Fraction Calculator Guide: Add, Subtract, Multiply, Divide, Simplify, and Convert Fractions
A fraction represents a ratio of two numbers. In the fraction a/b, the numerator a tells how many parts are being considered and the denominator b describes the size of the equal partition or reference unit.
The denominator cannot be zero. Division by zero is undefined, so expressions such as 3/0 are not fractions with an extremely large value—they are mathematically undefined.
Fractions can often be represented in several equivalent forms. For example, 1/2, 2/4, 50/100, 0.5, and 50% all represent the same numerical value even though their notation differs.
Exact fraction arithmetic is particularly useful because many rational numbers cannot be represented exactly with a finite decimal. One third, for example, is exactly 1/3 but has the repeating decimal representation 0.3333….
Addition and subtraction require a common denominator because the parts being combined must represent the same-sized units. Multiplication does not require a common denominator, and division is performed by multiplying by the reciprocal of the divisor.
Fraction simplification reduces numerator and denominator by their greatest common divisor. A simplified fraction is numerically identical to the original fraction but uses the smallest equivalent integer numerator and denominator.
Mixed numbers combine a whole-number part with a proper fraction, such as 2 1/3. Arithmetic is usually easier and less error-prone when mixed numbers are first converted into improper fractions, the operation is performed, and the result is converted back if desired.
Negative signs also need consistent handling. The forms −3/4, 3/−4, and −(3/4) represent the same negative rational number, but conventional simplified notation places the negative sign in the numerator or in front of the fraction rather than leaving a negative denominator.
This calculator therefore preserves exact fractional results first and treats decimal or percentage values as alternate representations rather than replacing exact fractions automatically.
How to Calculate Fractions Exactly and Convert Between Fraction Forms
- Choose the operation: Select addition, subtraction, multiplication, division, simplification, conversion, comparison, reciprocal, or common denominator.
- Enter numerators and denominators: Denominators must be nonzero integers for ordinary fraction arithmetic.
- Convert mixed numbers when needed: The calculator converts mixed numbers to improper fractions before performing arithmetic.
- Use a common denominator for addition and subtraction: The calculator can use the least common denominator to keep intermediate numbers smaller.
- Multiply directly for multiplication: Multiply numerators together and denominators together, then simplify.
- Use the reciprocal for division: Invert the second nonzero fraction and multiply.
- Simplify the final result: Divide numerator and denominator by their greatest common divisor.
- Choose the preferred output format: Display the result as a simplified fraction, improper fraction, mixed number, decimal, percentage, or several forms together.
Formula and variables
For addition and subtraction, the two fractions must be expressed over a common denominator. Cross multiplication constructs a valid common denominator bd, though a least common denominator can reduce intermediate values. Multiplication multiplies numerators and denominators directly. Division multiplies by the reciprocal of the second fraction, provided that second fraction is not zero.
Addition: a/b + c/d = (ad + bc)/bd; Subtraction: a/b − c/d = (ad − bc)/bd; Multiplication: a/b × c/d = ac/bd; Division: a/b ÷ c/d = ad/bc- a — First numerator
- The numerator of the first fraction.
- b — First denominator
- The nonzero denominator of the first fraction.
- c — Second numerator
- The numerator of the second fraction.
- d — Second denominator
- The nonzero denominator of the second fraction.
- GCD — Greatest common divisor
- The largest positive integer dividing both numerator and denominator, used for simplification.
- LCD — Least common denominator
- The least common multiple of the denominators used as the smallest common denominator for addition or subtraction.
Scenario 1: Add 3/4 and 5/6
The denominators 4 and 6 are different, so the fractions must be converted to a common denominator before addition.
- First fraction
- 3/4
- Second fraction
- 5/6
- The least common denominator of 4 and 6 is 12.
- Convert 3/4 to 9/12.
- Convert 5/6 to 10/12.
- Add numerators: 9 + 10 = 19.
- Result = 19/12.
- 19 and 12 share no common factor greater than 1, so the fraction is already simplified.
- As a mixed number, 19/12 = 1 7/12.
Result: 3/4 + 5/6 = 19/12 = 1 7/12.
The improper fraction 19/12 and mixed number 1 7/12 are exactly equivalent representations of the same result.
