Numeric Ratio Guards and Same-Unit Partitions
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261011-advance-20261015-r02-ratio-proportion-foundations · 2026-10-11 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=0, b=-5≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(0/1, No, undefined, undefined)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (0/1, No, undefined, undefined).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
2. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=1, b=-2≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(-1/2, Yes, -2/1, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (-1/2, Yes, -2/1, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
3. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=1, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(1/3, Yes, 3/1, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (1/3, Yes, 3/1, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
4. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=2, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(2/3, Yes, 3/2, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (2/3, Yes, 3/2, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
5. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=3, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(1/1, Yes, 1/1, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (1/1, Yes, 1/1, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
6. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=4, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(4/3, Yes, 3/4, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (4/3, Yes, 3/4, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
7. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=5, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(5/3, Yes, 3/5, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (5/3, Yes, 3/5, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
8. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=6, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(2/1, Yes, 1/2, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (2/1, Yes, 1/2, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
9. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=7, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(7/3, Yes, 3/7, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (7/3, Yes, 3/7, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
10. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=8, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(8/3, Yes, 3/8, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (8/3, Yes, 3/8, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
11. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=9, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(3/1, Yes, 1/3, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (3/1, Yes, 1/3, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
12. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=10, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(10/3, Yes, 3/10, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (10/3, Yes, 3/10, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
13. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=11, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(11/3, Yes, 3/11, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (11/3, Yes, 3/11, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
14. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=12, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(4/1, Yes, 1/4, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (4/1, Yes, 1/4, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
15. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=13, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(13/3, Yes, 3/13, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (13/3, Yes, 3/13, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
16. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=14, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(14/3, Yes, 3/14, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (14/3, Yes, 3/14, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
17. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=15, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(5/1, Yes, 1/5, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (5/1, Yes, 1/5, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
18. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=16, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(16/3, Yes, 3/16, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (16/3, Yes, 3/16, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
19. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=17, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(17/3, Yes, 3/17, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (17/3, Yes, 3/17, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
20. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=18, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(6/1, Yes, 1/6, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (6/1, Yes, 1/6, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
21. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=19, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(19/3, Yes, 3/19, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (19/3, Yes, 3/19, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
22. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=20, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(20/3, Yes, 3/20, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (20/3, Yes, 3/20, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
23. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=21, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(7/1, Yes, 1/7, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (7/1, Yes, 1/7, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
24. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=22, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(22/3, Yes, 3/22, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (22/3, Yes, 3/22, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
25. Forward Ratio and Independent Reverse Guard
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=23, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).
View answer and explanation
(23/3, Yes, 3/23, 1/1)
The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (23/3, Yes, 3/23, 1/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-forward-reverse · candidate
26. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/1, complete ordered pairs (x,y)=[[0, 0], [1, 1], [2, 2]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 0/1, 0/1], [Yes, Yes, Yes], [undefined, 1/1, 1/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 0/1, 0/1], [Yes, Yes, Yes], [undefined, 1/1, 1/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
27. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/1, complete ordered pairs (x,y)=[[0, 1], [1, 1], [2, 2]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([1/1, 0/1, 0/1], [No, Yes, Yes], [undefined, 1/1, 1/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([1/1, 0/1, 0/1], [No, Yes, Yes], [undefined, 1/1, 1/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
28. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-10/2, complete ordered pairs (x,y)=[[0, 0], [1, 2], [2, -1]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 7/1, 9/1], [Yes, No, No], [undefined, 2/1, -1/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 7/1, 9/1], [Yes, No, No], [undefined, 2/1, -1/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
29. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-9/2, complete ordered pairs (x,y)=[[0, 0], [1, 3], [2, -2]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 15/2, 7/1], [Yes, No, No], [undefined, 3/1, -1/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 15/2, 7/1], [Yes, No, No], [undefined, 3/1, -1/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
30. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-8/2, complete ordered pairs (x,y)=[[0, 0], [1, 4], [2, -3]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 8/1, 5/1], [Yes, No, No], [undefined, 4/1, -3/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 8/1, 5/1], [Yes, No, No], [undefined, 4/1, -3/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
31. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-7/2, complete ordered pairs (x,y)=[[0, 0], [1, 5], [2, -4]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 17/2, 3/1], [Yes, No, No], [undefined, 5/1, -2/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 17/2, 3/1], [Yes, No, No], [undefined, 5/1, -2/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
32. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-6/2, complete ordered pairs (x,y)=[[0, 0], [1, 6], [2, -5]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 9/1, 1/1], [Yes, No, No], [undefined, 6/1, -5/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 9/1, 1/1], [Yes, No, No], [undefined, 6/1, -5/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
33. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-5/2, complete ordered pairs (x,y)=[[0, 0], [1, 7], [2, -6]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 19/2, -1/1], [Yes, No, No], [undefined, 7/1, -3/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 19/2, -1/1], [Yes, No, No], [undefined, 7/1, -3/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
34. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-4/2, complete ordered pairs (x,y)=[[0, 0], [1, 8], [2, -7]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 10/1, -3/1], [Yes, No, No], [undefined, 8/1, -7/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 10/1, -3/1], [Yes, No, No], [undefined, 8/1, -7/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
35. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-3/2, complete ordered pairs (x,y)=[[0, 0], [1, 9], [2, -8]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 21/2, -5/1], [Yes, No, No], [undefined, 9/1, -4/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 21/2, -5/1], [Yes, No, No], [undefined, 9/1, -4/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
36. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-2/2, complete ordered pairs (x,y)=[[0, 0], [1, 10], [2, -9]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 11/1, -7/1], [Yes, No, No], [undefined, 10/1, -9/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 11/1, -7/1], [Yes, No, No], [undefined, 10/1, -9/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
37. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-1/2, complete ordered pairs (x,y)=[[0, 0], [1, 11], [2, -10]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 23/2, -9/1], [Yes, No, No], [undefined, 11/1, -5/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 23/2, -9/1], [Yes, No, No], [undefined, 11/1, -5/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
38. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=0/2, complete ordered pairs (x,y)=[[0, 0], [1, 12], [2, -11]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 12/1, -11/1], [Yes, No, No], [undefined, 12/1, -11/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 12/1, -11/1], [Yes, No, No], [undefined, 12/1, -11/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
39. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/2, complete ordered pairs (x,y)=[[0, 0], [1, 13], [2, -12]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 25/2, -13/1], [Yes, No, No], [undefined, 13/1, -6/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 25/2, -13/1], [Yes, No, No], [undefined, 13/1, -6/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
40. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=2/2, complete ordered pairs (x,y)=[[0, 0], [1, 14], [2, -13]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 13/1, -15/1], [Yes, No, No], [undefined, 14/1, -13/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 13/1, -15/1], [Yes, No, No], [undefined, 14/1, -13/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
41. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=3/2, complete ordered pairs (x,y)=[[0, 0], [1, 15], [2, -14]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 27/2, -17/1], [Yes, No, No], [undefined, 15/1, -7/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 27/2, -17/1], [Yes, No, No], [undefined, 15/1, -7/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
42. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=4/2, complete ordered pairs (x,y)=[[0, 0], [1, 16], [2, -15]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 14/1, -19/1], [Yes, No, No], [undefined, 16/1, -15/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 14/1, -19/1], [Yes, No, No], [undefined, 16/1, -15/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
43. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=5/2, complete ordered pairs (x,y)=[[0, 0], [1, 17], [2, -16]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 29/2, -21/1], [Yes, No, No], [undefined, 17/1, -8/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 29/2, -21/1], [Yes, No, No], [undefined, 17/1, -8/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
44. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=6/2, complete ordered pairs (x,y)=[[0, 0], [1, 18], [2, -17]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 15/1, -23/1], [Yes, No, No], [undefined, 18/1, -17/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 15/1, -23/1], [Yes, No, No], [undefined, 18/1, -17/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
45. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=7/2, complete ordered pairs (x,y)=[[0, 0], [1, 19], [2, -18]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 31/2, -25/1], [Yes, No, No], [undefined, 19/1, -9/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 31/2, -25/1], [Yes, No, No], [undefined, 19/1, -9/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
46. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=8/2, complete ordered pairs (x,y)=[[0, 0], [1, 20], [2, -19]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 16/1, -27/1], [Yes, No, No], [undefined, 20/1, -19/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 16/1, -27/1], [Yes, No, No], [undefined, 20/1, -19/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
47. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=9/2, complete ordered pairs (x,y)=[[0, 0], [1, 21], [2, -20]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 33/2, -29/1], [Yes, No, No], [undefined, 21/1, -10/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 33/2, -29/1], [Yes, No, No], [undefined, 21/1, -10/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
48. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=10/2, complete ordered pairs (x,y)=[[0, 0], [1, 22], [2, -21]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 17/1, -31/1], [Yes, No, No], [undefined, 22/1, -21/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 17/1, -31/1], [Yes, No, No], [undefined, 22/1, -21/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
49. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=11/2, complete ordered pairs (x,y)=[[0, 0], [1, 23], [2, -22]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 35/2, -33/1], [Yes, No, No], [undefined, 23/1, -11/1])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 35/2, -33/1], [Yes, No, No], [undefined, 23/1, -11/1]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
50. Complete Three-Point Proportion Law and Ratio Values
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=12/2, complete ordered pairs (x,y)=[[0, 0], [1, 24], [2, -23]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.
