acalculator

How do I simplify fractions?

Type the top and bottom of a fraction. The calculator divides both by their greatest common factor and shows each step.

Your numbers

Simplest form
3/4

42/56 in simplest form is 3/4.

Greatest common factor
14
As a decimal
0.75
Fraction
42/56
Working
56 = 1 × 42 + 14; 42 = 3 × 14 + 0; GCF(42, 56) = 14; Divide the top and the bottom by 14: 42 ÷ 14 = 3 and 56 ÷ 14 = 4; 42/56 = 3/4

Simplest form: 3/4. 42/56 in simplest form is 3/4.

Euclid's algorithm, step by step

How to calculate

Reduces a fraction to lowest terms by dividing the numerator and the denominator by their greatest common factor, with the working.

Example with the default inputs (Numerator (top) 42, Denominator (bottom) 56): 42/56 in simplest form is 3/4.

Method: a/b = (a ÷ g)/(b ÷ g), where g = GCF(a, b) is found with Euclid's algorithm: divide the larger number by the smaller, then the smaller by the remainder, until the remainder is 0; the last divisor is g.

  • The numerator and the denominator are whole numbers; the arithmetic is exact for numbers of any size up to 30 digits.
  • A minus sign on the denominator moves to the numerator, so the simplest form has a positive denominator.
  • The denominator cannot be 0.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Numerator (top) 42, Denominator (bottom) 56 gives Simplest form 3/4, Greatest common factor 14, As a decimal 0.75, Working 56 = 1 × 42 + 14; 42 = 3 × 14 + 0; GCF(42, 56) = 14; Divide the top and the bottom by 14: 42 ÷ 14 = 3 and 56 ÷ 14 = 4; 42/56 = 3/4.Source: Method from OpenStax Prealgebra 2e, section 4.2: https://openstax.org/books/prealgebra-2e/pages/4-2-multiply-and-divide-fractions
  2. Numerator (top) 1071, Denominator (bottom) 462 gives Simplest form 51/22, Greatest common factor 21, As a mixed number 2 7/22, As a decimal 2.318182.
  3. Numerator (top) 18, Denominator (bottom) -24 gives Simplest form -3/4, Greatest common factor 6, Working Move the minus sign to the top: 18/-24 = -18/24; 24 = 1 × 18 + 6; 18 = 3 × 6 + 0; GCF(18, 24) = 6; Divide the top and the bottom by 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4; 18/-24 = -3/4.
  4. Numerator (top) 100, Denominator (bottom) 25 gives Simplest form 4, Greatest common factor 25, As a decimal 4.
  5. Numerator (top) 7, Denominator (bottom) 13 gives Simplest form 7/13, Greatest common factor 1.

How it works

The calculator takes a whole-number numerator a and a whole-number denominator b (b is not 0) and finds g, the greatest common factor (GCF) of |a| and |b|, with Euclid's algorithm. The simplest form is (a ÷ g)/(b ÷ g), with any minus sign on the numerator. All arithmetic is exact with whole numbers, so there is no rounding. The numerator and the denominator must each be from −10^30 to 10^30; a number past that has no answer.

Euclid's algorithm. Start with the larger of |a| and |b| as x and the smaller as y (when they are equal, either). Write x = q × y + r, where q is the whole-number quotient and r the remainder (0 ≤ r < y). Then replace x with y and y with r, and repeat until r is 0. The last y (the last non-zero remainder) is the GCF.

The calculator shows:

  • Simplest form: the fraction in lowest terms, written p/q with a positive denominator q and a minus sign (a plain hyphen) on p when the fraction is negative. When q is 1 it shows only the whole number, such as 4. A numerator of 0 gives 0.

  • Greatest common factor: g, the GCF of |a| and |b|. When a is 0, the GCF is |b|.

  • As a mixed number: shown only when the simplest form is a fraction (q is not 1) whose absolute value is more than 1. It is the whole-number part, a space, and the proper fraction: 2 7/22, or -2 7/22 for a negative fraction.

