Simplifying Fractions – Reducing to Lowest Terms
Simplifying a fraction means writing it in its simplest form — where the numerator and denominator share no common factor other than 1. The value stays the same; the numbers just get smaller and easier to work with.
Simplifying is really just an application of the Fundamental Theorem of Arithmetic — every whole number has one unique prime factorisation, so cancelling the shared prime factors of a numerator and denominator is guaranteed to work and always gives the same simplest form no matter which common factor you divide out first. In practice, unsimplified fractions show up constantly: a survey result of “24 out of 36 people,” a recipe scaled up to “10/15 of a cup,” or exam scores like “42/60” are all far easier to interpret once reduced (2/3, 2/3, 7/10 respectively).
The Method
- Find the GCF of the numerator and denominator.
- Divide both by the GCF.
- Check: if GCF of new numerator and denominator = 1, you are done.
GCF(18,24) = 6. 18 ÷ 6 = 3. 24 ÷ 6 = 4. Answer: 3/4.
Check: GCF(3,4) = 1 ✓
GCF(36,60) = 12. 36 ÷ 12 = 3. 60 ÷ 12 = 5. Answer: 3/5.
GCF(45,120) = 15. 45 ÷ 15 = 3. 120 ÷ 15 = 8. Answer: 3/8.
Step-by-Step Without Finding GCF First
Divide by any common factor at each step — just repeat until no common factors remain.
When Is a Fraction Already in Lowest Terms?
A fraction a/b is in lowest terms when GCF(a,b) = 1. These pairs of numbers are called coprime.
| Fraction | GCF | Already simplified? |
|---|---|---|
| 5/7 | 1 | Yes |
| 4/6 | 2 | No → 2/3 |
| 11/13 | 1 | Yes (both prime) |
| 15/25 | 5 | No → 3/5 |
Key Takeaways
- Divide numerator and denominator by their GCF.
- A fraction is fully simplified when GCF = 1.
- Simplification does not change the value — only the appearance.
- Always simplify your final answer unless told otherwise.