Simplifying Fractions with the GCF (Including Mixed Numbers and Algebra)

Reduce any fraction to lowest terms in one step using the greatest common factor, with examples for improper fractions, mixed numbers and algebraic fractions.

By the Dayob Editorial Team · Updated · 2 min read

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Try the GCF Calculator – Enter two or more whole numbers to get their greatest common factor, least common multiple, prime factorizations and the Euclidean algorithm worked out.

A fraction is in lowest terms when the numerator and denominator have no common factor except 1. Simplifying makes fractions easier to compare and is usually expected in final answers. The fastest way is to divide the top and bottom by their greatest common factor.

One-step method

  1. Find the GCF of the numerator and denominator.
  2. Divide both by it.

Example: 42/56. GCF(42, 56) = 14. Dividing gives 3/4.

You can also divide in several smaller steps (42/56 → 21/28 → 3/4), which is fine but easy to stop too early. Using the GCF guarantees you're done in one step. The GCF calculator finds it for large numbers.

More examples

FractionGCFSimplified
18/2463/4
35/4975/7
144/360722/5
91/119713/17
17/51171/3

Is it already simplified?

If the GCF is 1, the fraction is already in lowest terms. 15/28 is simplified because 15 = 3 × 5 and 28 = 2² × 7 share no prime.

Improper fractions and mixed numbers

Simplify an improper fraction exactly the same way: 84/36 has a GCF of 12, giving 7/3, which is 2⅓ as a mixed number. For a mixed number, only the fractional part needs simplifying: 5 6/8 becomes 5 3/4.

Decimals to fractions

Write the decimal over a power of ten, then simplify. 0.375 = 375/1000; GCF(375, 1000) = 125, so 0.375 = 3/8. And 2.45 = 245/100 = 49/20.

Algebraic fractions

The same idea applies to variables: cancel the greatest common factor of the numerator and denominator.

  • (12x²) / (18x) = 2x/3, since the GCF is 6x (with x ≠ 0).
  • (6x + 9) / 15 = 3(2x + 3) / 15 = (2x + 3)/5.

Only cancel factors of the whole numerator and denominator, never individual terms. (x + 6)/6 does not simplify to x. Factoring first is covered in factoring out the GCF.

Simplifying before multiplying

When multiplying fractions, you can cancel common factors diagonally before multiplying to keep numbers small: (14/15) × (25/21). Cancel 7 from 14 and 21 (giving 2 and 3), and 5 from 25 and 15 (giving 5 and 3): (2/3) × (5/3) = 10/9.

Why simplify?

  • Comparing fractions is easier: 18/24 and 15/20 are both 3/4.
  • Answers are easier to check and read.
  • Ratios work the same way; see how to simplify ratios.

Further reading from official sources

More gcf guides

A student writing formulas on a chalkboardGCF vs. LCM: What's the Difference and When to Use EachThe greatest common factor and least common multiple are easy to confuse. Learn what each one means, how they're related, and which one a word problem needs.3 min readA man standing in front of a chalkboard covered in equationsThe Euclidean Algorithm: The Fastest Way to Find the GCFHow Euclid's division method finds the greatest common factor of large numbers in a few steps, why it works, and how to extend it to three or more numbers.3 min readA young boy writing complex formulas on a chalkboardPrime Factorization: Factor Trees, Division Ladders and Divisibility RulesBreak any number into prime factors using factor trees or the ladder method, with divisibility shortcuts and examples of using primes to find the GCF and LCM.3 min read