This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-r01-fraction-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.
For integer numerator p=0 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.
No
A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=0 and denominator q=5, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=5≠0, so 0/5 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-10 and denominator q=3, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=3≠0, so -10/3 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-9 and denominator q=-4, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=-4≠0, so -9/-4 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-8 and denominator q=5, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=5≠0, so -8/5 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-7 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.
No
A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-6 and denominator q=-7, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=-7≠0, so -6/-7 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-5 and denominator q=1, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=1≠0, so -5/1 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-4 and denominator q=2, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=2≠0, so -4/2 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-3 and denominator q=-3, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=-3≠0, so -3/-3 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-2 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.
No
A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=-1 and denominator q=5, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=5≠0, so -1/5 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=0 and denominator q=-6, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=-6≠0, so 0/-6 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=1 and denominator q=7, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=7≠0, so 1/7 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=2 and denominator q=1, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=1≠0, so 2/1 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=3 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.
No
A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=4 and denominator q=3, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=3≠0, so 4/3 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=5 and denominator q=4, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=4≠0, so 5/4 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=6 and denominator q=-5, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=-5≠0, so 6/-5 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=7 and denominator q=6, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=6≠0, so 7/6 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=8 and denominator q=0, is p/q a valid rational-number representation? Answer Yes or No.
No
A valid denominator must be nonzero. Here q=0, so the representation is invalid even if the numerator is zero.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=9 and denominator q=-1, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=-1≠0, so 9/-1 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=10 and denominator q=2, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=2≠0, so 10/2 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=11 and denominator q=3, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=3≠0, so 11/3 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
For integer numerator p=12 and denominator q=-4, is p/q a valid rational-number representation? Answer Yes or No.
Yes
q=-4≠0, so 12/-4 is a valid rational-number representation.
Reviewed step-by-step derivation
Fraction · qf-fraction-representation-validity · candidate
The valid fractions are -12/(1) and 24/(-2), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are 24 and 24; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -11/(-2) and -32/(-6), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are 66 and 64; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -10/(3) and -40/(12), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are -120 and -120; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -9/(-4) and 46/(20), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are -180 and -184; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -8/(5) and -16/(10), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are -80 and -80; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -7/(-6) and -20/(-18), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are 126 and 120; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -6/(7) and 24/(-28), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are 168 and 168; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -5/(-1) and -24/(-5), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are 25 and 24; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -4/(2) and -8/(4), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are -16 and -16; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -3/(-3) and 10/(9), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are -27 and -30; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -2/(4) and -8/(16), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are -32 and -32; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are -1/(-5) and -4/(-25), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are 25 and 20; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 0/(6) and 0/(-12), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are 0 and 0; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 1/(-7) and 4/(-21), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are -21 and -28; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 2/(1) and 8/(4), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are 8 and 8; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 3/(-2) and -14/(10), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are 30 and 28; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 4/(3) and 8/(6), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are 24 and 24; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 5/(-4) and 16/(-12), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are -60 and -64; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 6/(5) and -24/(-20), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are -120 and -120; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 7/(-6) and 36/(-30), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are -210 and -216; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 8/(7) and 16/(14), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are 112 and 112; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 9/(-1) and -26/(3), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are 27 and 26; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 10/(2) and 40/(8), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are 80 and 80; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 11/(-3) and 56/(-15), with both denominators nonzero. Do they represent the same rational number?
No
The two cross products are -165 and -168; they are different, so the answer is No.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
The valid fractions are 12/(4) and -24/(-8), with both denominators nonzero. Do they represent the same rational number?
Yes
The two cross products are -96 and -96; they are equal, so the answer is Yes.
Reviewed step-by-step derivation
Fraction · qf-fraction-equivalence · candidate
Reduce the valid integer fraction -24/(2) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-12/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -12/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -33/(-6) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
11/2
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 11/2; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -40/(12) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-10/3
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -10/3; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -45/(-20) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
9/4
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 9/4; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -16/(10) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-8/5
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -8/5; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -21/(-18) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
7/6
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 7/6; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -24/(28) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-6/7
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -6/7; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -25/(-5) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
5/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 5/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -8/(4) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-2/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -2/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -9/(-9) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
1/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 1/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -8/(16) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-1/2
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -1/2; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction -5/(-25) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
1/5
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 1/5; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 0/(12) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
0/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 0/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 3/(-21) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-1/7
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -1/7; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 8/(4) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
2/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 2/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 15/(-10) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-3/2
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -3/2; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 8/(6) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
4/3
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 4/3; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 15/(-12) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-5/4
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -5/4; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 24/(20) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
6/5
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 6/5; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 35/(-30) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-7/6
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -7/6; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 16/(14) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
8/7
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 8/7; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 27/(-3) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-9/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -9/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 40/(8) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
5/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 5/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 55/(-15) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
-11/3
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is -11/3; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Reduce the valid integer fraction 24/(8) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.
3/1
Normalize the denominator sign and divide numerator and denominator by their greatest common divisor. The canonical result is 3/1; zero is written 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-reduction · candidate
Compute 0/(1)+(0)/(-1) exactly and write the result in lowest terms with a positive denominator.
0/1
Use the common-denominator numerator 0×-1+(0)×1 and denominator 1×-1, then normalize and reduce to 0/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -11/(-2)+(9)/(2) exactly and write the result in lowest terms with a positive denominator.
