Question 7089: 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=14, 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=14, 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(14, 2)

positive weight sum = 16

total times first weight := mul(5/2, 14)

total times first weight = 35

first proportional part x := div(35, 16)

first proportional part x = 35/16

second part from remaining total := sub(5/2, 35/16)

second part from remaining total = 5/16

sum of allocated parts := sum(35/16, 5/16)

sum of allocated parts = 5/2

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

allocation sum residual = 0

allocation b times x := mul(2, 35/16)

allocation b times x = 35/8

allocation a times y := mul(14, 5/16)

allocation a times y = 35/8

allocation proportional residual := sub(35/8, 35/8)

allocation proportional residual = 0

actual allocated ratio x/y := div(35/16, 5/16)

actual allocated ratio x/y = 7

T=5/2;parts=(35/16,5/16);residuals=(0,0)

actual ratio legal=Yes;x/y=7/1

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

(35/16, 5/16, 0/1, 0/1, Yes, 7/1)

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.