Understanding your results
Simplified fraction
This is the exact rational result reduced so the numerator and denominator have no common factor greater than 1.
It is usually the preferred canonical fraction form.
Improper fraction
An improper fraction has numerator magnitude greater than or equal to denominator magnitude.
It is often the most convenient form for further arithmetic.
Mixed number
A mixed number combines an integer and proper fraction.
It can be easier to read in measurement or everyday contexts but is less convenient for algebraic manipulation.
Decimal representation
Some fractions terminate as decimals while others repeat indefinitely.
A displayed decimal can therefore be an approximation even when the fraction is exact.
Percentage representation
Multiply the fraction value by 100 to express it as a percentage.
Repeating fractions can produce repeating percentages.
Assumptions
- Numerators and denominators are integers unless a conversion mode explicitly accepts decimal input.
- Denominators are nonzero.
- Fraction results are simplified using the greatest common divisor.
- Negative signs are normalized so denominators are positive in canonical output.
- Exact fractional arithmetic is preserved before decimal approximation.
- Mixed numbers are converted to improper fractions for arithmetic.
- Decimal conversions use sufficient precision to identify terminating or repeating behavior where supported.
Limitations
- A denominator of zero is undefined and cannot produce a valid ordinary fraction.
- A zero numerator is valid when the denominator is nonzero and represents the number zero.
- Decimal-to-fraction conversion can be exact for terminating decimals entered exactly, but conversion from rounded floating-point values can recover only the entered approximation.
- Repeating decimal conversion requires knowledge of the repeating pattern; a truncated decimal such as 0.333 cannot be assumed automatically to mean exactly 1/3.
- Very large numerators and denominators can exceed ordinary JavaScript Number integer precision if arbitrary-precision integer arithmetic is not used.
- Using floating-point arithmetic internally for exact fraction operations can introduce rounding error. Integer or BigInt-based rational arithmetic is preferable.
- Negative mixed-number notation can be ambiguous if implemented carelessly. For example, −2 1/3 conventionally means −(2 + 1/3), or −7/3.
- Fraction comparison should avoid unnecessary decimal conversion because exact cross multiplication can compare rational values without rounding.
Common mistakes
- Adding denominators when adding fractions.
- Subtracting denominators when subtracting fractions.
- Using a common denominator for multiplication when none is required.
- Dividing numerators and denominators directly when dividing fractions.
- Forgetting to invert the second fraction during division.
- Trying to divide by the fraction 0/n.
- Failing to simplify the final fraction.
- Simplifying numerator and denominator by different factors.
- Treating 3/0 as infinity.
- Assuming 0/3 is undefined.
- Treating a rounded decimal as an exact repeating decimal.
- Misinterpreting a negative mixed number such as −2 1/3.
Practical use cases
Scenario 2: Subtract fractions
Calculate 7/8 − 1/6.
Using denominator 24 gives 21/24 − 4/24 = 17/24.
Scenario 3: Multiply fractions
Calculate 3/5 × 10/9.
The product is 30/45, which simplifies to 2/3.
Scenario 4: Divide fractions
Calculate 4/7 ÷ 2/3.
Multiply 4/7 by the reciprocal 3/2 to get 12/14 = 6/7.
Scenario 5: Convert mixed number to improper fraction
Convert 3 2/5.
3 × 5 + 2 = 17, giving 17/5.
Scenario 6: Convert decimal to fraction
Convert 0.375.
0.375 = 375/1000, which simplifies to 3/8.
Planning and decision guide
A fraction is a quotient
The notation a/b represents a divided by b.
The denominator must therefore be nonzero.
Scenario 7: Zero numerator
0/5 = 0.
Any zero numerator over a nonzero denominator represents zero.
Zero denominator is undefined
The expression 5/0 asks for a number that, when multiplied by zero, would produce five.
No real or ordinary complex number satisfies that requirement.
Equivalent fractions represent the same ratio
Multiplying numerator and denominator by the same nonzero number preserves the fraction’s value.
For example, 2/3 = 4/6 = 10/15.
Scenario 8: Generate equivalent fraction
Multiply 3/7 by 4/4.
The equivalent fraction is 12/28.
Simplification reverses equivalent scaling
Divide numerator and denominator by a shared factor.
The greatest common divisor produces the fully reduced form in one step.
