Question 7098: Same-Unit Weighted Partition and Actual Ratio Guard

Original question and givens

Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=23, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).

• G001 — Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=23, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).

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-ratio-proportion

Step-by-step derivation

exact rational numerical ratios;each denominator needs its own guard

cross-multiplied equality does not legalize a zero ratio denominator

positive weight sum := sum(23, 2)

positive weight sum = 25

total times first weight := mul(5/2, 23)

total times first weight = 115/2

first proportional part x := div(115/2, 25)

first proportional part x = 23/10

second part from remaining total := sub(5/2, 23/10)

second part from remaining total = 1/5

sum of allocated parts := sum(23/10, 1/5)

sum of allocated parts = 5/2

allocation sum residual := sub(5/2, 5/2)

allocation sum residual = 0

allocation b times x := mul(2, 23/10)

allocation b times x = 23/5

allocation a times y := mul(23, 1/5)

allocation a times y = 23/5

allocation proportional residual := sub(23/5, 23/5)

allocation proportional residual = 0

actual allocated ratio x/y := div(23/10, 1/5)

actual allocated ratio x/y = 23/2

T=5/2;parts=(23/10,1/5);residuals=(0,0)

actual ratio legal=Yes;x/y=23/2

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute actual allocated ratio x/y with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute allocation proportional residual with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute allocation a times y with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute allocation b times x with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute allocation sum residual with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute sum of allocated parts with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute second part from remaining total with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute first proportional part x with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute total times first weight with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute positive weight sum with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(23/10, 1/5, 0/1, 0/1, Yes, 23/2)

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.