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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-20260930-r02-ratio-proportion-foundations · 2026-09-30 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
In the rational numerical-ratio setting, find the exact value of -12:(1). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-12/1
Divide the antecedent by the consequent in the stated order and normalize the result to -12/1.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -11:(-2). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
11/2
Divide the antecedent by the consequent in the stated order and normalize the result to 11/2.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -10:(3). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-10/3
Divide the antecedent by the consequent in the stated order and normalize the result to -10/3.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -9:(-4). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
9/4
Divide the antecedent by the consequent in the stated order and normalize the result to 9/4.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -8:(5). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-8/5
Divide the antecedent by the consequent in the stated order and normalize the result to -8/5.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -7:(-6). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
7/6
Divide the antecedent by the consequent in the stated order and normalize the result to 7/6.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -6:(7). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-6/7
Divide the antecedent by the consequent in the stated order and normalize the result to -6/7.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -5:(-8). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
5/8
Divide the antecedent by the consequent in the stated order and normalize the result to 5/8.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -4:(1). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-4/1
Divide the antecedent by the consequent in the stated order and normalize the result to -4/1.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -3:(-2). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
3/2
Divide the antecedent by the consequent in the stated order and normalize the result to 3/2.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -2:(3). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-2/3
Divide the antecedent by the consequent in the stated order and normalize the result to -2/3.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of -1:(-4). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
1/4
Divide the antecedent by the consequent in the stated order and normalize the result to 1/4.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 0:(5). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
0/1
Divide the antecedent by the consequent in the stated order and normalize the result to 0/1.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 1:(-6). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-1/6
Divide the antecedent by the consequent in the stated order and normalize the result to -1/6.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 2:(7). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
2/7
Divide the antecedent by the consequent in the stated order and normalize the result to 2/7.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 3:(-8). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-3/8
Divide the antecedent by the consequent in the stated order and normalize the result to -3/8.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 4:(1). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
4/1
Divide the antecedent by the consequent in the stated order and normalize the result to 4/1.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 5:(-2). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-5/2
Divide the antecedent by the consequent in the stated order and normalize the result to -5/2.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 6:(3). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
2/1
Divide the antecedent by the consequent in the stated order and normalize the result to 2/1.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 7:(-4). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-7/4
Divide the antecedent by the consequent in the stated order and normalize the result to -7/4.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 8:(5). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
8/5
Divide the antecedent by the consequent in the stated order and normalize the result to 8/5.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 9:(-6). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-3/2
Divide the antecedent by the consequent in the stated order and normalize the result to -3/2.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 10:(7). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
10/7
Divide the antecedent by the consequent in the stated order and normalize the result to 10/7.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 11:(-8). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
-11/8
Divide the antecedent by the consequent in the stated order and normalize the result to -11/8.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, find the exact value of 12:(1). The consequent is nonzero; answer as a reduced p/q with a positive denominator.
12/1
Divide the antecedent by the consequent in the stated order and normalize the result to 12/1.
Reviewed step-by-step derivation
Ratio and proportion · qf-ratio-exact-value · candidate
In the rational numerical-ratio setting, is -12:(-1)=-24:(-2) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are 24 and 24; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -11:(2)=34:(-6) a valid proportion? Both consequents are nonzero.
No
The exact cross products are 66 and 68; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -10:(3)=-40:(12) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are -120 and -120; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -9:(-4)=46:(20) a valid proportion? Both consequents are nonzero.
No
The exact cross products are -180 and -184; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -8:(5)=-48:(30) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are -240 and -240; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -7:(6)=15:(-12) a valid proportion? Both consequents are nonzero.
No
The exact cross products are 84 and 90; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -6:(-7)=-18:(-21) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are 126 and 126; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -5:(8)=21:(-32) a valid proportion? Both consequents are nonzero.
No
The exact cross products are 160 and 168; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -4:(1)=-20:(5) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are -20 and -20; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -3:(-2)=19:(12) a valid proportion? Both consequents are nonzero.
