=6/6≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(25/2, 3/4, 47/4, 1, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (25/2, 3/4, 47/4, 1, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
63. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=27/3, reference=4/1, tolerance=1/7≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(9, 4, 5, 1/7, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 4, 5, 1/7, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
64. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=29/4, reference=5/2, tolerance=2/8≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(29/4, 5/2, 19/4, 1/4, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (29/4, 5/2, 19/4, 1/4, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
65. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=31/5, reference=6/3, tolerance=3/2≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(31/5, 2, 21/5, 3/2, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (31/5, 2, 21/5, 3/2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
66. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=33/1, reference=7/4, tolerance=4/3≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(33, 7/4, 125/4, 4/3, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (33, 7/4, 125/4, 4/3, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
67. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=35/2, reference=8/1, tolerance=5/4≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(35/2, 8, 19/2, 5/4, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (35/2, 8, 19/2, 5/4, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
68. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=37/3, reference=9/2, tolerance=6/5≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(37/3, 9/2, 47/6, 6/5, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (37/3, 9/2, 47/6, 6/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
69. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=39/4, reference=1/3, tolerance=1/6≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(39/4, 1/3, 113/12, 1/6, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (39/4, 1/3, 113/12, 1/6, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
70. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=41/5, reference=2/4, tolerance=2/7≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(41/5, 1/2, 77/10, 2/7, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (41/5, 1/2, 77/10, 2/7, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
71. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=43/1, reference=3/1, tolerance=3/8≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(43, 3, 40, 3/8, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (43, 3, 40, 3/8, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
72. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=45/2, reference=4/2, tolerance=4/2≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(45/2, 2, 41/2, 2, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (45/2, 2, 41/2, 2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
73. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=47/3, reference=5/3, tolerance=5/3≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(47/3, 5/3, 14, 5/3, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (47/3, 5/3, 14, 5/3, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
74. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=49/4, reference=6/4, tolerance=6/4≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(49/4, 3/2, 43/4, 3/2, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (49/4, 3/2, 43/4, 3/2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
75. Inclusive Absolute-Error Tolerance
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=51/5, reference=7/1, tolerance=1/5≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
View answer and explanation
(51/5, 7, 16/5, 1/5, No)
The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (51/5, 7, 16/5, 1/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-tolerance-audit · candidate
76. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=1/1, tolerance=1/2≥0, candidate=1/1. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(1, 1/2, 1/2, 3/2, 1, Yes)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1, 1/2, 1/2, 3/2, 1, Yes).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
77. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=2/2, tolerance=2/3≥0, candidate=3/2. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(1, 2/3, 1/3, 5/3, 3/2, Yes)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1, 2/3, 1/3, 5/3, 3/2, Yes).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
78. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=3/3, tolerance=3/4≥0, candidate=5/3. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(1, 3/4, 1/4, 7/4, 5/3, Yes)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1, 3/4, 1/4, 7/4, 5/3, Yes).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
79. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=4/4, tolerance=4/5≥0, candidate=7/4. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(1, 4/5, 1/5, 9/5, 7/4, Yes)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1, 4/5, 1/5, 9/5, 7/4, Yes).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
80. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=5/1, tolerance=5/6≥0, candidate=9/5. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(5, 5/6, 25/6, 35/6, 9/5, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 5/6, 25/6, 35/6, 9/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
81. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=6/2, tolerance=6/7≥0, candidate=11/1. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(3, 6/7, 15/7, 27/7, 11, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 6/7, 15/7, 27/7, 11, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
82. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=7/3, tolerance=1/8≥0, candidate=13/2. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(7/3, 1/8, 53/24, 59/24, 13/2, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7/3, 1/8, 53/24, 59/24, 13/2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
83. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=8/4, tolerance=2/2≥0, candidate=15/3. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(2, 1, 1, 3, 5, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 1, 1, 3, 5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
84. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=9/1, tolerance=3/3≥0, candidate=17/4. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(9, 1, 8, 10, 17/4, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 1, 8, 10, 17/4, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
85. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=1/2, tolerance=4/4≥0, candidate=19/5. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(1/2, 1, -1/2, 3/2, 19/5, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/2, 1, -1/2, 3/2, 19/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
86. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=2/3, tolerance=5/5≥0, candidate=21/1. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(2/3, 1, -1/3, 5/3, 21, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2/3, 1, -1/3, 5/3, 21, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
87. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=3/4, tolerance=6/6≥0, candidate=23/2. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(3/4, 1, -1/4, 7/4, 23/2, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/4, 1, -1/4, 7/4, 23/2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
88. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=4/1, tolerance=1/7≥0, candidate=25/3. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(4, 1/7, 27/7, 29/7, 25/3, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, 1/7, 27/7, 29/7, 25/3, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
89. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=5/2, tolerance=2/8≥0, candidate=27/4. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(5/2, 1/4, 9/4, 11/4, 27/4, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 1/4, 9/4, 11/4, 27/4, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
90. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=6/3, tolerance=3/2≥0, candidate=29/5. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(2, 3/2, 1/2, 7/2, 29/5, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 3/2, 1/2, 7/2, 29/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
91. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=7/4, tolerance=4/3≥0, candidate=31/1. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(7/4, 4/3, 5/12, 37/12, 31, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7/4, 4/3, 5/12, 37/12, 31, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
92. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=8/1, tolerance=5/4≥0, candidate=33/2. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(8, 5/4, 27/4, 37/4, 33/2, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 5/4, 27/4, 37/4, 33/2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
93. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=9/2, tolerance=6/5≥0, candidate=35/3. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(9/2, 6/5, 33/10, 57/10, 35/3, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9/2, 6/5, 33/10, 57/10, 35/3, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
94. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=1/3, tolerance=1/6≥0, candidate=37/4. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(1/3, 1/6, 1/6, 1/2, 37/4, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/3, 1/6, 1/6, 1/2, 37/4, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
95. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=2/4, tolerance=2/7≥0, candidate=39/5. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(1/2, 2/7, 3/14, 11/14, 39/5, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/2, 2/7, 3/14, 11/14, 39/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
96. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=3/1, tolerance=3/8≥0, candidate=41/1. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(3, 3/8, 21/8, 27/8, 41, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 3/8, 21/8, 27/8, 41, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
97. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=4/2, tolerance=4/2≥0, candidate=43/2. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(2, 2, 0, 4, 43/2, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 2, 0, 4, 43/2, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
98. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=5/3, tolerance=5/3≥0, candidate=45/3. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(5/3, 5/3, 0, 10/3, 15, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/3, 5/3, 0, 10/3, 15, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
99. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=6/4, tolerance=6/4≥0, candidate=47/4. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(3/2, 3/2, 0, 3, 47/4, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, 3/2, 0, 3, 47/4, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
100. Inclusive Error Interval and a Candidate Boundary
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=7/1, tolerance=1/5≥0, candidate=49/5. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).
View answer and explanation
(7, 1/5, 34/5, 36/5, 49/5, No)
Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 1/5, 34/5, 36/5, 49/5, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-app20-error-error-interval · candidate
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