Question 1837: Net Compound Interest Can Be Negative

Original question and givens

Use one fixed exact rate per declared period and no deposits, withdrawals, fees, taxes or rounding. Rate and period conventions agree; balances/principal are nonnegative and 1+rate>0. This is not investment advice, a quote or a market forecast. 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. principal=155/4, rate=-1/20, periods=2. Give (principal, rate, periods, future value, future value−principal). A permitted negative rate may produce negative net interest.

• G001 — Use one fixed exact rate per declared period and no deposits, withdrawals, fees, taxes or rounding. Rate and period conventions agree; balances/principal are nonnegative and 1+rate>0. This is not investment advice, a quote or a market forecast. 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. principal=155/4, rate=-1/20, periods=2. Give (principal, rate, periods, future value, future value−principal). A permitted negative rate may produce negative net interest.

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-compound-interest

Step-by-step derivation

accumulation factor := sum(1, -1/20)

accumulation factor = 19/20

per-period rate=-1/20; factor=19/20>0

fixed-period exact recurrence B_(k+1)=(1+r)B_k

B₀=155/4; periods=2

0 periods leave the principal unchanged

B_1 := mul(155/4, 19/20)

B_1 = 589/16

B_2 := mul(589/16, 19/20)

B_2 = 11191/320

B₀=P; B_(k+1)=(1+r)B_k

B_n=P(1+r)^n by induction

(19/20)^2

(1+r)^n = 361/400

future balance := mul(155/4, 361/400)

future balance = 11191/320

net interest := sub(11191/320, 155/4)

net interest = -1209/320

Reverse checks

• S009 · premise_and_result_audit — Recompute net interest with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute future balance with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute B_2 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute B_1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute accumulation factor with exact rational arithmetic, not rounded decimals.

Result

(155/4, -1/20, 2, 11191/320, -1209/320)

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.