# Exact Rational Errors and Inclusive Tolerance Bounds

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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-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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r03-numerical-approximation-error-0100/)

Numerical approximation and error · qf-app20-error-error-interval · candidate



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