No
The exact cross products are -36 and -38; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -2:(3)=-4:(6) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are -12 and -12; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is -1:(4)=4:(-12) a valid proportion? Both consequents are nonzero.
No
The exact cross products are 12 and 16; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 0:(-5)=0:(-20) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are 0 and 0; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 1:(6)=-4:(-30) a valid proportion? Both consequents are nonzero.
No
The exact cross products are -30 and -24; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 2:(7)=12:(42) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are 84 and 84; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 3:(-8)=-5:(16) a valid proportion? Both consequents are nonzero.
No
The exact cross products are 48 and 40; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 4:(1)=12:(3) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are 12 and 12; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 5:(2)=-19:(-8) a valid proportion? Both consequents are nonzero.
No
The exact cross products are -40 and -38; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 6:(-3)=30:(-15) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are -90 and -90; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 7:(4)=-41:(-24) a valid proportion? Both consequents are nonzero.
No
The exact cross products are -168 and -164; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 8:(5)=16:(10) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are 80 and 80; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 9:(-6)=-26:(18) a valid proportion? Both consequents are nonzero.
No
The exact cross products are 162 and 156; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 10:(7)=40:(28) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are 280 and 280; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 11:(8)=-54:(-40) a valid proportion? Both consequents are nonzero.
No
The exact cross products are -440 and -432; they are different, so the answer is No.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
In the rational numerical-ratio setting, is 12:(-1)=72:(-6) a valid proportion? Both consequents are nonzero.
Yes
The exact cross products are -72 and -72; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-verification · candidate
Solve the rational proportion -12/(1)=x/(-2). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
24/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=24/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -11/(-2)=x/(3). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
33/2
From bx=ad, x=ad/b. Exact substitution and reduction give x=33/2, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -10/(3)=x/(4). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-40/3
From bx=ad, x=ad/b. Exact substitution and reduction give x=-40/3, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -9/(-4)=x/(-5). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-45/4
From bx=ad, x=ad/b. Exact substitution and reduction give x=-45/4, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -8/(5)=x/(6). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-48/5
From bx=ad, x=ad/b. Exact substitution and reduction give x=-48/5, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -7/(-6)=x/(2). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
7/3
From bx=ad, x=ad/b. Exact substitution and reduction give x=7/3, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -6/(7)=x/(-3). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
18/7
From bx=ad, x=ad/b. Exact substitution and reduction give x=18/7, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -5/(-1)=x/(4). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
20/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=20/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -4/(2)=x/(5). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-10/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=-10/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -3/(-3)=x/(-6). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-6/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=-6/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -2/(4)=x/(2). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-1/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=-1/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion -1/(-5)=x/(3). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
3/5
From bx=ad, x=ad/b. Exact substitution and reduction give x=3/5, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 0/(6)=x/(-4). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
0/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=0/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 1/(-7)=x/(5). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-5/7
From bx=ad, x=ad/b. Exact substitution and reduction give x=-5/7, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 2/(1)=x/(6). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
12/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=12/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 3/(-2)=x/(-2). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
3/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=3/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 4/(3)=x/(3). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
4/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=4/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 5/(-4)=x/(4). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-5/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=-5/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 6/(5)=x/(-5). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-6/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=-6/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 7/(-6)=x/(6). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-7/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=-7/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 8/(7)=x/(2). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
16/7
From bx=ad, x=ad/b. Exact substitution and reduction give x=16/7, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 9/(-1)=x/(-3). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
27/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=27/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 10/(2)=x/(4). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
20/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=20/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 11/(-3)=x/(5). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-55/3
From bx=ad, x=ad/b. Exact substitution and reduction give x=-55/3, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
Solve the rational proportion 12/(4)=x/(-6). Both denominators are nonzero; give x as a reduced p/q with a positive denominator.
