Simplify Fractions
Simplify a fraction to its lowest terms, with every step: the greatest common divisor found with the Euclidean algorithm, the division of the top and bottom, and the answer as a fraction, a mixed number, a decimal and a percentage. Fraction bars show how the small pieces are put together into bigger ones.
What Does Simplifying Mean?
The same amount can be written in many ways: 2/3 = 4/6 = 84/126. A fraction is simplified (in lowest terms) when the numerator and the denominator have no common factor left except 1. Simplifying never changes the amount, only the size of the pieces: 84 pieces of 1/126 are exactly as much as 2 pieces of 1/3.
To simplify in one go, divide the top and the bottom by their greatest common divisor (GCD). The Euclidean algorithm finds it quickly, even for very big numbers: divide the bigger number by the smaller one and keep the remainder, then repeat with the smaller number and the remainder. The last remainder that is not 0 is the GCD.
Fraction Bars
barsStep-by-Step Solution
Worked Techniques
Divide, keep the remainder, repeat. The last remainder that is not 0 is the GCD, so 84/126 = (84 ÷ 42)/(126 ÷ 42) = 2/3.
Write the top and bottom as products of primes and cross out the common ones: 2·3·7 = 42 is the GCD. Handy for small numbers.
Dividing by small common factors one at a time also works; stop when no common factor is left. Dividing by the GCD does it in one step.
Sources & Verification
- K. H. Rosen, Elementary Number Theory and Its Applications, 6th ed., Pearson, 2011 — Ch. 3 (the greatest common divisor and the Euclidean algorithm).
- H.-H. Wu, Understanding Numbers in Elementary School Mathematics, American Mathematical Society, 2011 — Part 2 (Fractions: equivalent fractions and the fundamental fact of equivalent fractions).
Frequently Asked Questions
How do you simplify a fraction?
Divide the numerator and the denominator by their greatest common divisor (GCD). Example: 84/126.
1. Find the GCD of 84 and 126 with the Euclidean algorithm: 126 = 1 · 84 + 42, then 84 = 2 · 42 + 0. The last remainder that is not 0 is 42.
2. Divide the top and bottom by 42: 84 ÷ 42 = 2 and 126 ÷ 42 = 3.
3. So 84/126 = 2/3, and 2 and 3 have no common factor left.
How do you find the greatest common divisor?
With the Euclidean algorithm: divide the bigger number by the smaller one and keep the remainder, then repeat with the smaller number and the remainder until the remainder is 0. The last remainder that is not 0 is the GCD. For 12 and 16: 16 = 1 · 12 + 4 and 12 = 3 · 4 + 0, so the GCD is 4 and 12/16 = 3/4. It stays fast for very big numbers: the GCD of 123456789 and 987654321 is 9, found in two steps. The GCD & LCM calculator also handles more than two numbers.
How do I know a fraction is already in simplest form?
When the greatest common divisor of the top and bottom is 1. For 7/9: 9 = 1 · 7 + 2, 7 = 3 · 2 + 1, 2 = 2 · 1 + 0, so the GCD is 1 and 7/9 cannot be simplified. Two different prime numbers, or two numbers next to each other (like 8/9), never have a common factor.
How do you simplify an improper fraction or a mixed number?
An improper fraction is simplified in the same way and can then be written as a mixed number: 150/100 has GCD 50, so 150/100 = 3/2 = 1 1/2. For a mixed number, write it as an improper fraction first: 2 6/8 = 22/8, the GCD of 22 and 8 is 2, so 22/8 = 11/4 = 2 3/4.
Can I simplify a decimal or a negative fraction?
Yes. A decimal is first written as a fraction over 10, 100, 1000 and so on: 0.375 = 375/1000, the GCD is 125, so 0.375 = 3/8. For a negative fraction the minus sign stays and only the numbers are simplified: −18/24 has GCD 6, so −18/24 = −3/4.
Does simplifying change the value of a fraction?
No. Dividing the top and the bottom by the same number gives an equivalent fraction: the same amount, cut into fewer but bigger pieces. 84/126 and 2/3 are both 0.666…; in the fraction bars, every 42 small pieces are put together into one bigger piece.