Reducing fractions, simplify, and reduce to the lowest terms all refer to eliminating common factors between the numerator and the denominator.

2/4 = ½

18/72 = ¼

30/45 = 2/3

All of the above fractions have been reduced or simplified to its lowest terms. The most efficient way to do this is to

**1.**Find the greatest common factor (GCF) between the numerator and the denominator. (If necessary, refer to article,

*GCF – by Listing Factors*or

*GCF and LCM by Prime Factorization*in the related article section below)

**2.**Divide both the numerator and denominator by the GCF.

**Example 1: Simplify 2/4**

Step 1) 2/4 --- GCF = 2

Step 2) Divide the numerator, 2, by 2

Step 3) Divide the denominator, 4, by 2

**Answer: 2/4 = ½**

**Example 2: Reduce 18/72**

Step 1) 18/72 …… GCF = 18

Step 2) Divide the numerator, 18, by 18

Step 3) Divide the denominator, 72, by 18

**Answer: 18/72 = ¼**

**Example 3: Reduce 30/45 to the lowest terms**

Step 1 30/45…….GCF = 15

Step 2 Divide the numerator, 30, by 15

Step 3) Divide the denominator, 45, by 15

**Answer: 30/45 = 2/3**

**Common Questions**

**How do you know if the fraction requires reducing?**

A fraction requires reducing when the numerator and denominator have at least one common factor other than 1. For example 2/4, the factors for 2: 1, 2. the factors for 4: 1, 2, 4. What factors do they have in common? 1 and 2. Therefore, divide the numerator and denominator by 2, and the result is ½.

**Can I divide the fraction by any common factor besides the GCF?**

Yes, but remember as long as the numerator and denominator of a fraction have a factor in common, it has not been reduced to its lowest terms.

If you divide a fraction by its GCF, then it is reduced to its lowest terms in one step. Otherwise, reducing fractions can take several steps.

For example, let’s take another look at 30/45.

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Factors of 45: 1, 3, 5, 9, 15, 45

Common factors: 1, 3, 5, 15

Divide the numerator and denominator of 30/45 by the

**common factor 5.**The result is 6/ 9. It was reduced, but 30/45 was not reduced to its lowest terms. Why? The numerator, 6, and the denominator, 9, still have a common factor. In other words, 3 x 2 = 6, and 3 x 3 = 9. The

**common factor is 3**which is also in the above list. So, to simplify, use the common factor, 3, to divide the numerator and the denominator. Thus 6/9 = 2/3. Examine the fraction 2/3. Do 2 and 3 have any common factors? No. Thus, 30/45 = 2/3 has been reduced or simplified to its lowest terms. In summary, 30/45 = 2/3. Since the GCF was not used, it took two steps to get to the lowest terms.