Direct answer: To simplify a fraction to lowest terms, find the greatest common divisor (GCD) of its numerator and denominator, divide both numbers by that GCD, and stop when the numerator and denominator share no common divisor except 1. For example, 12/18 simplifies to 2/3, because the GCD of 12 and 18 is 6, and 12 ÷ 6 = 2 while 18 ÷ 6 = 3.
A fraction like 6/12 is correct as a value — it equals one half — but it is not finished. Teachers and exam mark schemes expect the simplest form, and nobody orders “6/12 of a pizza” when they mean half. Simplifying is the final step of almost every fraction problem.
What Does “Lowest Terms” Mean?
A fraction is in lowest terms when its numerator and denominator have no common divisor except 1. That means no whole number larger than 1 divides into both of them. The fraction 2/3 is in lowest terms because the only common divisor of 2 and 3 is 1. The fraction 12/18 is not, because 2, 3, and 6 all divide into both 12 and 18.
Simplifying does not change a fraction’s value. Dividing top and bottom by the same number is dividing by 1 in disguise — 6/6 = 1 — so the value stays the same while the numbers shrink. That is why 12/18, 6/9, 4/6, and 2/3 are the same number, and 2/3 is its simplest form.
What Is the Greatest Common Divisor (GCD)?
The greatest common divisor of two numbers is the largest whole number that divides into both of them without a remainder. For 12 and 18, the divisors of 12 are 1, 2, 3, 4, 6, and 12, and the divisors of 18 are 1, 2, 3, 6, 9, and 18. The common divisors are 1, 2, 3, and 6, and the greatest is 6. So gcd(12, 18) = 6.
For small numbers, listing divisors works well. Calcrio’s fraction calculator finds the GCD and shows the division step for any fraction you enter.
How to Simplify a Fraction, Step by Step
Step 1: Find the GCD of the numerator and denominator. List divisors, or use prime factorization, and take the largest common divisor. For 12/18, the GCD is 6.
Step 2: Divide the numerator by the GCD. For 12/18, 12 ÷ 6 = 2. This is the new numerator.
Step 3: Divide the denominator by the GCD. For 12/18, 18 ÷ 6 = 3. This is the new denominator.
Step 4: Check that the result is in lowest terms. The new numerator and denominator should share no common divisor except 1. For 2/3, the only common divisor is 1, so the work is done. If they still share a divisor, repeat the process with the new GCD.
That is the whole method. If the GCD is 1 from the start, the fraction is already in lowest terms and there is nothing to do — 5/6, 11/12, and 7/10 are all already simplified.
Worked Example 1: Simplify 12/18 = 2/3
This is the example every student should be able to do by heart. Step 1: common divisors of 12 and 18 are 1, 2, 3, and 6, so the GCD is 6. Step 2: 12 ÷ 6 = 2. Step 3: 18 ÷ 6 = 3. Step 4: 2 and 3 share only 1, so 2/3 is final.
Shortcut: simplify in stages instead of finding the GCD at once. Divide 12/18 by 2, giving 6/9, then by 3, giving 2/3. Keep going until no common divisor larger than 1 remains; stopping at 6/9 leaves the answer unfinished.
Worked Example 2: Simplify 8/12 = 2/3
Step 1: the divisors of 8 are 1, 2, 4, 8, and the divisors of 12 are 1, 2, 3, 4, 6, 12. The common divisors are 1, 2, and 4, so the GCD is 4. Step 2: 8 ÷ 4 = 2. Step 3: 12 ÷ 4 = 3. Step 4: 2 and 3 share only 1, so 2/3 is final. Notice that 8/12 and 12/18 both simplify to 2/3 — different starting points, same simplest form. Lowest terms reveal when two different-looking fractions are actually equal.
Worked Example 3: Simplify 15/25 = 3/5
Step 1: the divisors of 15 are 1, 3, 5, 15, and the divisors of 25 are 1, 5, 25. The common divisors are 1 and 5, so the GCD is 5. Step 2: 15 ÷ 5 = 3. Step 3: 25 ÷ 5 = 5. Step 4: 3 and 5 share only 1, so 3/5 is final. Shortcut: numbers ending in 5 or 0 often share a factor of 5, so test 5 first.
Worked Example 4: Simplify 9/12 = 3/4
Step 1: the divisors of 9 are 1, 3, 9, and the divisors of 12 are 1, 2, 3, 4, 6, 12. The common divisors are 1 and 3, so the GCD is 3. Step 2: 9 ÷ 3 = 3. Step 3: 12 ÷ 3 = 4. Step 4: 3 and 4 share only 1, so 3/4 is final. Size check: 9/12 = 0.75 and 3/4 = 0.75, so the value is unchanged, as it must be.
