¹/2 and ²/4

These fractions are equivalent fractions. They both represent the same amount. One half of an orange is equal to two quarters of an orange. However, only one of these fractions is written in lowest terms.

A fraction is in lowest terms when the numerator and denominator have no common factor other than 1.

The factors of 2 are 1 and 2.
The factors of 4 are 1, 2, and 4.
2 and 4 share a common factor: 2.

We can reduce this fraction by dividing both the numerator and denominator by their common factor, 2.

^{2 ÷ 2}/4 ÷ 2 = ¹/2

1 and 2 have no common factor other than 1, so the fraction is in lowest terms.

Method #1: Common Factors
(a slow and steady method)

Let's try another example:

³⁰/36

Do 30 and 36 share any factors other than 1?

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
30 and 36 have three common factors: 2, 3, and 6.

Let's see what happens if we divide the numerator and denominator by their lowest common factor, 2. (In fact, we'd know that they have 2 as a common factor without having to work out all their factors, because both 30 and 36 are even numbers.)

^{30 ÷ 2}/36 ÷ 2 = ¹⁵/18

Are we done? Do 15 and 18 share any factors other than 1?

The factors of 15 are 1, 3, 5, 15.
The factors of 18 are 1, 2, 3, 6, 9, 18.
15 and 18 have one common factor: 3.

Once again, we divide the numerator and denominator by their common factor, 3.

^{15 ÷ 3}/18 ÷ 3 = ⁵/6

Are we done? Do 5 and 6 share any factors other than 1?

The factors of 5 are 1 and 5.
The factors of 6 are 1, 2, 3, and 6.
5 and 6 have no common factors other than 1.

This method will reduce a fraction to its lowest terms, but it can take several steps until you reach that point. What would have happened if, instead of dividing the numerator and denominator by their lowest common factor, we had started with their greatest common factor?

Method #2: Greatest Common Factor
(a more efficient method)

Let's try it again:

³⁰/36

Do 30 and 36 share any factors other than 1?

The factors of 30 are 1, 2, 3, 5, 6, 10, 15.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18.
30 and 36 have three common factors: 2, 3, and 6.
The greatest common factor is 6.

Divide the numerator and denominator by the greatest common factor:

^{30 ÷ 6}/36 ÷ 6 = ⁵/6

This time, it takes only one step to get to the same result. To reduce a fraction to its lowest terms, divide the numerator and denominator by the greatest common factor.

Method #3: Prime Factors
(an even more efficient method)

Another way to reduce fractions is to break the numerator and denominator down to their prime factors, and remove every prime factor the two have in common. Let's do that example one more time, using this method.

³⁰/36
The prime factors of 30 are 2 x 3 x 5.
The prime factors of 36 are 2 x 2 x 3 x 3.
^{2 x 3 x 5}/2 x 2 x 3 x 3
We remove the 2 x 3 the numerator and denominator have in common:
⁵/2 x 3 = ⁵/6

(If you think about it, this works the same way as the last method. The greatest common factor of two numbers is the same as the product of the prime factors they have in common.)