View answer and explanation
([0/1, 18/1, -35/1], [Yes, No, No], [undefined, 24/1, -23/2])
Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 18/1, -35/1], [Yes, No, No], [undefined, 24/1, -23/2]).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-complete-point-law · candidate
51. Same-Unit Weighted Partition and Actual Ratio Guard
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=1, b=1, same-unit nonnegative total T=0/1. 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).
View answer and explanation
(0/1, 0/1, 0/1, 0/1, No, undefined)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (0/1, 0/1, 0/1, 0/1, No, undefined).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
52. Same-Unit Weighted Partition and Actual Ratio Guard
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=1, b=3, same-unit nonnegative total T=8/1. 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).
View answer and explanation
(2/1, 6/1, 0/1, 0/1, Yes, 1/3)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (2/1, 6/1, 0/1, 0/1, Yes, 1/3).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
53. Same-Unit Weighted Partition and Actual Ratio Guard
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=3, 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).
View answer and explanation
(3/2, 1/1, 0/1, 0/1, Yes, 3/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (3/2, 1/1, 0/1, 0/1, Yes, 3/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
54. Same-Unit Weighted Partition and Actual Ratio Guard
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=4, 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).
View answer and explanation
(5/3, 5/6, 0/1, 0/1, Yes, 2/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (5/3, 5/6, 0/1, 0/1, Yes, 2/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
55. Same-Unit Weighted Partition and Actual Ratio Guard
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=5, 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).
View answer and explanation
(25/14, 5/7, 0/1, 0/1, Yes, 5/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (25/14, 5/7, 0/1, 0/1, Yes, 5/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
56. Same-Unit Weighted Partition and Actual Ratio Guard
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=6, 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).
View answer and explanation
(15/8, 5/8, 0/1, 0/1, Yes, 3/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (15/8, 5/8, 0/1, 0/1, Yes, 3/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
57. Same-Unit Weighted Partition and Actual Ratio Guard
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=7, 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).
View answer and explanation
(35/18, 5/9, 0/1, 0/1, Yes, 7/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (35/18, 5/9, 0/1, 0/1, Yes, 7/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
58. Same-Unit Weighted Partition and Actual Ratio Guard
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=8, 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).
View answer and explanation
(2/1, 1/2, 0/1, 0/1, Yes, 4/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (2/1, 1/2, 0/1, 0/1, Yes, 4/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
59. Same-Unit Weighted Partition and Actual Ratio Guard
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=9, 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).
View answer and explanation
(45/22, 5/11, 0/1, 0/1, Yes, 9/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (45/22, 5/11, 0/1, 0/1, Yes, 9/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
60. Same-Unit Weighted Partition and Actual Ratio Guard
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=10, 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).
View answer and explanation
(25/12, 5/12, 0/1, 0/1, Yes, 5/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (25/12, 5/12, 0/1, 0/1, Yes, 5/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
61. Same-Unit Weighted Partition and Actual Ratio Guard
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=11, 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).
View answer and explanation
(55/26, 5/13, 0/1, 0/1, Yes, 11/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (55/26, 5/13, 0/1, 0/1, Yes, 11/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
62. Same-Unit Weighted Partition and Actual Ratio Guard
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=12, 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).
View answer and explanation
(15/7, 5/14, 0/1, 0/1, Yes, 6/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (15/7, 5/14, 0/1, 0/1, Yes, 6/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
63. Same-Unit Weighted Partition and Actual Ratio Guard
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=13, 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).
View answer and explanation
(13/6, 1/3, 0/1, 0/1, Yes, 13/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (13/6, 1/3, 0/1, 0/1, Yes, 13/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
64. Same-Unit Weighted Partition and Actual Ratio Guard
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).
View answer and explanation
(35/16, 5/16, 0/1, 0/1, Yes, 7/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (35/16, 5/16, 0/1, 0/1, Yes, 7/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
65. Same-Unit Weighted Partition and Actual Ratio Guard
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=15, 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).
View answer and explanation
(75/34, 5/17, 0/1, 0/1, Yes, 15/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (75/34, 5/17, 0/1, 0/1, Yes, 15/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
66. Same-Unit Weighted Partition and Actual Ratio Guard
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=16, 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).
View answer and explanation
(20/9, 5/18, 0/1, 0/1, Yes, 8/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (20/9, 5/18, 0/1, 0/1, Yes, 8/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
67. Same-Unit Weighted Partition and Actual Ratio Guard
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=17, 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).
View answer and explanation
(85/38, 5/19, 0/1, 0/1, Yes, 17/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (85/38, 5/19, 0/1, 0/1, Yes, 17/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
68. Same-Unit Weighted Partition and Actual Ratio Guard
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=18, 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).
