Fractions
How to Simplify Fractions
To simplify a fraction, divide the numerator and the denominator by their greatest common factor. The result is the same value written with the smallest possible whole numbers, called lowest terms. This guide covers three ways to find the factor to divide by, how to simplify improper fractions and mixed numbers, how to tell when you are finished, and the mistakes that give wrong answers.
How to simplify fractions
Find the largest whole number that divides both the top and the bottom exactly. That is the greatest common factor, or GCF. Divide both numbers by it. You are done.
Example: 24/36
2 and 3 share no factor except 1, so 2/3 is in lowest terms.
The simplify fractions calculator does this for any fraction and shows the GCF it used, along with the decimal and percent. The rest of this guide is about doing it by hand, and about why it works.
Why simplifying does not change the value
Dividing the top and the bottom by the same number is the same as dividing the whole fraction by 12/12, and 12/12 is 1. Dividing by 1 changes nothing. So 24/36 and 2/3 are the same number, written two ways.
Picture a pizza cut into 8 slices with 6 of them left. That is 6/8 of the pizza. Now picture the same pizza cut into 4 larger slices with 3 left. That is 3/4. The amount of pizza is identical; only the size of the slices changed. Simplifying is choosing the biggest slices that still describe the amount exactly.
Fractions that name the same value are called equivalent fractions. Every fraction has infinitely many of them (3/4, 6/8, 9/12, 12/16 and so on) but exactly one in lowest terms. That is why simplified answers are the standard: two people who get 6/8 and 9/12 can see at once that they agree once both write 3/4.
Method 1: divide by the greatest common factor
Listing factors, as in the 24/36 example, works well for numbers up to about 100. For bigger numbers, Euclid’s method finds the GCF without any factoring.
- Divide the larger number by the smaller one and note the remainder.
- Replace the larger number with the smaller one, and the smaller one with the remainder.
- Repeat until the remainder is 0. The last divisor is the GCF.
Example: 391/493
No small number divides both, and the fraction looks finished. Euclid’s method shows it is not.
The method works because any number that divides both originals also divides their remainder, so the GCF is carried down through every line. The GCF calculator shows Euclid’s working alongside the prime-factor method, and how to find the GCF and LCM covers the ladder method too.
Method 2: divide by small numbers until you cannot
If you do not want to find the GCF first, divide by any common factor you can spot, then look again. Keep going until nothing but 1 divides both. This takes more steps but needs no factor lists.
The GCF of 72 and 120 is 24, which is 2 × 2 × 2 × 3: exactly the four divisions above, multiplied together. Repeated division reaches the GCF in pieces.
These quick tests tell you which small numbers to try:
| Divisible by | Test | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6 or 8 | 120 |
| 3 | Sum of digits divides by 3 | 72: 7 + 2 = 9 |
| 4 | Last two digits divide by 4 | 136: 36 ÷ 4 = 9 |
| 5 | Last digit is 0 or 5 | 45 |
| 6 | Passes the tests for 2 and 3 | 54 |
| 9 | Sum of digits divides by 9 | 126: 1 + 2 + 6 = 9 |
| 10 | Last digit is 0 | 840 |
There is no simple test for 7, 11, 13 or 17, which is how fractions like 391/493 slip through. When the quick tests all fail and the numbers are large, switch to Euclid’s method.
Method 3: prime factors
Write the numerator and denominator as products of primes, cross out each prime that appears on both, and multiply what is left.
This method shows you why a fraction cannot be simplified further: once no prime appears on both sides, there is nothing left to divide by. The prime factorization calculator breaks any number into primes with a factor tree, and the guide what is a prime number lists the primes you will need most.
Example: 360/840
Simplifying improper fractions, mixed numbers and negatives
Improper fractions
Simplify exactly as you would a proper fraction, then convert to a mixed number if that form is wanted.
Simplify first and the conversion uses smaller numbers. The improper fraction calculator handles both steps.
Mixed numbers
Only the fraction part needs simplifying. The whole number stays as it is.
If you have converted a mixed number to an improper fraction for some other calculation, the mixed number calculator turns it back.
Negative fractions
Ignore the sign, simplify, then put it back. -15/35 = -3/7, dividing by the GCF of 5. A fraction with one negative part is negative wherever the minus sign sits: -3/7, 3/-7 and -(3/7) are the same number.
How to know a fraction is in simplest form
A fraction is in simplest form when the GCF of its numerator and denominator is 1. Some fractions show this at a glance:
- The numerator is 1. 1/7, 1/12, 1/100.
- The numbers are consecutive. 4/5, 11/12, 99/100. Any factor shared by two numbers would also divide their difference, and the difference here is 1.
- The numerator is prime and is not a factor of the denominator. 13/20, 7/30.
Anything else deserves a check. One odd and one even number is not enough: 15/42 still reduces by 3 to 5/14. Two odd numbers can also share a factor, as 391/493 shows. When in doubt, run Euclid’s method; if the GCF it finds is 1, you are finished.