10/1
Use the common-denominator numerator -11×2+(9)×-2 and denominator -2×2, then normalize and reduce to 10/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -10/(3)+(6)/(3) exactly and write the result in lowest terms with a positive denominator.
-4/3
Use the common-denominator numerator -10×3+(6)×3 and denominator 3×3, then normalize and reduce to -4/3.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -9/(-4)+(3)/(-4) exactly and write the result in lowest terms with a positive denominator.
3/2
Use the common-denominator numerator -9×-4+(3)×-4 and denominator -4×-4, then normalize and reduce to 3/2.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -8/(5)+(0)/(5) exactly and write the result in lowest terms with a positive denominator.
-8/5
Use the common-denominator numerator -8×5+(0)×5 and denominator 5×5, then normalize and reduce to -8/5.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -7/(-6)+(-3)/(1) exactly and write the result in lowest terms with a positive denominator.
-11/6
Use the common-denominator numerator -7×1+(-3)×-6 and denominator -6×1, then normalize and reduce to -11/6.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -6/(7)+(-6)/(-2) exactly and write the result in lowest terms with a positive denominator.
15/7
Use the common-denominator numerator -6×-2+(-6)×7 and denominator 7×-2, then normalize and reduce to 15/7.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -5/(-1)+(-9)/(3) exactly and write the result in lowest terms with a positive denominator.
2/1
Use the common-denominator numerator -5×3+(-9)×-1 and denominator -1×3, then normalize and reduce to 2/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -4/(2)+(-12)/(4) exactly and write the result in lowest terms with a positive denominator.
-5/1
Use the common-denominator numerator -4×4+(-12)×2 and denominator 2×4, then normalize and reduce to -5/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -3/(-3)+(10)/(-5) exactly and write the result in lowest terms with a positive denominator.
-1/1
Use the common-denominator numerator -3×-5+(10)×-3 and denominator -3×-5, then normalize and reduce to -1/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -2/(4)+(7)/(1) exactly and write the result in lowest terms with a positive denominator.
13/2
Use the common-denominator numerator -2×1+(7)×4 and denominator 4×1, then normalize and reduce to 13/2.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute -1/(-5)+(4)/(2) exactly and write the result in lowest terms with a positive denominator.
11/5
Use the common-denominator numerator -1×2+(4)×-5 and denominator -5×2, then normalize and reduce to 11/5.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 0/(6)+(1)/(-3) exactly and write the result in lowest terms with a positive denominator.
-1/3
Use the common-denominator numerator 0×-3+(1)×6 and denominator 6×-3, then normalize and reduce to -1/3.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 1/(-7)+(-2)/(4) exactly and write the result in lowest terms with a positive denominator.
-9/14
Use the common-denominator numerator 1×4+(-2)×-7 and denominator -7×4, then normalize and reduce to -9/14.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 2/(1)+(-5)/(5) exactly and write the result in lowest terms with a positive denominator.
1/1
Use the common-denominator numerator 2×5+(-5)×1 and denominator 1×5, then normalize and reduce to 1/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 3/(-2)+(-8)/(-1) exactly and write the result in lowest terms with a positive denominator.
13/2
Use the common-denominator numerator 3×-1+(-8)×-2 and denominator -2×-1, then normalize and reduce to 13/2.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 4/(3)+(-11)/(2) exactly and write the result in lowest terms with a positive denominator.
-25/6
Use the common-denominator numerator 4×2+(-11)×3 and denominator 3×2, then normalize and reduce to -25/6.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 5/(-4)+(11)/(3) exactly and write the result in lowest terms with a positive denominator.
29/12
Use the common-denominator numerator 5×3+(11)×-4 and denominator -4×3, then normalize and reduce to 29/12.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 6/(5)+(8)/(-4) exactly and write the result in lowest terms with a positive denominator.
-4/5
Use the common-denominator numerator 6×-4+(8)×5 and denominator 5×-4, then normalize and reduce to -4/5.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 7/(-6)+(5)/(5) exactly and write the result in lowest terms with a positive denominator.
-1/6
Use the common-denominator numerator 7×5+(5)×-6 and denominator -6×5, then normalize and reduce to -1/6.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 8/(7)+(2)/(1) exactly and write the result in lowest terms with a positive denominator.
22/7
Use the common-denominator numerator 8×1+(2)×7 and denominator 7×1, then normalize and reduce to 22/7.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 9/(-1)+(-1)/(-2) exactly and write the result in lowest terms with a positive denominator.
-17/2
Use the common-denominator numerator 9×-2+(-1)×-1 and denominator -1×-2, then normalize and reduce to -17/2.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 10/(2)+(-4)/(3) exactly and write the result in lowest terms with a positive denominator.
11/3
Use the common-denominator numerator 10×3+(-4)×2 and denominator 2×3, then normalize and reduce to 11/3.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 11/(-3)+(-7)/(4) exactly and write the result in lowest terms with a positive denominator.
-65/12
Use the common-denominator numerator 11×4+(-7)×-3 and denominator -3×4, then normalize and reduce to -65/12.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
Compute 12/(4)+(-10)/(-5) exactly and write the result in lowest terms with a positive denominator.
5/1
Use the common-denominator numerator 12×-5+(-10)×4 and denominator 4×-5, then normalize and reduce to 5/1.
Reviewed step-by-step derivation
Fraction · qf-fraction-exact-addition · candidate
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