# Reducing Fractions to Lowest Terms - Find the Equivalent Fraction

Sometimes, the numerator and denominator will have common factors. This fraction is considered Not Simplified. We can SIMPLIFY it by dividing the numerator and the denominator by the greatest common factor.

As you can see, you must be familiar with finding the greatest common factor of two whole numbers. If not, I suggest you visit this lesson before you set out to simplify fractions.

Dividing the numerator and the denominator by one of their common divisors does not necessarily give us the reduced fraction to the lowest term, but rather another fraction called equivalent fraction.

But dividing the numerator and the denominator by their greatest common factor (GCF) does always give us the reduced fraction to the lowest term and also this fraction is the equivalent fraction.

Let's explain what we said above with a mathematical text:

### Note:

Let represent a fraction, and represent a whole number.

if is one of the common divisors of and , then we can simplify the fraction using the following formula:

if is the greatest common factor of and (i.e ), then the formula above give us the fration reduced to the lowest term.

This simple example shows us how we went from the unsimplified fraction to the reduced fraction to the lowest term, and that we always get an equivalent fraction.

Another way to reduce a fraction is repeated division, where we divide the denominator and numerator of the fraction each time by the least common divisor of them until the least common divisor is 1. This method is good for beginners only and important to understand the basis of reducing fractions to an equivalent fractions.

## Examples of How to Reducing Fractions to the lowest terms

#### Example1:

Reduce the fraction to the lowest term.

#### Solution:

Using Repeated Divisions Method:

In this example, the numerator and the denominator are both divisibles by 2.

• Divide 6 and 30 by 2 we get:

The resulting fraction is still not reduced to the lowest term, because its numerator and its denominator are both divisible by 3.

• Divide 3 and 15 by 3 we get:

The least common divisor of 1 and 5 is 1, so, this resulting fraction is in its simplest form. Finally:

Using Greatest Common Factor Method:

In the above method, we have simplified the fraction by two successive divisions, the first one is by 2 and the second one is by 3. Is there any shortcut method without repeating the division? Sure, the shortcut method is dividing the denominator and the numerator by the greatest common factor which is the product of the divisors obtained in the previous method (i.e ). Thus,

#### Example2:

Reduce the fraction to the lowest term.

#### Solution:

Using Repeated Divisions Method:

In this example, the numerator and the denominator are both divisibles by 2.

• Divide 8 and 24 by 2 we get:

The resulting fraction is still not reduced to the lowest term, because its numerator and its denominator are both divisible by 2.

• Divide 4 and 12 by 2 we get:

The resulting fraction is still not reduced to the lowest term, because its numerator and its denominator are both divisible by 2.

• Divide 2 and 6 by 2 we get:

The least common divisor of 1 and 5 is 1, so, this resulting fraction is in its simplest form. Finally:

Using Greatest Common Factor Method:

The greatest common factor of 8 and 24 is 8 (the product of the divisors obtained in the above method. i.e ). Thus,

#### Example3:

Reduce the fraction to the lowest term.

#### Solution:

Using Repeated Divisions Method:

In this example, the numerator and the denominator are both divisibles by 2.

• Divide 12 and 28 by 2 we get:

The resulting fraction is still not reduced to the lowest term, because its numerator and its denominator are both divisible by 2.

• Divide 6 and 14 by 2 we get:

The least common divisor of 3 and 7 is 1, so, this resulting fraction is in its simplest form. Finally:

Using Greatest Common Factor Method:

The greatest common factor of 12 and 28 is 4 (the product of the divisors obtained in the above method. i.e ). Thus,

#### Example5:

Reduce the fraction to the lowest term.

#### Solution:

Using Greatest Common Factor Method:

The greatest common factor of 28 and 49 is 7. Thus,

The resulting fraction is reduced to the lowest term.

#### Example6:

Reduce the fraction to the lowest term.

#### Solution:

Using Greatest Common Factor Method:

The greatest common factor of 6 and 39 is 3. Thus,

The resulting fraction is reduced to the lowest term.