Scenario 9: Simplify 42/56
gcd(42, 56) = 14.
42 ÷ 14 = 3 and 56 ÷ 14 = 4, so the result is 3/4.
Euclid’s algorithm efficiently finds the GCD
Repeatedly replace the larger pair with the divisor and remainder until the remainder becomes zero.
The final nonzero divisor is the greatest common divisor.
Least common denominator comes from the LCM
The least common denominator of two fractions is the least common multiple of their denominators.
Using the LCD minimizes the scale of intermediate numerators.
LCM and GCD are related
For positive integers a and b, lcm(a,b) × gcd(a,b) = a × b.
This provides an efficient way to compute the least common multiple.
Scenario 10: LCD of 8 and 12
gcd(8,12) = 4.
lcm = 8 × 12 ÷ 4 = 24.
Addition requires like-sized parts
Fractions such as thirds and fifths cannot have their numerators added directly because the units differ.
They must first be expressed using a common denominator.
Scenario 11: Why 1/2 + 1/3 is not 2/5
Halves and thirds are different-sized partitions.
Using denominator 6 gives 3/6 + 2/6 = 5/6.
Subtraction follows the same denominator rule
Convert both fractions to equivalent fractions with a common denominator.
Then subtract the numerators and simplify.
Multiplication is structurally simpler
Multiply numerator by numerator and denominator by denominator.
Common denominators are unnecessary.
Cross-canceling can simplify before multiplication
Common factors can be canceled between a numerator and the opposite denominator before multiplying.
This keeps intermediate integers smaller while preserving the exact result.
Scenario 12: 14/15 × 25/21
Cancel 14 with 21 by 7, giving 2 and 3.
Cancel 25 with 15 by 5, giving 5 and 3.
Product = 10/9.
Division means multiplication by the reciprocal
For nonzero c/d, dividing by c/d is equivalent to multiplying by d/c.
The reciprocal exists only when c is nonzero.
Scenario 13: Divide by zero fraction
3/4 ÷ 0/5 is undefined.
The reciprocal of 0/5 would require 5/0, which is undefined.
A reciprocal swaps numerator and denominator
The reciprocal of a nonzero fraction a/b is b/a.
A fraction multiplied by its reciprocal equals 1.
Scenario 14: Reciprocal of −3/7
The reciprocal is −7/3.
Their product is 1.
Proper fractions have smaller numerator magnitude
For positive fractions, a proper fraction has numerator less than denominator.
Its value lies between zero and one.
Improper fractions are still valid fractions
An improper fraction has numerator magnitude at least as large as denominator magnitude.
It can represent a whole number or a value greater than one in magnitude.
Scenario 15: 12/4
12/4 is an improper fraction.
It simplifies exactly to the integer 3.
Mixed numbers are notation, not a separate numeric type
2 3/4 means 2 + 3/4.
It can always be represented as the improper fraction 11/4.
Convert mixed number by multiplying and adding
For whole number w and fraction n/d, improper numerator = wd + n.
The denominator remains d.
Scenario 16: 5 3/8
5 × 8 + 3 = 43.
Therefore 5 3/8 = 43/8.
Convert improper fraction to mixed number by division
Integer division gives the whole-number part.
The remainder becomes the new numerator over the original denominator.
Scenario 17: 29/6
29 ÷ 6 = 4 remainder 5.
Therefore 29/6 = 4 5/6.
Negative mixed numbers require explicit grouping
The conventional meaning of −2 1/4 is −(2 + 1/4) = −9/4.
It should not be parsed as −2 + 1/4 = −7/4.
Normalize negative denominators
A fraction such as 3/−5 is usually written −3/5.
Keeping denominators positive creates consistent canonical output.
Scenario 18: −6/−8
The two negative signs cancel.
6/8 simplifies to 3/4.
Fractions compare exactly by cross multiplication
For positive denominators, compare ad with bc for fractions a/b and c/d.
This avoids decimal rounding.
Scenario 19: Compare 7/12 and 5/8
7 × 8 = 56.
5 × 12 = 60.
Since 56 < 60, 7/12 < 5/8.
Decimal conversion is division
Divide numerator by denominator.
The result either terminates or repeats for rational numbers.
