To simplify a fraction, divide the numerator and the denominator by their greatest common factor (GCF). For 84/126, the largest number that divides both is 42, so 84 ÷ 42 = 2 and 126 ÷ 42 = 3, and the fraction simplifies to 2/3. When the top and bottom share no factor other than 1, the fraction is in lowest terms.
Simplifying never changes a fraction’s value. Dividing the numerator and denominator by the same number is the same as dividing by 42/42, which equals 1. What changes is how readable the number is: 2/3 is much easier to picture and compare than 84/126.
Method 1: Divide by the greatest common factor
This is the fastest method when you can find the GCF.
- Find the GCF of the numerator and denominator.
- Divide both by the GCF.
- Write the result. It is guaranteed to be in lowest terms.
For small numbers you can list factors. For larger ones, Euclid’s algorithm finds the GCF quickly: divide the larger number by the smaller, then keep dividing the previous divisor by the remainder until the remainder is 0. The last nonzero remainder is the GCF.
GCF(126, 84): 126 ÷ 84 = 1 remainder 42; 84 ÷ 42 = 2 remainder 0 → GCF = 42
84/126 = (84 ÷ 42)/(126 ÷ 42) = 2/3
The GCF calculator shows this process for any pair of numbers, and how to find the GCF and LCM explains each GCF method in more depth.
Method 2: Divide by small primes repeatedly
If the GCF is not obvious, divide by any common factor you can spot, then repeat until nothing else divides both numbers. Starting with the smallest primes, 2, 3, 5 and 7, works well.
84/126 → both even, divide by 2 → 42/63
42/63 → digit sums 6 and 9 are both divisible by 3, divide by 3 → 14/21
14/21 → both divisible by 7, divide by 7 → 2/3
This takes more steps, but each step is easy and you cannot go wrong as long as you keep going until no common factor remains. Multiplying the divisors you used, 2 × 3 × 7 = 42, gives you the GCF as a by-product.
Method 3: Prime factorization
Break both numbers into primes, cancel the primes they share, and multiply what is left.
84 = 2 × 2 × 3 × 7
126 = 2 × 3 × 3 × 7
Cancel one 2, one 3 and one 7 from each: 84/126 = 2/3
This method is ideal for large numbers and for fractions that include variables. A second example: 210/495. Since 210 = 2 × 3 × 5 × 7 and 495 = 3 × 3 × 5 × 11, the shared primes are 3 and 5, so the GCF is 15 and the fraction simplifies to 14/33. The prime factorization calculator produces the factor trees for you.
Divisibility rules that speed things up
Recognizing common factors at a glance is most of the work. These tests apply to whole numbers:
| Divisible by | Test | Example |
|---|---|---|
| 2 | Last digit is even | 9,876 |
| 3 | Digit sum is divisible by 3 | 462 (4 + 6 + 2 = 12) |
| 4 | Last two digits form a multiple of 4 | 1,316 (16) |
| 5 | Last digit is 0 or 5 | 495 |
| 6 | Divisible by both 2 and 3 | 462 |
| 8 | Last three digits form a multiple of 8 | 7,128 (128) |
| 9 | Digit sum is divisible by 9 | 1,071 (1 + 0 + 7 + 1 = 9) |
| 10 | Last digit is 0 | 2,520 |
| 11 | Alternating digit sum is divisible by 11 | 1,001 (1 − 0 + 0 − 1 = 0) |
The divisibility rules checker applies every test at once if you want to confirm.
Simplifying improper fractions and mixed numbers
Improper fractions simplify exactly like proper ones. 45/12 has a GCF of 3, so it simplifies to 15/4. You can leave it there or convert it to a mixed number by dividing: 15 ÷ 4 = 3 remainder 3, so 15/4 = 3 3/4. The improper fraction to mixed number calculator does that conversion.
Mixed numbers keep their whole-number part; you only simplify the fraction. In 5 18/24, the GCF of 18 and 24 is 6, so the answer is 5 3/4.
Fractions with variables follow the same logic, with one extra condition: you cannot divide by something that could be zero. For example, 12x²/18x = 2x/3, valid for every x except 0.
Common mistakes
Canceling digits instead of factors. 16/64 happens to equal 1/4 if you strike out both 6s, and 19/95 equals 1/5 if you strike out the 9s, but these are coincidences. Try the same trick on 12/24 and you get 1/4, while the right answer is 1/2. Only factors, never individual digits, may be canceled.
Canceling terms in a sum. (2 + 6)/2 is 8/2 = 4, not 6. You can only cancel a factor that multiplies the entire numerator and the entire denominator.
Stopping too early. Dividing 72/96 by 2 gives 36/48, which is still reducible. Keep going, or use the GCF of 24 in one step, to reach 3/4.
Dividing only one part. Whatever you do to the numerator, you must do to the denominator. Dividing just one of them changes the value.
Simplifying measurements
Fractions of an inch are a place where simplifying is routine. Tape measures mark sixteenths, so a reading of 12 sixteenths is written 12/16 inch and reduced to 3/4 inch. Because the denominators are powers of 2, you only ever divide by 2: 12/16 → 6/8 → 3/4. A reading of 10/16 reduces to 5/8, and 9/16 is already in lowest terms because 9 is odd.
| Sixteenths | Simplified |
|---|---|
| 2/16 | 1/8 |
| 4/16 | 1/4 |
| 6/16 | 3/8 |
| 8/16 | 1/2 |
| 10/16 | 5/8 |
| 12/16 | 3/4 |
| 14/16 | 7/8 |
Kitchen measures work the same way: 6/8 of a cup is 3/4 cup, and 4/12 of a recipe batch is 1/3.
Why it matters
Simplified fractions are easier to compare, add and check. You will simplify at the end of nearly every fraction calculation, including when you add fractions with unlike denominators or convert a decimal to a fraction. The simplifying fractions calculator shows the GCF and every step, and the equivalent fractions calculator works in the opposite direction, building larger fractions with the same value.
Frequently asked questions
How do you simplify a fraction?
Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. For 84/126 the GCF is 42, so 84 ÷ 42 = 2 and 126 ÷ 42 = 3, giving 2/3.
How do I know a fraction is fully simplified?
A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. For example, 35/48 is already simplified: 35 = 5 × 7 and 48 = 2 × 2 × 2 × 2 × 3 have no prime factor in common.
Do I have to simplify an improper fraction to a mixed number?
Not necessarily. 15/4 is fully simplified as it stands. Converting it to 3 3/4 is a change of format, not simplification. Algebra usually prefers improper fractions, while measurements and recipes usually prefer mixed numbers.
Does simplifying a fraction change its value?
No. Dividing the numerator and denominator by the same nonzero number is the same as dividing the fraction by 1, so 84/126 and 2/3 are exactly equal. Only the way the number is written changes.
How do you simplify a negative fraction?
Simplify the numbers as usual and keep a single minus sign, usually in front of the fraction or in the numerator. −6/8, 6/−8 and −(6/8) all simplify to −3/4.