How two fractions combine into one reduced exact result
Fraction arithmetic keeps the answer exact as an integer numerator over a nonzero integer denominator. A single-fraction conversion shows its decimal value and whether that decimal terminates or repeats.
The input fractions use integer numerators and nonzero integer denominators. For addition or subtraction, cross-products create a common denominator. Multiplication pairs numerators and denominators directly; division flips the second fraction before multiplying. To convert a single fraction, select Convert to decimal and leave the second fraction out of the workflow. Reduction by an integer greatest common divisor keeps the rational value exact in either case.
Fraction operations use exact integer arithmetic and greatest-common-divisor reduction. The equation keeps the raw step visible while a separate rounded decimal aids comparison.
How numerators, denominators and the operation set the answer
Every box takes a whole number, and the operation decides how the four numbers are cross-multiplied. Order and sign matter more here than in most calculators.
- First numerator and denominator
- The top and bottom of the first fraction, each a whole number from −1,000,000 to 1,000,000, with a nonzero denominator. A sign may sit on either part, and 1/−2 is normalized to −1/2. There is no box for a mixed number, so enter 1 1/2 as the improper fraction 3/2; a decimal such as 1.5 is rejected.
- Operation
- Add and subtract cross-multiply over the product of the denominators, multiply pairs tops and bottoms, and divide multiplies by the second fraction's reciprocal. Convert ignores the second fraction completely. Picking the wrong operation changes the answer sharply: 3/4 ÷ 3/8 is 2, while 3/4 × 3/8 is 9/32.
- Second numerator and denominator
- The fraction acted on by the first, used by every operation except Convert. Its denominator cannot be zero, and in division its numerator cannot be zero either, because dividing by 0/d has no value. Order matters for subtraction and division: 1/2 − 1/3 is 1/6, but 1/3 − 1/2 is −1/6.
- Reduction and mixed form
- The raw result is divided by its greatest common divisor, so 3/2 × 4/9 shows a raw 12/18 and a removed divisor of 6 before reaching 2/3. The mixed-number fact keeps the sign on the whole part: −7/2 reads −3 1/2, meaning negative three and a half, not −3 plus one-half.
- Decimal display
- The decimal approximation is printed to eight places, so 2/3 appears as 0.66666667 and is labeled as repeating. An exact zero result is written 0/1, as when 1/7 − 1/7 gives a raw 0/49 that reduces by 49. Carry the fraction, not the rounded decimal, into any further calculation.
Steps
- Enter a numerator and nonzero denominator for the first fraction.
- Choose an arithmetic operation and enter the second fraction, or choose Convert to decimal to work with the first fraction alone.
- Read the simplified exact result, decimal display and its terminating or repeating status.
Reading tip: Check the operation symbol before reading the answer. In single-fraction mode, an equals sign marks a terminating decimal and an approximation sign marks a repeating expansion rounded for display.
Where exact fraction arithmetic beats a rounded decimal
Check homework
Show the unreduced operation and the exact simplified rational answer.
Divide portions
Use reciprocal multiplication to divide one nonzero fraction by another.
Compare with decimals
Read an approximate decimal without replacing the exact fraction.
Normalize signs
Keep negative results on the numerator while the denominator stays positive.
How the calculation works
- a / b
- First integer numerator and nonzero integer denominator.
- c / d
- Second integer numerator and nonzero integer denominator.
- gcd
- Greatest common divisor used to remove all shared integer factors.
- raw result
- Numerator and denominator from the selected operation before simplification.
The main answer remains an exact rational number; decimal output is only an eight-place approximation when necessary.
What every part means
- a / b
- First integer numerator and nonzero integer denominator.
- c / d
- Second integer numerator and nonzero integer denominator.
- gcd
- Greatest common divisor used to remove all shared integer factors.
- raw result
- Numerator and denominator from the selected operation before simplification.
Calculate it step by step
Use the selected arithmetic rule, with b and d nonzero where used, and c nonzero for division.
Keep any negative sign on the numerator.
The decimal terminates exactly when the reduced denominator has no prime factors other than 2 or 5; otherwise the displayed value is rounded.
Integer arithmetic uses BigInt, avoiding floating-point errors while forming and reducing fractions. Only the secondary decimal conversion is rounded.
See how Calculation Guide tests examples and handles calculation limits.
See it in practice
- First numerator
- 1
- First denominator
- 2
- Operation
- Add (+)
- Second numerator
- 1
- Second denominator
- 3
1/2 + 1/3 = (1×3 + 1×2)/(2×3) = 5/6. The greatest common divisor of 5 and 6 is 1, so the exact answer stays 5/6.
- Mixed-number form5/6
- Decimal approximation0.83333333
Interpretation & limitations
How to read your result
The primary result is an exact improper or proper fraction. The mixed form can be easier to read. In conversion mode, 1/8 = 0.125 terminates, while 1/3 ≈ 0.33333333 repeats; the exact fraction remains authoritative.
Where this estimate stops
Inputs must be whole numbers within the stated bounds. This tool does not accept decimals directly; express a decimal as a fraction first.
Frequently asked questions
01Why is 2/4 shown as 1/2?
Both parts are divided by their greatest common divisor, 2; the value stays equal.
02Can I divide by zero?
No. A denominator of zero is invalid, and division by a second fraction with zero numerator is also invalid.
03Can a numerator be negative?
Yes. The sign is carried through the operation and shown on the final numerator.
04Is 0/5 allowed?
Yes, as a fraction value. It is invalid as the divisor in a division operation.
05Do I need a second fraction to get a decimal?
No. Choose Convert first fraction to decimal; only the first numerator and denominator are used.
06Is a repeating decimal exact?
The exact answer is the simplified fraction; the decimal shown to eight places is approximate.