Question 32139: Exact Fraction Addition

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0063-32139 · q-20260930-r01-fraction-exact-addition-0014

Original question and givens

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

G001 — Compute 1/(-7)+(-2)/(4) 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.

mko-fraction

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 = [1, -7, [-2, 4]]

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(1, 4)

After

Fraction ORIGINAL p times s = 4

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(-2, -7)

After

Fraction ORIGINAL r times q = 14

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(4, 14)

After

Fraction addition ORIGINAL cross numerator sum = 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, S004

S006 · calculation

Before

Fraction addition ORIGINAL denominator product := mul(-7, 4)

After

Fraction addition ORIGINAL denominator product = -28

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 = [1, -7, -2, 4]

After

derived = [18, -28]

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 · calculation

Before

Fraction addition normalization ORIGINAL negate numerator for denominator sign := sub(0, 18)

After

Fraction addition normalization ORIGINAL negate numerator for denominator sign = -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, S007

S009 · calculation

Before

Fraction addition normalization ORIGINAL negate denominator := sub(0, -28)

After

Fraction addition normalization ORIGINAL negate denominator = 28

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 · implies

Before

source roles = [Fraction addition normalization, 18, -28]

After

derived = [-18, 28]

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, S009

S011 · calculation

Before

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(0, 28)

After

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 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, S010

S012 · calculation

Before

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(18, 0)

After

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 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, S011

S013 · implies

Before

source roles = [Fraction addition normalization, 0, 18, 28, 0]

After

derived = [0, 18]

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, S012

S014 · calculation

Before

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(1, 18)

After

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 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, S013

S015 · calculation

Before

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(28, 18)

After

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 10

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 · implies

Before

source roles = [Fraction addition normalization, 1, 28, 18, 1]

After

derived = [18, 10]

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, S015

S017 · calculation

Before

Fraction addition normalization gcd step2 ORIGINAL quotient times divisor := mul(1, 10)

After

Fraction addition normalization gcd step2 ORIGINAL quotient times divisor = 10

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 gcd step2 ORIGINAL dividend minus product := sub(18, 10)

After

Fraction addition normalization gcd step2 ORIGINAL dividend minus product = 8

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, 2, 18, 10, 1]

After

derived = [10, 8]

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, S018

S020 · calculation

Before

Fraction addition normalization gcd step3 ORIGINAL quotient times divisor := mul(1, 8)

After

Fraction addition normalization gcd step3 ORIGINAL quotient times divisor = 8

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 · calculation

Before

Fraction addition normalization gcd step3 ORIGINAL dividend minus product := sub(10, 8)

After

Fraction addition normalization gcd step3 ORIGINAL dividend minus 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, S020

S022 · implies

Before

source roles = [Fraction addition normalization, 3, 10, 8, 1]

After

derived = [8, 2]

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, S021

S023 · calculation

Before

Fraction addition normalization gcd step4 ORIGINAL quotient times divisor := mul(4, 2)

After

Fraction addition normalization gcd step4 ORIGINAL quotient times divisor = 8

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, S022

S024 · calculation

Before

Fraction addition normalization gcd step4 ORIGINAL dividend minus product := sub(8, 8)

After

Fraction addition normalization gcd step4 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, S023

S025 · implies

Before

source roles = [Fraction addition normalization, 4, 8, 2, 4]

After

derived = [8, 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, S024

S026 · calculation

Before

Fraction addition normalization ORIGINAL numerator over positive gcd := div(-18, 2)

After

Fraction addition normalization ORIGINAL numerator over positive gcd = -9

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, S025

S027 · calculation

Before

Fraction addition normalization ORIGINAL denominator over positive gcd := div(28, 2)

After

Fraction addition normalization ORIGINAL denominator over positive gcd = 14

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, S026

S028 · calculation

Before

Fraction addition normalization reverse reduced numerator times gcd := mul(-9, 2)

After

Fraction addition normalization reverse reduced numerator times gcd = -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, S027

S029 · calculation

Before

Fraction addition normalization reverse reduced denominator times gcd := mul(14, 2)

After

Fraction addition normalization reverse reduced denominator times gcd = 28

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, S028

S030 · implies

Before

source roles = [Fraction addition normalization, -18, 28, 2]

After

derived = [-9, 14, -18, 28, -9/14]

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, S029

S031 · calculation

Before

Fraction addition reverse ORIGINAL fraction sum := sum(-1/7, -1/2)

After

Fraction addition reverse ORIGINAL fraction sum = -9/14

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, S030

S032 · implies

Before

source roles = [-1/7, -1/2]

After

derived = [-9/14, -9/14]

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, S031

Reverse checks

S032 · 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.
S031 · premise_and_result_audit — Recompute Fraction addition reverse ORIGINAL fraction sum with exact rational arithmetic, not rounded decimals.
S030 · 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.
S029 · premise_and_result_audit — Recompute Fraction addition normalization reverse reduced denominator times gcd with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute Fraction addition normalization reverse reduced numerator times gcd with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Fraction addition normalization ORIGINAL denominator over positive gcd with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute Fraction addition normalization ORIGINAL numerator over positive gcd with exact rational arithmetic, not rounded decimals.
S025 · 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.
S024 · premise_and_result_audit — Recompute Fraction addition normalization gcd step4 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute Fraction addition normalization gcd step4 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
S022 · 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.
S021 · premise_and_result_audit — Recompute Fraction addition normalization gcd step3 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute Fraction addition normalization gcd step3 ORIGINAL quotient times divisor 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 gcd step2 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute Fraction addition normalization gcd step2 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
S016 · 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.
S015 · premise_and_result_audit — Recompute Fraction addition normalization gcd step1 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute Fraction addition normalization gcd step1 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
S013 · 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.
S012 · premise_and_result_audit — Recompute Fraction addition normalization gcd step0 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Fraction addition normalization gcd step0 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
S010 · 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.
S009 · premise_and_result_audit — Recompute Fraction addition normalization ORIGINAL negate denominator with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute Fraction addition normalization ORIGINAL negate numerator for denominator sign with exact rational arithmetic, not rounded decimals.
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

-9/14

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: 828aa31ce892e85698c6a5565cc56051801430829c8e65c6d17490bfd10b2bb9
candidate record SHA-256: 962533a135d8e2a7af2a3d1da73539956a405bfe4d664c1bd728103708c1085b
semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
Lean: NOT_RUN · human review: NOT_PERFORMED

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