# Question 32148: Exact Fraction Addition [Return to catalogue page 161](https://ocme.evemisslab.com/en/questions/page/161/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260930-r01-fraction-foundations/#q-20260930-r01-fraction-exact-addition-0023) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0063-32148 · q-20260930-r01-fraction-exact-addition-0023 ## Original question and givens Compute 10/(2)+(-4)/(3) exactly and write the result in lowest terms with a positive denominator. - G001 — Compute 10/(2)+(-4)/(3) 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](https://ocme.evemisslab.com/en/mko/mko-fraction/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001 S002 · impliesBefore source roles = [10, 2, [-4, 3]] 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 dependenciesRule: Fraction_ALL_RAW_SAFE_integer_components_even_zero_den_validity · Uses: G001, S001 S003 · calculationBefore Fraction ORIGINAL p times s := mul(10, 3) After Fraction ORIGINAL p times s = 30 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Fraction ORIGINAL r times q := mul(-4, 2) After Fraction ORIGINAL r times q = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Fraction addition ORIGINAL cross numerator sum := sum(30, -8) After Fraction addition ORIGINAL cross numerator sum = 22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Fraction addition ORIGINAL denominator product := mul(2, 3) After Fraction addition ORIGINAL denominator product = 6 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [10, 2, -4, 3] After derived = [22, 6] 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 dependenciesRule: Fraction_ADDITION_common_den_ALL_cross_terms_not_naive_components · Uses: G001, S006 S008 · impliesBefore source roles = [Fraction addition normalization, 22, 6] After derived = [22, 6] 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 dependenciesRule: Fraction_DEN_nonzero_sign_normalized_BEFORE_gcd · Uses: G001, S007 S009 · calculationBefore Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(3, 6) After Fraction addition normalization gcd step0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(22, 18) After Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [Fraction addition normalization, 0, 22, 6, 3] After derived = [18, 4] Conditions: none Use x=qy+r,0≤r