-18/1
From bx=ad, x=ad/b. Exact substitution and reduction give x=-18/1, uniquely because b≠0.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-missing-term · candidate
In the rational setting, the fixed factor is k=-12. Does the explicit point (x,y)=(-3,36) satisfy y=kx? Check only this point.
Yes
kx=36 and the stated y is 36; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-11. Does the explicit point (x,y)=(-2,23) satisfy y=kx? Check only this point.
No
kx=22 and the stated y is 23; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-10. Does the explicit point (x,y)=(-1,10) satisfy y=kx? Check only this point.
Yes
kx=10 and the stated y is 10; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-9. Does the explicit point (x,y)=(0,1) satisfy y=kx? Check only this point.
No
kx=0 and the stated y is 1; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-8. Does the explicit point (x,y)=(1,-8) satisfy y=kx? Check only this point.
Yes
kx=-8 and the stated y is -8; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-7. Does the explicit point (x,y)=(2,-13) satisfy y=kx? Check only this point.
No
kx=-14 and the stated y is -13; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-6. Does the explicit point (x,y)=(3,-18) satisfy y=kx? Check only this point.
Yes
kx=-18 and the stated y is -18; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-5. Does the explicit point (x,y)=(-3,16) satisfy y=kx? Check only this point.
No
kx=15 and the stated y is 16; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-4. Does the explicit point (x,y)=(-2,8) satisfy y=kx? Check only this point.
Yes
kx=8 and the stated y is 8; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-3. Does the explicit point (x,y)=(-1,4) satisfy y=kx? Check only this point.
No
kx=3 and the stated y is 4; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-2. Does the explicit point (x,y)=(0,0) satisfy y=kx? Check only this point.
Yes
kx=0 and the stated y is 0; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=-1. Does the explicit point (x,y)=(1,0) satisfy y=kx? Check only this point.
No
kx=-1 and the stated y is 0; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=0. Does the explicit point (x,y)=(2,0) satisfy y=kx? Check only this point.
Yes
kx=0 and the stated y is 0; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=1. Does the explicit point (x,y)=(3,4) satisfy y=kx? Check only this point.
No
kx=3 and the stated y is 4; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=2. Does the explicit point (x,y)=(-3,-6) satisfy y=kx? Check only this point.
Yes
kx=-6 and the stated y is -6; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=3. Does the explicit point (x,y)=(-2,-5) satisfy y=kx? Check only this point.
No
kx=-6 and the stated y is -5; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=4. Does the explicit point (x,y)=(-1,-4) satisfy y=kx? Check only this point.
Yes
kx=-4 and the stated y is -4; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=5. Does the explicit point (x,y)=(0,1) satisfy y=kx? Check only this point.
No
kx=0 and the stated y is 1; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=6. Does the explicit point (x,y)=(1,6) satisfy y=kx? Check only this point.
Yes
kx=6 and the stated y is 6; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=7. Does the explicit point (x,y)=(2,15) satisfy y=kx? Check only this point.
No
kx=14 and the stated y is 15; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=8. Does the explicit point (x,y)=(3,24) satisfy y=kx? Check only this point.
Yes
kx=24 and the stated y is 24; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=9. Does the explicit point (x,y)=(-3,-26) satisfy y=kx? Check only this point.
No
kx=-27 and the stated y is -26; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=10. Does the explicit point (x,y)=(-2,-20) satisfy y=kx? Check only this point.
Yes
kx=-20 and the stated y is -20; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=11. Does the explicit point (x,y)=(-1,-10) satisfy y=kx? Check only this point.
No
kx=-11 and the stated y is -10; comparison gives No and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
In the rational setting, the fixed factor is k=12. Does the explicit point (x,y)=(0,0) satisfy y=kx? Check only this point.
Yes
kx=0 and the stated y is 0; comparison gives Yes and does not justify any claim about other inputs.
Reviewed step-by-step derivation
Ratio and proportion · qf-proportion-fixed-factor · candidate
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