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Fraction Calculator

Add, subtract, multiply, divide and simplify fractions; fraction ↔ recurring decimal

2
Operation
34
Write fractions as 3/4, mixed numbers as 1 2/3 (whole number, space, fraction), and recurring decimals with brackets: 0.(3) = 0.333…, 0.1(6) = 0.1666….
Answer
3712=3
Lowest terms
37/12
Mixed number
3 1/12
Decimal (recurring)
3.083
Percentage
308.3333%
Recurring decimal: the digits under the bar repeat for ever. In bracket form it is 3.08(3) ≈ 3.083333.
Step-by-step working
  1. Write each number as a fraction
    2 1/3 = (2 × 3 + 1)/3 = 7/3
  2. Find the LCM of the denominators
    LCM(3, 4) = 12
  3. Make the denominators equal
    7/3 = (7 × 4)/(3 × 4) = 28/12
    3/4 = (3 × 3)/(4 × 3) = 9/12
  4. Add the numerators
    28/12 + 9/12 = (28 + 9)/12 = 37/12
  5. Write as a mixed number
    37 ÷ 12 = 3 remainder 1, so 37/12 = 3 1/12

Runs entirely in your browser. Nothing you type or upload is sent to us or stored.

Results are for general guidance. Please check anything important before relying on it. Terms of use

How to Use

  1. Choose Calculate, Simplify or Decimal ↔ Fraction.
  2. Type fractions as 3/4, mixed numbers as 2 1/3 (whole number, space, fraction) and recurring decimals with brackets, such as 0.(3) for 0.333…
  3. In Calculate mode, pick + − × ÷ and enter the second number. Tap swap to change the order, which matters for − and ÷. The answer appears instantly as a fraction in lowest terms, a mixed number and a decimal.
  4. Scroll down to the step-by-step working: the LCM of the denominators, the equivalent fractions and the HCF used to simplify.

Fractions, mixed numbers and recurring decimals, step by step

This fraction calculator adds, subtracts, multiplies and divides fractions and mixed numbers exactly, with no rounding, and writes out the working your teacher expects: converting mixed numbers to improper fractions, finding the LCM of the denominators, and simplifying with the HCF. It also simplifies a single fraction and converts between fractions and decimals, including recurring decimals such as 0.333… and 0.1666….

The rules

Mixed → improper: w a/b = (w × b + a) / b
Add/subtract: a/b ± c/d = (a × L/b ± c × L/d) / L, L = LCM(b, d)
Multiply: a/b × c/d = (a × c) / (b × d)
Divide: a/b ÷ c/d = a/b × d/c (multiply by the reciprocal)
Simplify: divide the numerator and denominator by their HCF

Worked example: 2 1/3 + 1 3/4

  1. Make them improper: 2 1/3 = (2 × 3 + 1)/3 = 7/3 and 1 3/4 = (1 × 4 + 3)/4 = 7/4.
  2. LCM of 3 and 4 is 12, so 7/3 = 28/12 and 7/4 = 21/12.
  3. Add the numerators: 28/12 + 21/12 = 49/12.
  4. HCF(49, 12) = 1, so 49/12 is already in lowest terms. As a mixed number, 49 ÷ 12 = 4 remainder 1, giving 4 1/12, which is 4.08333… = 4.08(3).

Using the LCM instead of multiplying the denominators keeps the numbers small. Multiplying 3 × 4 also works here, but for 5/12 + 7/18 the LCM is 36, much smaller than 12 × 18 = 216.

Terminating and recurring decimals

Every fraction becomes a decimal that either stops (terminates) or repeats a block of digits for ever. Write the fraction in lowest terms and look at the denominator: if its only prime factors are 2 and 5, the decimal terminates; otherwise it recurs.

FractionDenominatorDecimal
3/82³0.375 (terminates)
7/202² × 50.35 (terminates)
1/330.333… = 0.(3)
1/62 × 30.1666… = 0.1(6)
1/770.(142857)
22/773.(142857), not π

The repeating block is usually written with a bar over it. On a keyboard, type it in brackets: 0.(3) or 0.1(6).

Recurring decimal to fraction (algebra method)

Take x = 0.1666…. Multiply by 10 to bring the point to where the repetition starts (10x = 1.666…), and by 100 to move it past one repeating block (100x = 16.666…). Subtracting cancels the endless tail: 90x = 15, so x = 15/90 = 1/6. For a pure recurring decimal like 0.333…, 10x − x = 3 gives x = 3/9 = 1/3. The calculator writes out these lines for any decimal you enter, and shows that 0.(9) is exactly 1.

Tips

  • A negative mixed number such as −1 2/3 means −(1 + 2/3) = −5/3.
  • Dividing by a fraction is the same as multiplying by its reciprocal: 3/4 ÷ 3/8 = 3/4 × 8/3 = 2.
  • Numbers can have up to 18 digits; all the arithmetic is exact.

Frequently Asked Questions

Find the LCM of the denominators, rewrite each fraction with that denominator, then add the numerators. For 1/2 + 1/3: LCM(2, 3) = 6, so 1/2 = 3/6 and 1/3 = 2/6, giving 5/6. The calculator shows each of these steps.

Type the whole number, a space, then the fraction: 2 1/3 means two and one third. A minus sign in front applies to the whole number, so −1 2/3 is −5/3.

Choose Decimal ↔ Fraction and type the repeating digits in brackets: 0.(3). Let x = 0.333…; then 10x = 3.333…, and subtracting gives 9x = 3, so x = 1/3. For 0.1666… type 0.1(6) to get 1/6.

Write it in lowest terms. If the denominator's only prime factors are 2 and/or 5, the decimal terminates (3/8 = 0.375). If it has any other prime factor, the decimal recurs (1/6 = 0.1666…).

Divide the numerator and denominator by their HCF. For 126/54 the HCF is 18, so 126/54 = 7/3. In Simplify mode the calculator finds the HCF step by step and shows the division.

Multiply the first fraction by the reciprocal of the second (turn it upside down). 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2. You can't divide by a fraction equal to 0.