Worked Example 5: Simplify 6/12 = 1/2
This is the case that appears at the end of multiplication problems: 3/4 × 2/3 = 6/12, and the answer is not finished until it is simplified. Step 1: the GCD of 6 and 12 is 6. Step 2: 6 ÷ 6 = 1. Step 3: 12 ÷ 6 = 2. Step 4: 1 and 2 share only 1, so 1/2 is final. A numerator of 1 always means lowest terms, since 1 shares no divisor larger than 1 with any number.
How Do You Simplify an Improper Fraction?
The same steps apply, and then you may convert the result to a mixed number. Take 15/12 as an example. Step 1: the GCD of 15 and 12 is 3. Step 2: 15 ÷ 3 = 5. Step 3: 12 ÷ 3 = 4, giving 5/4. Step 4: 5 and 4 share only 1, so 5/4 is in lowest terms. Convert to a mixed number: 5 ÷ 4 = 1 remainder 1, so 5/4 = 1 1/4. The order matters for clarity: simplify first, then convert, because converting an unsimplified fraction such as 15/12 directly gives 1 3/12, which still needs simplifying to 1 1/4.
To simplify a mixed number, simplify its fraction part alone: 2 6/12 becomes 2 1/2. The whole number never changes.
Simplifying After Each Operation
Simplifying is not a standalone topic; it is the last step of every fraction operation, and forgetting it is the most common way to lose marks on otherwise correct work.
After addition: 1/2 + 1/3 = 5/6, already simplified, but 1/3 + 1/6 = 3/6, which must be simplified by GCD 3 to 1/2. Our guide on how to add fractions covers the addition steps.
After subtraction: 2/3 − 1/4 = 5/12, already simplified, but 7/10 − 3/10 = 4/10, which simplifies by GCD 2 to 2/5.
After multiplication: 3/4 × 2/3 = 6/12 = 1/2. Multiplication answers almost always need simplifying, because multiplying denominators creates large numbers that usually share factors with the numerator.
After division: 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4. Division answers need the same final simplification, followed by conversion to a mixed number when the result is improper.
Check the GCD the moment an answer appears. Calcrio’s free fraction calculator simplifies automatically, so comparing its output with your work shows whether you stopped one step early.
Common Mistakes to Avoid
Dividing only the numerator or only the denominator. Simplifying 12/18 by dividing only the top by 6 gives 2/18, which is a different, smaller value. Both numbers must be divided by the same GCD, or the value changes.
Dividing by a common divisor but not the greatest one, then stopping. Dividing 12/18 by 2 gives 6/9, which is simpler but not simplest. Keep going until the only common divisor is 1.
Dividing by a number that is not a common divisor. Dividing 12/18 by 4 gives 3/4.5, which is not a fraction of whole numbers at all. The divisor must divide into both numbers evenly.
Thinking a fraction with a large numerator cannot be simplified. Size is irrelevant; only common divisors matter. The fraction 15/25 looks ordinary, but both numbers share 5, so it simplifies to 3/5.
Simplifying the whole number in a mixed number. In 2 6/12, only the fraction part simplifies: 2 1/2. The whole number 2 is untouched.
Practice Problems (with Answers)
Try these before reading the answers.
- Simplify 10/15. GCD 5: 10 ÷ 5 = 2, 15 ÷ 5 = 3, giving 2/3.
- Simplify 14/21. GCD 7: 14 ÷ 7 = 2, 21 ÷ 7 = 3, giving 2/3.
- Simplify 20/30. GCD 10: 20 ÷ 10 = 2, 30 ÷ 10 = 3, giving 2/3.
- Simplify 18/24. GCD 6: 18 ÷ 6 = 3, 24 ÷ 6 = 4, giving 3/4.
- Simplify 16/20. GCD 4: 16 ÷ 4 = 4, 20 ÷ 4 = 5, giving 4/5.
Frequently Asked Questions
How do you simplify a fraction to lowest terms?
Find the GCD of the numerator and denominator, then divide both by it. For 12/18, the GCD is 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3, giving 2/3.
What is the GCD of 12 and 18?
6. The divisors of 12 are 1, 2, 3, 4, 6, 12, and the divisors of 18 are 1, 2, 3, 6, 9, 18. Their greatest common divisor is 6.
How do you know when a fraction is fully simplified?
When the numerator and denominator share no common divisor except 1. For 2/3, the only common divisor is 1, so it is fully simplified. For 6/9, the common divisor 3 remains, so it is not.
Does simplifying change the value of a fraction?
No. Dividing the top and bottom by the same number does not change the value. 12/18, 6/9, 4/6, and 2/3 are all equal to about 0.667.
Can you simplify a fraction in stages?
Yes. Dividing 12/18 by 2 gives 6/9, then dividing by 3 gives 2/3. The result is the same as dividing by the GCD, 6, in one step — provided you keep going until no common divisor larger than 1 remains.
What is 6/12 simplified?
1/2. The GCD of 6 and 12 is 6, so 6 ÷ 6 = 1 and 12 ÷ 6 = 2. A numerator of 1 always means the fraction is in lowest terms.