View answer and explanation
(9/4, 1/4, 0/1, 0/1, Yes, 9/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (9/4, 1/4, 0/1, 0/1, Yes, 9/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
69. Same-Unit Weighted Partition and Actual Ratio Guard
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=19, 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).
View answer and explanation
(95/42, 5/21, 0/1, 0/1, Yes, 19/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (95/42, 5/21, 0/1, 0/1, Yes, 19/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
70. Same-Unit Weighted Partition and Actual Ratio Guard
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=20, 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).
View answer and explanation
(25/11, 5/22, 0/1, 0/1, Yes, 10/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (25/11, 5/22, 0/1, 0/1, Yes, 10/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
71. Same-Unit Weighted Partition and Actual Ratio Guard
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=21, 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).
View answer and explanation
(105/46, 5/23, 0/1, 0/1, Yes, 21/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (105/46, 5/23, 0/1, 0/1, Yes, 21/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
72. Same-Unit Weighted Partition and Actual Ratio Guard
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=22, 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).
View answer and explanation
(55/24, 5/24, 0/1, 0/1, Yes, 11/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (55/24, 5/24, 0/1, 0/1, Yes, 11/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
73. Same-Unit Weighted Partition and Actual Ratio Guard
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).
View answer and explanation
(23/10, 1/5, 0/1, 0/1, Yes, 23/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (23/10, 1/5, 0/1, 0/1, Yes, 23/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
74. Same-Unit Weighted Partition and Actual Ratio Guard
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=24, 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).
View answer and explanation
(30/13, 5/26, 0/1, 0/1, Yes, 12/1)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (30/13, 5/26, 0/1, 0/1, Yes, 12/1).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
75. Same-Unit Weighted Partition and Actual Ratio Guard
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=25, 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).
View answer and explanation
(125/54, 5/27, 0/1, 0/1, Yes, 25/2)
Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (125/54, 5/27, 0/1, 0/1, Yes, 25/2).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-proportional-partition · candidate
76. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=0, b=0, d=0, x=0/1. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(No, undefined, 0/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (No, undefined, 0/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
77. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=1, b=0, d=2, x=3/1. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(No, undefined, 2/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (No, undefined, 2/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
78. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=2, b=3, d=0, x=1/3. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(No, undefined, -1/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (No, undefined, -1/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
79. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=1, b=2, d=3, x=3/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 3/2, 0/1, Yes)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 3/2, 0/1, Yes).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
80. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-8, b=3, d=2, x=5/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -16/3, -47/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -16/3, -47/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
81. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-7, b=3, d=2, x=6/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -14/3, -23/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -14/3, -23/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
82. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-6, b=3, d=2, x=7/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -4/1, -45/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -4/1, -45/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
83. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-5, b=3, d=2, x=8/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -10/3, -22/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -10/3, -22/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
84. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-4, b=3, d=2, x=9/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -8/3, -43/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -8/3, -43/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
85. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-3, b=3, d=2, x=10/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -2/1, -21/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -2/1, -21/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
86. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-2, b=3, d=2, x=11/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -4/3, -41/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -4/3, -41/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
87. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-1, b=3, d=2, x=12/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, -2/3, -20/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -2/3, -20/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
88. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=0, b=3, d=2, x=13/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 0/1, -39/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 0/1, -39/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
89. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=1, b=3, d=2, x=14/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 2/3, -19/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 2/3, -19/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
90. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=2, b=3, d=2, x=15/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 4/3, -37/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 4/3, -37/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
91. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=3, b=3, d=2, x=16/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 2/1, -18/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 2/1, -18/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
92. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=4, b=3, d=2, x=17/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 8/3, -35/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 8/3, -35/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
93. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=5, b=3, d=2, x=18/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 10/3, -17/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 10/3, -17/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
94. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=6, b=3, d=2, x=19/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 4/1, -33/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 4/1, -33/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
95. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=7, b=3, d=2, x=20/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 14/3, -16/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 14/3, -16/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
96. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=8, b=3, d=2, x=21/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 16/3, -31/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 16/3, -31/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
97. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=9, b=3, d=2, x=22/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 6/1, -15/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 6/1, -15/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
98. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=10, b=3, d=2, x=23/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 20/3, -29/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 20/3, -29/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
99. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=11, b=3, d=2, x=24/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 22/3, -14/1, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 22/3, -14/1, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
100. Candidate Proportion Guards and Cross-Product Defect
Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=12, b=3, d=2, x=25/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).
View answer and explanation
(Yes, 8/1, -27/2, No)
Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 8/1, -27/2, No).
Reviewed step-by-step derivation
Ratio and proportion · qf-density15-ratio-missing-component-guard · candidate
Canonical question JSONL · English JSONL · Validation report