∿Mixed Number Lab

Simplification fundamentals

Simplify Fractions

Learn how to simplify fractions to lowest terms using the GCD method, prime factorization, and repeated division. Includes a free instant simplifier and 12 worked examples covering proper fractions, improper fractions, and mixed numbers.

5 min readGrade 4-7Instant simplifier12 examplesProper & improperMixed numbers

Written by

Mixed Number Lab Editorial Team

Focus

Reduce to lowest terms

Updated

2026-05-11

Core rule: Find the GCD, then divide

Quick Preview

Reduce any fraction to its simplest form

Input

12/18

GCD

6

Output

2/3

12/18 -> 2/3. GCD(12, 18) = 6, so divide both parts by 6.

Instant Simplifier

Reduce any fraction to lowest terms instantly

Reduce any fraction to lowest terms instantly, then expand the full GCD steps below.

Instant Simplifier

Enter a numerator and denominator to see the simplest form, the GCD, and the full reduction path.

Fraction

In This Guide

Quick Refresher

What does it mean to simplify a fraction?

A fraction is in simplest form, also called lowest terms, when the numerator and denominator share no common factor other than 1. Simplifying does not change the value of the fraction. It only changes how the fraction is written.

12/18 and 2/3 are the same amount. But 2/3 is simpler to read, easier to compare, and the expected final form in most math classes.

Not Simplified

12/18

GCD is 6, so this fraction can still be reduced.

Simplified

2/3

GCD is 1, so this is already in lowest terms.

Same Value

12/18 = 2/3

12/18 = 2/3 = 0.666... The value never changes.

Key idea: A fraction is fully simplified when GCD(numerator, denominator) = 1. Dividing both parts by the GCD gives the simplest form in one step.

Why Simplify

Why simplifying fractions matters

Simplest form is easier to compare, easier to use in later operations, and usually required as the final answer in classwork and tests.

Adding Mixed Numbers

Useful

Simplified fractions make common denominators easier to spot.

Learn addition

Comparing Fractions

Helpful

You can only compare fractions reliably when both are fully reduced.

Compare fractions

Converting to Decimals

Cleaner

A simplified fraction divides more cleanly.

Convert to decimals

Final Answers

Required

Most teachers and tests require the answer in simplest form.

Use the calculator

Method 1

Divide both parts by the Greatest Common Divisor

The GCD method is the fastest and most reliable approach. Find the largest number that divides both the numerator and denominator evenly, then divide both by it. One step gives you the simplest form.

1

Find the GCD of the numerator and denominator

List the factors of both numbers and choose the largest shared factor.

textFactors of 12: 1,2,3,4,6,12quad textFactors of 18: 1,2,3,6,9,18quad textGCD(12,18)=6
2

Divide the numerator by the GCD

The numerator shrinks by the largest shared factor.

12 div 6 = 2
3

Divide the denominator by the GCD

The denominator must be divided by the same number so the value stays equal.

18 div 6 = 3quad frac1218=frac23
Tip: If the GCD is 1, the fraction is already in simplest form. No further steps needed.

Quick Formula

simplified fraction = (numerator / GCD) / (denominator / GCD)

simplified fraction = (numerator / GCD) / (denominator / GCD)

Example: 12/18 -> (12 / 6) / (18 / 6) = 2/3

Memory Rule

Find the biggest shared factor, divide both parts by it, done.

Example 1

12/18

The numerator and denominator share a greatest common divisor of 6.

  1. 1.GCD(12, 18) = 6.
  2. 2.12 ÷ 6 = 2.
  3. 3.18 ÷ 6 = 3.

Answer: 2/3

Example 2

8/12

Both parts divide evenly by 4.

  1. 1.GCD(8, 12) = 4.
  2. 2.8 ÷ 4 = 2.
  3. 3.12 ÷ 4 = 3.

Answer: 2/3

Example 3

15/25

The shared factor 5 reduces the fraction in one step.

  1. 1.GCD(15, 25) = 5.
  2. 2.15 ÷ 5 = 3.
  3. 3.25 ÷ 5 = 5.

Answer: 3/5

Example 4

7/14

The numerator itself is the GCD.

  1. 1.GCD(7, 14) = 7.
  2. 2.7 ÷ 7 = 1.
  3. 3.14 ÷ 7 = 2.

Answer: 1/2

Method 2

Cancel common prime factors from numerator and denominator

Prime factorization shows exactly why the GCD works. Write both numbers as products of primes, then cancel every factor that appears in both. What remains is the simplest form.

1

Write the prime factorization of each number

Break the numerator and denominator into prime-number building blocks.

12 = 2 times 2 times 3quad 18 = 2 times 3 times 3
2

Cancel common prime factors

Remove every prime factor that appears in both the numerator and denominator.

frac2 times 2 times 32 times 3 times 3=frac23
3

Multiply the remaining factors

The factors left behind form the simplified fraction.

frac1218=frac23
When to use: Prime factorization is best when the GCD is not obvious at first glance, especially with larger numbers.

Prime Example 1

12/18

Prime factors make the shared 2 x 3 visible.

  1. 1.12 = 2^2 x 3.
  2. 2.18 = 2 x 3^2.
  3. 3.Cancel 2 x 3.

