# Question 29536: Compound and Discount with the Same Factor and Period Count [Return to catalogue page 148](https://ocme.evemisslab.com/en/questions/page/148/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r02-present-value-discounting-foundations/#q-20261013-advance-20261022-r02-present-value-discounting-0061) 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. R0058-29536 · q-20261013-advance-20261022-r02-present-value-discounting-0061 ## Original question and givens Use a fixed rate i=-2/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. principal=70/2, periods=1. Give (principal,rate,periods,compounded future amount,recovered present amount,exact recovery?). - G001 — Use a fixed rate i=-2/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. principal=70/2, periods=1. Give (principal,rate,periods,compounded future amount,recovered present amount,exact recovery?). ## 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-present-value-discounting](https://ocme.evemisslab.com/en/mko/mko-present-value-discounting/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-str22-presentvalue-forward-backward-audit, presentvalue-forward-backward-audit] After derived = [[principal, rate, periods, future, recovered, roundtrip]] Conditions: none Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention. Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001 S002 · impliesBefore source roles = [Present value rate, [-2, 20]] After derived = [-1/10] Conditions: none Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks. Step identifiers and dependenciesRule: STR22_input_RAW_pair1e6_before_sign_gcd_normalization · Uses: G001, S001 S003 · calculationBefore Present value ORIGINAL positive accumulation factor := sum(1, -1/10) After Present value ORIGINAL positive accumulation factor = 9/10 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Present value ORIGINAL positive accumulation factor, sum, [1, -1/10]] After derived = [[1], [9, 10], 9/10] Conditions: none SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold. Step identifiers and dependenciesRule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S003 S005 · impliesBefore source roles = [-1/10, 9/10] After derived = [Yes] Conditions: none Use the SAME fixed per-period rate/period convention without cash flows,fees,taxes or rounding.Require1+rate>0;the rate may be negative.This is not a market forecast or investment advice. Step identifiers and dependenciesRule: Present_value_FIXED_model_same_rate_period_NOcashflows · Uses: G001, S004 S006 · impliesBefore source roles = [9/10, 1] After derived = [1, 9/10, 1, out*base^exp=factor^n] Conditions: none Initialize out=1,base=factor,exp=n with invariant out*base^exp=factor^n.At n=0 return the empty product1,but STILL qualify the original factor. Step identifiers and dependenciesRule: Present_value_BINARY_power_initial_emptyproduct_invariant · Uses: G001, S005 S007 · calculationBefore Present value ORIGINAL binary odd multiply0 := mul(1, 9/10) After Present value ORIGINAL binary odd multiply0 = 9/10 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [Present value ORIGINAL binary odd multiply0, mul, [1, 9/10]] After derived = [[1, 1], [9, 10], 9/10] Conditions: none SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold. Step identifiers and dependenciesRule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S007 S009 · impliesBefore source roles = [9/10, 1, 0, [1, 9/10, 1]] After derived = [Yes, 9/10, 9/10, 0, Yes] Conditions: none Write old exp=2q+r.Multiply by base when r=1,and square the base ONLY if q>0.This preserves the SAME exact power without an unused final square that could cause a spurious source hold.Do not replace the SOURCE binary sequence with linear multiplication and shrink its domain. Step identifiers and dependenciesRule: Present_value_EACH_binary_exponent_step_no_unused_final_square · Uses: G001, S008 S010 · impliesBefore source roles = [Present value principal, [70, 2]] After derived = [35] Conditions: none Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks. Step identifiers and dependenciesRule: STR22_input_RAW_pair1e6_before_sign_gcd_normalization · Uses: G001, S009 S011 · calculationBefore Present value ORIGINAL compound principal := mul(35, 9/10) After Present value ORIGINAL compound principal = 63/2 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore source roles = [Present value ORIGINAL compound principal, mul, [35, 9/10]] After derived = [[5, 1], [63, 2], 63/2] Conditions: none SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold. Step identifiers and dependenciesRule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S011 S013 · calculationBefore Present value ORIGINAL discount SAME compounded amount := div(63/2, 9/10) After Present value ORIGINAL discount SAME compounded amount = 35 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [Present value ORIGINAL discount SAME compounded amount, div, [63/2, 9/10]] After derived = [[9, 2], [35, 1], 35] Conditions: none SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold. Step identifiers and dependenciesRule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S013 S015 · impliesBefore source roles = [35, -1/10, 1, 9/10] After derived = [63/2, 35, Yes] Conditions: none The SAME positive accumulation gives(principal*acc)/acc=principal.Retain both actual calculations,including n=0.This exact-model algebraic roundtrip makes no claim about realized future returns. Step identifiers and dependenciesRule: Present_value_roundtrip_cancel_SAME_positive_power · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S014 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S013 · premise_and_result_audit — Recompute Present value ORIGINAL discount SAME compounded amount with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S011 · premise_and_result_audit — Recompute Present value ORIGINAL compound principal with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S009 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S008 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S007 · premise_and_result_audit — Recompute Present value ORIGINAL binary odd multiply0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S005 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S004 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S003 · premise_and_result_audit — Recompute Present value ORIGINAL positive accumulation factor with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. ## Result (35, -1/10, 1, 63/2, 35, Yes) 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: a0e247615ebbe0de5423f450b35caabfe8ada62d93973eb2971bc2c85cfdbee4 - candidate record SHA-256: bcd3fc2d4f060cb466a8a61ac8552cb9567c9f71d6e149dd29f7a1731e3ab9c7 - semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df - question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r02-present-value-discounting-0061.json)