  • As a decimal: a ÷ b, shown to at most 6 decimal places.

  • Fraction: the fraction as typed, a/b (for example 18/-24).

  • Working: the steps, joined by a semicolon and a space, in this order:

    1. When b is negative: Move the minus sign to the top: a/b = (−a)/(−b), for example Move the minus sign to the top: 18/-24 = -18/24.
    2. One line per division of Euclid's algorithm, x = q × y + r, ending with the line whose remainder is 0.
    3. GCF(|a|, |b|) = g.
    4. When g is 1: The GCF is 1, so the fraction is already in lowest terms. Otherwise: Divide the top and the bottom by g: |a| ÷ g = … and |b| ÷ g = ….
    5. a/b = followed by the simplest form.

    When a is 0, the working is only GCF(0, |b|) = |b|; 0/b = 0.

Assumptions

  • The numerator and denominator are whole numbers from −10^30 to 10^30.
  • The denominator cannot be 0.
  • Numbers in the working and in the fractions are written without thousands separators, with a plain hyphen for a minus sign.

Worked examples by hand

42/56. Euclid: 56 = 1 × 42 + 14, then 42 = 3 × 14 + 0, so the GCF is 14. 42 ÷ 14 = 3 and 56 ÷ 14 = 4, so 42/56 = 3/4 (0.75).

1071/462. This is Euclid's own classic example. 1071 = 2 × 462 + 147, 462 = 3 × 147 + 21, 147 = 7 × 21 + 0, so the GCF is 21. 1071 ÷ 21 = 51 and 462 ÷ 21 = 22, so 1071/462 = 51/22, which is 2 7/22 as a mixed number (51 = 2 × 22 + 7), about 2.318182.

18/−24. Move the minus sign to the top: −18/24. Euclid on 24 and 18: 24 = 1 × 18 + 6, 18 = 3 × 6 + 0, so the GCF is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18/−24 = −3/4.

100/25. 100 = 4 × 25 + 0, so the GCF is 25, and 100/25 = 4/1, which is the whole number 4.

7/13. 13 = 1 × 7 + 6, 7 = 1 × 6 + 1, 6 = 6 × 1 + 0, so the GCF is 1 and 7/13 is already in simplest form.

Other questions people ask

How do I simplify a fraction?

Find the greatest common factor (GCF) of the numerator and the denominator, then divide both by it. For 42/56 the GCF is 14, and 42 ÷ 14 = 3 and 56 ÷ 14 = 4, so 42/56 = 3/4.

How do I find the greatest common factor of two numbers?

Use Euclid's algorithm. Divide the larger number by the smaller one and keep the remainder. Then divide the smaller number by that remainder, and repeat until the remainder is 0. The last number you divided by is the GCF. For 56 and 42: 56 = 1 × 42 + 14, then 42 = 3 × 14 + 0, so the GCF is 14.

How do I know a fraction is in simplest form?

A fraction is in simplest form (lowest terms) when the only whole number that divides both the numerator and the denominator is 1. 7/13 is in simplest form; 6/8 is not, because 2 divides both.

What happens to a negative fraction?

The minus sign goes on the top. 18/−24 and −18/24 are the same number, and both simplify to −3/4. A negative divided by a negative is positive: −6/−8 = 3/4.

What if the simplified fraction is bigger than 1?

An improper fraction such as 51/22 is already in simplest form when its GCF is 1. The calculator also writes it as a mixed number: 51 ÷ 22 = 2 remainder 7, so 51/22 = 2 7/22.

Can the bottom number be 0?

No. A fraction with a denominator of 0 has no value, because nothing times 0 gives a number other than 0. The calculator asks for another denominator.

Does simplifying change the value of the fraction?

No. Dividing the top and the bottom by the same number is the same as dividing the fraction by 1, so 42/56 and 3/4 are equal: both are 0.75.