Terminating decimals have restricted denominator factors
After simplification, a fraction has a terminating decimal exactly when its denominator contains no prime factors other than 2 and 5.
This follows from the base-10 place-value system.
Scenario 20: 3/40 terminates
40 = 2³ × 5.
Therefore 3/40 has a terminating decimal: 0.075.
Scenario 21: 1/6 repeats
6 contains the prime factor 3.
Therefore its decimal expansion does not terminate: 0.1666….
Repeating decimals can represent exact rational numbers
The repeating decimal 0.333… equals exactly 1/3.
The ellipsis or repeating bar is essential; 0.333 without repetition notation is only 333/1000.
Scenario 22: 0.333 versus 0.333…
0.333 = 333/1000 exactly.
0.333… = 1/3 exactly.
Terminating decimal to fraction uses place value
Write the decimal digits over a power of ten and simplify.
For example, 0.125 = 125/1000 = 1/8.
Repeating decimal conversion uses algebra
A known repeating block can be isolated by multiplying by an appropriate power of 10 and subtracting.
This converts the repeating decimal exactly into a fraction.
Scenario 23: Convert 0.272727…
Let x = 0.272727….
100x = 27.272727….
Subtract x: 99x = 27.
x = 27/99 = 3/11.
Fraction to percentage multiplies by 100
First evaluate the fraction as a ratio.
Then multiply the value by 100%.
Use the Percentage Calculator for deeper percentage relationships
The Fraction Calculator can display the equivalent percentage.
The Percentage Calculator handles percentage change, reverse percentages, successive changes, percentage points, discounts, and related problems.
Scenario 24: 7/20 as a percentage
7 ÷ 20 = 0.35.
0.35 × 100 = 35%.
Percent to fraction reverses the same relationship
Write p% as p/100 and simplify.
For decimals inside the percentage, first remove decimal places consistently.
Scenario 25: 12.5% as a fraction
12.5% = 12.5/100 = 125/1000.
Simplify to 1/8.
Fractions can represent ratios larger than one
A fraction is not required to lie between zero and one.
For example, 7/3 is a valid rational number greater than two.
Fractions can represent negative quantities
A negative sign can apply to the numerator, denominator, or entire fraction.
Canonical form should place only one negative sign.
Zero has infinitely many fraction representations
0/1, 0/5, and 0/100 all equal zero.
Canonical simplification typically returns 0/1.
Scenario 26: Simplify 0/18
The value is zero.
A normalized rational representation can be 0/1.
A fraction equal to an integer can simplify fully
If denominator divides numerator exactly, the result is an integer.
The calculator can still display the equivalent fraction if requested.
Scenario 27: 36/9
36 ÷ 9 = 4.
The simplest result is the integer 4.
Division signs and fraction bars represent the same fundamental operation
3/4 and 3 ÷ 4 have the same numerical meaning.
Fractions package that quotient into exact rational notation.
Order of operations still applies in compound expressions
Fraction bars can group entire numerator and denominator expressions.
For advanced expressions, use the Scientific Calculator or Algebra Calculator.
Scenario 28: (1 + 2)/(3 + 4)
Evaluate the grouped numerator and denominator first.
The result is 3/7, not 1 + 2/3 + 4.
Exact arithmetic should use integers internally
Represent each rational number as a normalized numerator and denominator pair.
This avoids binary floating-point errors in calculations such as 1/10 + 2/10.
BigInt is preferable for large exact fractions
JavaScript Number cannot exactly represent every integer above 2^53 − 1.
BigInt-based numerator and denominator arithmetic preserves exactness for much larger integer values.
Scenario 29: Large denominators
Cross multiplication can create intermediate integers much larger than either original denominator.
GCD reduction and arbitrary-precision arithmetic prevent silent precision loss.
Cross-cancel before multiplication to control growth
Reduce common factors before multiplying large fractions.
This is mathematically equivalent and computationally more efficient.
The Basic Calculator is the general arithmetic companion
Fractions are exact rational arithmetic.
Use the Basic Calculator for unrestricted decimal and whole-number arithmetic where exact fraction representation is not required.
The strongest result preserves both exact and approximate forms
Show simplified fraction first, then mixed number where relevant, decimal approximation, and percentage if useful.
This lets users choose the representation appropriate to the problem without losing exactness.
Frequently asked questions
How do I add fractions?
Find a common denominator, convert both fractions to equivalent fractions with that denominator, add the numerators, and simplify.
How do I subtract fractions?
Find a common denominator, subtract the numerators, keep the denominator, and simplify.
How do I multiply fractions?
Multiply numerators together and denominators together, then simplify.
How do I divide fractions?
Multiply the first fraction by the reciprocal of the second fraction.
Why do fractions need a common denominator for addition?
Because the quantities must represent equal-sized parts before their numerators can be combined.
Do fractions need a common denominator for multiplication?
No. Multiply numerators and denominators directly.
What is the least common denominator?
It is the least common multiple of the denominators.
How do I simplify a fraction?
Divide numerator and denominator by their greatest common divisor.
What is the simplest form of a fraction?
A fraction is in simplest form when numerator and denominator have no common positive factor greater than 1.
What is a proper fraction?
For positive values, it is a fraction whose numerator is smaller than its denominator.
What is an improper fraction?
It is a fraction whose numerator magnitude is greater than or equal to the denominator magnitude.
What is a mixed number?
A mixed number combines an integer and a proper fraction, such as 3 1/2.
How do I convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.
How do I convert an improper fraction to a mixed number?
Divide numerator by denominator. The quotient is the whole number and the remainder becomes the new numerator.
What is a reciprocal?
The reciprocal of a nonzero fraction a/b is b/a.
What is the reciprocal of zero?
Zero has no reciprocal because its reciprocal would require division by zero.
Can a fraction have a zero numerator?
Yes. Zero divided by any nonzero denominator equals zero.
Can a fraction have a zero denominator?
No. Division by zero is undefined.
Is 3/0 infinity?
No. 3/0 is undefined in ordinary arithmetic.
Is 0/0 equal to zero?
No. 0/0 is an indeterminate form and is not assigned an ordinary numerical value.
How do I compare two fractions?
For positive denominators, compare cross products or convert both fractions to a common denominator.
How do I convert a fraction to a decimal?
Divide the numerator by the denominator.
How do I convert a decimal to a fraction?
For a terminating decimal, write it over the appropriate power of 10 and simplify.
Can repeating decimals be converted to fractions?
Yes. A repeating decimal represents a rational number and can be converted exactly when the repeating pattern is known.
What is 0.5 as a fraction?
1/2.
What is 0.25 as a fraction?
1/4.
What is 0.125 as a fraction?
1/8.
What is 0.333... as a fraction?
Exactly 1/3 when the 3 repeats indefinitely.
What is 0.333 as a fraction?
Exactly 333/1000, which is different from the repeating decimal 0.333....
How do I convert a fraction to a percentage?
Divide numerator by denominator and multiply by 100. Use the Percentage Calculator for additional percentage operations.
What is 1/2 as a percentage?
50%.
What is 3/4 as a percentage?
75%.
What is 1/3 as a percentage?
33.333...%, repeating.
How do I convert a percentage to a fraction?
Write the percentage over 100 and simplify.
What is 25% as a fraction?
25/100 = 1/4.
Can fractions be negative?
Yes. A negative sign can be placed before the fraction or in the numerator.
Where should the negative sign go in a simplified fraction?
Conventionally the denominator is kept positive and the negative sign is placed in the numerator or before the fraction.
What does −2 1/3 mean?
Conventionally it means the negative of the entire mixed number, which equals −7/3.
Why is exact fraction arithmetic better than decimals sometimes?
Fractions can represent rational values exactly, while repeating decimals must be rounded when stored with finite digits.
How accurate is a fraction calculator?
With integer or arbitrary-precision rational arithmetic, fraction operations can be exact for valid integer inputs rather than approximate.
Sources and review
- Fractions — OpenStax. Accessed 2026-09-01.
- Multiply and Divide Fractions — OpenStax. Accessed 2026-09-01.
- Add and Subtract Fractions With Common Denominators — OpenStax. Accessed 2026-09-01.
- Add and Subtract Fractions With Different Denominators — OpenStax. Accessed 2026-09-01.
- Rational Numbers — Encyclopaedia of Mathematics. Accessed 2026-09-01.
Reviewed 2026-09-01 by Dr Akawak Ejigu, DBA.