Answer: 2/3

Prime Example 2

24/36

Both numbers share 2^2 x 3.

  1. 1.24 = 2^3 x 3.
  2. 2.36 = 2^2 x 3^2.
  3. 3.Cancel 2^2 x 3.

Answer: 2/3

Prime Example 3

30/42

Cancel the shared 2 x 3 and keep what remains.

  1. 1.30 = 2 x 3 x 5.
  2. 2.42 = 2 x 3 x 7.
  3. 3.Cancel 2 x 3.

Answer: 5/7

Method 3

Divide by small common factors step by step

If you cannot spot the GCD right away, divide both parts by any common factor you can see, then repeat until no common factors remain. This method takes more steps but never requires you to find the GCD upfront.

1

Divide by a small common factor

Start with any shared factor you can spot.

frac2436div 2 = frac1218
2

Repeat while a common factor remains

Keep reducing both parts by the same number.

frac1218div 2 = frac69
3

Stop when the GCD is 1

The final fraction has no shared factor other than 1.

frac69div 3 = frac23,quad textGCD(2,3)=1
Tip: You always reach the same answer as the GCD method. The only difference is how many steps it takes.

Example Path

frac2436rightarrowfrac1218rightarrowfrac69rightarrowfrac23

Repeated Division Example 1

24/36

Small reductions eventually reach the same 2/3 result.

  1. 1.Divide by 2: 24/36 -> 12/18.
  2. 2.Divide by 2 again: 12/18 -> 6/9.
  3. 3.Divide by 3: 6/9 -> 2/3.

Answer: 2/3

Repeated Division Example 2

16/24

You can mix small and larger common factors.

  1. 1.Divide by 2: 16/24 -> 8/12.
  2. 2.Divide by 4: 8/12 -> 2/3.
  3. 3.GCD(2, 3) = 1, so stop.

Answer: 2/3

Repeated Division Example 3

48/60

The method works even when the GCD is not obvious at first.

  1. 1.Divide by 2: 48/60 -> 24/30.
  2. 2.Divide by 6: 24/30 -> 4/5.
  3. 3.GCD(4, 5) = 1, so stop.

Answer: 4/5

Compare the three simplification methods

Pick the method that matches the goal: speed, factor understanding, or a beginner-friendly path.

CompareMethod 1: GCDMethod 2: Prime FactorizationMethod 3: Repeated Division
SpeedFastestMediumSlowest
Best forAll fractionsLarge or unfamiliar numbersWhen GCD is not obvious
Skill neededFinding GCDPrime factorizationBasic division
Error riskLowLowMedium (easy to stop too early)
RecommendationBest defaultBest for understandingGood for beginners

Special Cases

Already-simple fractions, improper fractions, and mixed numbers

These cases cause common formatting mistakes. The value stays the same, but the final format depends on the kind of fraction you started with.

Special Case

Already simplest

Some fractions have no common factor to remove.

Rule: If GCD(numerator, denominator) = 1, return the original fraction.

3/7

  1. 1.GCD(3, 7) = 1.
  2. 2.There is no shared factor greater than 1.
  3. 3.Keep the fraction unchanged.

Answer: 3/7

Special Case

Numerator is 1

A numerator of 1 means the fraction is already reduced.

Rule: When the numerator is 1, the GCD must be 1.

1/6

  1. 1.GCD(1, 6) = 1.
  2. 2.No common factor can be removed.
  3. 3.Keep the original fraction.

Answer: 1/6

Special Case

Improper fractions

Reduce first, then decide whether mixed-number form is needed.

Rule: Simplify the fraction before optionally converting it to a mixed number.

18/12

  1. 1.GCD(18, 12) = 6.
  2. 2.18/12 simplifies to 3/2.
  3. 3.Optionally convert 3/2 to 1 1/2.

Answer: 3/2 or 1 1/2

Special Case

Mixed numbers

The whole number stays exactly the same.

Rule: When simplifying a mixed number, only the fraction part changes.

2 4/6

  1. 1.Keep the whole number 2.
  2. 2.Simplify the fraction part: 4/6 -> 2/3.
  3. 3.Attach the simplified fraction.

Answer: 2 2/3

When simplifying a mixed number, only the fraction part changes. The whole number stays exactly the same.

Built-in Calculator

Try the fraction simplifier

Enter any fraction below and see the simplest form, the GCD, and the full step-by-step reduction.

Instant Simplifier

Reduce any fraction to lowest terms instantly

Reduce any fraction to lowest terms instantly, then expand the full GCD steps below.

Instant Simplifier

Enter a numerator and denominator to see the simplest form, the GCD, and the full reduction path.

Fraction

FAQ

Simplify fractions FAQ

How do you simplify a fraction?

Find the greatest common divisor of the numerator and denominator, then divide both parts by that number.

How do I know a fraction is in simplest form?

A fraction is in simplest form when the numerator and denominator share no common factor other than 1.

Do you simplify the whole number in a mixed number?

No. In a mixed number, the whole number stays the same and only the fraction part is simplified.

Continue Learning

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Convert decimals into mixed numbers, then simplify the fraction part.

Mixed Number Calculator

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