OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 32147: Exact Fraction Addition

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 161

← Question derivation index · Original question batch

R0063-32147 · q-20260930-r01-fraction-exact-addition-0022

Original question and givens

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

  • G001 — Compute 9/(-1)+(-1)/(-2) exactly and write the result in lowest terms with a positive denominator.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-fraction-exact-addition, addition, no-required-domain, []]

After

derived = [[reduced_fraction_RETAIN1_STRING]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependencies

Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001

S002 · implies

Before

source roles = [9, -1, [-1, -2]]

After

derived = [Yes]

Conditions: none

Validate EVERY raw SAFE integer component,including the numerator at q0/0/0 eligibility.A validity task may legally return No;actual arithmetic requires ALL used denominators nonzero.

Step identifiers and dependencies

Rule: Fraction_ALL_RAW_SAFE_integer_components_even_zero_den_validity · Uses: G001, S001

S003 · calculation

Before

Fraction ORIGINAL p times s := mul(9, -2)

After

Fraction ORIGINAL p times s = -18

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Fraction ORIGINAL r times q := mul(-1, -1)

After

Fraction ORIGINAL r times q = 1

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Fraction addition ORIGINAL cross numerator sum := sum(-18, 1)

After

Fraction addition ORIGINAL cross numerator sum = -17

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Fraction addition ORIGINAL denominator product := mul(-1, -2)

After

Fraction addition ORIGINAL denominator product = 2

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [9, -1, -1, -2]

After

derived = [-17, 2]

Conditions: none

Use the common denominator qs with numerator ps+rq,retaining BOTH cross terms.This is not componentwise addition.Derived large integers are not SAFE-capped.

Step identifiers and dependencies

Rule: Fraction_ADDITION_common_den_ALL_cross_terms_not_naive_components · Uses: G001, S006

S008 · implies

Before

source roles = [Fraction addition normalization, -17, 2]

After

derived = [-17, 2]

Conditions: none

A denominator must FIRST be nonzero.If negative,negate numerator AND denominator together.Derived numerator/denominator are not RAW-SAFE recapped;changing only the denominator would change the value.

Step identifiers and dependencies

Rule: Fraction_DEN_nonzero_sign_normalized_BEFORE_gcd · Uses: G001, S007

S009 · calculation

Before

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(8, 2)

After

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 16

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(17, 16)

After

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 1

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [Fraction addition normalization, 0, 17, 2, 8]

After

derived = [16, 1]

Conditions: none

Use x=qy+r,0≤r<y,to replace gcd(x,y)by gcd(y,r),preserving common divisors BOTH ways.Divide ONLY by nonzero y;remainders strictly decrease to0.

Step identifiers and dependencies

Rule: Fraction_EUCLID_all_remainders_and_common_divisor_invariant · Uses: G001, S010

S012 · calculation

Before

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(2, 1)

After

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 2

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(2, 2)

After

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 0

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [Fraction addition normalization, 1, 2, 1, 2]

After

derived = [2, 0]

Conditions: none

Use x=qy+r,0≤r<y,to replace gcd(x,y)by gcd(y,r),preserving common divisors BOTH ways.Divide ONLY by nonzero y;remainders strictly decrease to0.

Step identifiers and dependencies

Rule: Fraction_EUCLID_all_remainders_and_common_divisor_invariant · Uses: G001, S013

S015 · calculation

Before

Fraction addition normalization ORIGINAL numerator over positive gcd := div(-17, 1)

After

Fraction addition normalization ORIGINAL numerator over positive gcd = -17

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Fraction addition normalization ORIGINAL denominator over positive gcd := div(2, 1)

After

Fraction addition normalization ORIGINAL denominator over positive gcd = 2

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Fraction addition normalization reverse reduced numerator times gcd := mul(-17, 1)

After

Fraction addition normalization reverse reduced numerator times gcd = -17

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Fraction addition normalization reverse reduced denominator times gcd := mul(2, 1)

After

Fraction addition normalization reverse reduced denominator times gcd = 2

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

source roles = [Fraction addition normalization, -17, 2, 1]

After

derived = [-17, 2, -17, 2, -17/2]

Conditions: none

Divide BOTH components by the positive gcd,retaining value and a positive lowest-terms denominator.Multiply back to recover both components.Zero is0/1;integers RETAIN/1,not decimals.

Step identifiers and dependencies

Rule: Fraction_REDUCED_positive_den_ZERO0over1_RETAIN_integer1 · Uses: G001, S018

S020 · calculation

Before

Fraction addition reverse ORIGINAL fraction sum := sum(-9, 1/2)

After

Fraction addition reverse ORIGINAL fraction sum = -17/2

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

source roles = [-9/1, 1/2]

After

derived = [-17/2, -17/2]

Conditions: none

Reverse-check by summing the TWO ORIGINAL fractions with their own denominators.Compare the reduced result exactly without rewriting original parameters.

Step identifiers and dependencies

Rule: Fraction_ADDITION_reverse_original_two_values · Uses: G001, S020

Reverse checks

  • S021 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S020 · premise_and_result_audit — Recompute Fraction addition reverse ORIGINAL fraction sum with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S018 · premise_and_result_audit — Recompute Fraction addition normalization reverse reduced denominator times gcd with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Fraction addition normalization reverse reduced numerator times gcd with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Fraction addition normalization ORIGINAL denominator over positive gcd with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Fraction addition normalization ORIGINAL numerator over positive gcd with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Recompute Fraction addition normalization gcd step1 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Fraction addition normalization gcd step1 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Recompute Fraction addition normalization gcd step0 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Fraction addition normalization gcd step0 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Recompute Fraction addition ORIGINAL denominator product with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Fraction addition ORIGINAL cross numerator sum with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Fraction ORIGINAL r times q with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Fraction ORIGINAL p times s with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

-17/2

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: b1e778f8324410ff004ca8b7803bad307fac73852a3a425971f9f1d8fd3f5752
  • candidate record SHA-256: 0e3c91bce78d963dfae0e081ea0c0cdbef9aa69b362bafead0c04f8e3b7c64af
  • semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
  • question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON