What simplifying means
To simplify a fraction means to write an equivalent fraction using the smallest possible numbers. The value doesn’t change; only how it looks.
18/24 = 3/4
A fraction is in simplest form (or lowest terms) when the numerator and denominator share no common factor other than 1.
Method 1: Divide by common factors step by step
Keep dividing the top and bottom by any common factor until you can’t anymore.
Example: 36/48
- Both even: ÷ 2 → 18/24
- Both even: ÷ 2 → 9/12
- Both divisible by 3: ÷ 3 → 3/4
Method 2: Divide by the GCF in one step
Find the greatest common factor and divide once.
Example: 36/48
- GCF(36, 48) = 12
- 36 ÷ 12 = 3; 48 ÷ 12 = 4 → 3/4
Faster, but you need to find the GCF first.
Method 3: Prime factorization
Write both numbers as primes and cancel matching ones.
Example: 42/70
- 42 = 2 × 3 × 7
- 70 = 2 × 5 × 7
- Cancel 2 and 7 → 3/5
Divisibility rules help
| If both numbers… | Divide by |
|---|---|
| are even | 2 |
| have digit sums divisible by 3 | 3 |
| end in 0 or 5 | 5 |
| end in 0 | 10 |
Simplifying improper fractions and mixed numbers
- 20/8 → ÷ 4 → 5/2 = 2 1/2
- 3 6/9 → simplify the fraction part → 3 2/3
Is it already simplest?
If the numerator is 1, or the numerator and denominator are consecutive numbers (like 7/8), or both are different primes (like 5/7), the fraction is already in simplest form.
Worked examples
Example 1: Simplify 12/18.
GCF = 6 → 2/3.
Example 2: Simplify 45/60.
÷ 5 → 9/12 → ÷ 3 → 3/4.
Example 3: Is 14/21 in simplest form?
Both divisible by 7 → 2/3. No.
Common mistakes
- Stopping too early: 18/24 → 9/12 is still not simplest.
- Dividing only the numerator or only the denominator.
- Subtracting instead of dividing (18/24 is not 18 − 6 over 24 − 6).
- "Cancelling" digits: crossing out the 6s in 16/64 happens to give 1/4, but the same trick would turn 12/24 into 1/4, which is wrong (12/24 = 1/2). Only cancel common factors, never digits.
Try it yourself
- Simplify 8/12.
- Simplify 25/100.
- Simplify 56/72.
- Simplify 30/12 as a mixed number.
- Is 9/16 in simplest form?
Answers: 1) 2/3 2) 1/4 3) 7/9 4) 2 1/2 5) yes
Quick recap
- Simplify by dividing top and bottom by a common factor.
- Dividing by the greatest common factor gets there in one step.
- Simplest form means the only common factor is 1.