Simplifying in other settings
The same division turns up whenever a fraction is hiding in another form:
- Decimals. 0.45 is 45/100, which simplifies by 5 to 9/20. The decimal to fraction calculator does this for any decimal, including repeating ones.
- Percentages. 35% is 35/100, which simplifies by 5 to 7/20.
- Ratios. 12:18 simplifies by 6 to 2:3, exactly like 12/18. The ratio calculator simplifies ratios with any number of parts.
- Answers to fraction arithmetic. Sums, differences, products and quotients usually need a final simplification. The fraction calculator simplifies every answer automatically. When multiplying, cancelling before you multiply does this simplifying early; how to multiply fractions shows how.
Simplified fractions table
| Fraction | GCF | Simplest form |
|---|---|---|
| 4/10 | 2 | 2/5 |
| 6/9 | 3 | 2/3 |
| 10/12 | 2 | 5/6 |
| 12/20 | 4 | 3/5 |
| 14/35 | 7 | 2/5 |
| 18/30 | 6 | 3/5 |
| 20/48 | 4 | 5/12 |
| 21/28 | 7 | 3/4 |
| 27/36 | 9 | 3/4 |
| 30/42 | 6 | 5/7 |
| 32/40 | 8 | 4/5 |
| 35/56 | 7 | 5/8 |
| 40/64 | 8 | 5/8 |
| 44/121 | 11 | 4/11 |
| 51/85 | 17 | 3/5 |
| 60/96 | 12 | 5/8 |
| 66/99 | 33 | 2/3 |
Several rows have a GCF of 7, 11 or 17, which none of the quick divisibility tests catch. Those are the fractions most often left half-simplified.
Common mistakes
- Dividing only one part. 24/36 → 2/36 is a different number. Top and bottom must both be divided by the same factor.
- Subtracting instead of dividing. Taking 3 off both parts of 6/9 gives 3/6, which is 1/2. The correct answer is 2/3. Only multiplication and division by the same number keep the value.
- Stopping too soon. Dividing 24/36 by 2 gives 12/18, which still reduces. Check again after every division, or divide by the GCF once.
- Cancelling digits. Striking the 6s in 16/64 leaves 1/4, which happens to be right; striking the 9s in 19/95 leaves 1/5, also right by coincidence. Striking the 2s in 12/24 leaves 1/4, which is wrong; the answer is 1/2. Digits are not factors.
- Cancelling across a plus sign. In
(4 + 6)/6the sixes do not cancel. Add first: 10/6, which simplifies to 5/3. - Simplifying the whole number of a mixed number. In 4 6/8 only 6/8 changes. The answer is 4 3/4, not 2 3/4.
Frequently asked questions
What is 24/36 in simplest form?
2/3. The greatest common factor of 24 and 36 is 12, and dividing both by it gives 24 ÷ 12 = 2 and 36 ÷ 12 = 3. If you divide by smaller factors instead, 24/36 becomes 12/18, then 6/9, then 2/3. Either way the answer is the same, and as a decimal it is about 0.6667.
What is 12/18 simplified?
2/3. The GCF of 12 and 18 is 6, so divide both numbers by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. A quick way to find that GCF is to notice both numbers are even (divisible by 2) and both digit sums divide by 3, so both divide by 6.
What is the simplest form of 15/45?
1/3. 15 divides into 45 exactly three times, so 15 is the greatest common factor and dividing gives 15 ÷ 15 = 1 and 45 ÷ 15 = 3. Whenever the numerator divides the denominator exactly, the simplified fraction has 1 on top. Its decimal value is 0.333…, repeating.
How do you simplify fractions quickly?
Check the quick divisibility tests first: both even means divide by 2, digit sums divisible by 3 means divide by 3, both ending in 0 or 5 means divide by 5. Repeat until none apply. For large numbers or when nothing obvious works, use Euclid’s method to find the greatest common factor in a few lines.
How do you simplify an improper fraction?
Divide the numerator and denominator by their greatest common factor, just as with a proper fraction. 54/24 has a GCF of 6 and simplifies to 9/4. If a mixed number is wanted, divide 9 by 4 to get 2 remainder 1, giving 2 1/4. Simplifying before converting keeps the division small.
How do you know if a fraction is in simplest form?
It is in simplest form when the only whole number that divides both the numerator and the denominator is 1. Fractions with 1 on top, consecutive numbers such as 7/8, or a prime numerator that does not divide the denominator are always finished. Otherwise, find the GCF; if it is 1, nothing more can be done.
Is simplifying a fraction the same as reducing it?
Yes. Simplifying, reducing, writing in lowest terms and writing in simplest form all describe the same process: dividing the top and bottom by common factors until only 1 divides both. The fraction’s value never changes, only the numbers used to write it. 18/24 and 3/4 are the same amount.
How do you simplify fractions with large numbers?
Use Euclid’s method. Divide the larger number by the smaller and keep the remainder, then divide the smaller number by that remainder, and repeat until the remainder is 0. The last divisor is the GCF. For 391/493 the steps lead to 17, which turns the fraction into 23/29 even though no quick test spots it.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/how-to-simplify-fractions/