# Simplify Fractions

Simplifying or reducing fractions means to convert the fraction into simplest form. A fraction is called the simplified fraction if 1 is the only common factor of its numerator and the denominator. To simplify a fraction, choose the highest number that can completely divide both the numerator and the denominator.

### How to Simplify Fraction

There are three methods to simplify a fraction:

• Using Repeated Division
• Using Greatest Common Factor (GCF)
• Using Prime Factor Tree

### Using Repeated Division

In this method, we choose a small number (such as 2, 3, 4, 5) to divide a fraction. The selection of a number is decided by the fraction by looking at the fraction.

Suppose, a fraction is given, and we have chosen the number 5 to divide the fraction. It would be a wrong selection of the number because it would not go into either number. Instead of 5, if we choose 3, it would be a suitable number to use. Hence, the selection of the number must be appropriate.

To simplify the fraction, follow the steps given below:

• Pick a small number.
• Divide the numerator and the denominator by that small number. It generates a new fraction with the new numerator and the denominator.
• Continue the above step, if the newly generated fraction is still divisible by that small number. Else pick a different number to divide the fraction and move to the above step.
• Make sure that fraction has no common factor.

Example 1: Reduce the fraction .

Solution:

In the fraction , both the numerator and the denominator are even numbers, so we will pick 2 to divide the fraction. We get the fraction that is still divisible by 2. So, we will divide it again by 2. We get the fraction that is still divisible by 2. So, we will divide it again by 2. The fraction cannot be further simplified because 3 is a prime number and divisible by 1 and itself. The denominator is not divisible by 3.

Hence, the simplified fraction is .

Example 2: Simplify the fraction .

Solution:

In the fraction , both the numerator and the denominator are even numbers, so we will pick 2 to divide the fraction. We get the fraction that is still divisible by 2. So, we will divide it again by 2. We get the fraction in which the numerator is divisible by 2, but the denominator is not. So, we will pick such a different number that can divide both the numerator and the denominator. Hence, we will divide the fraction by 3. The fraction cannot be further simplified because 2 is a prime number and divisible by 1 and itself. The denominator is not divisible by 2.

Hence, the simplified fraction is .

Example 3: Simplify the fraction .

Solution:

In the fraction , both the numerator and the denominator are divisible by 5. So, we will pick the number 5 to divide the fraction. The fraction cannot be further simplified because 2 is a prime number and divisible by 1 and itself. The denominator is not divisible by 2.

Hence, the simplified fraction is .

### Using Greatest Common Factor (GCF)

The greatest common factor is the number that divides the number completely. While reducing the fraction, it is easy to repeatedly divide the numerator and the denominator by the greatest common factor. If the GCF of the numerator and denominator is 1, the fraction cannot be reduced further. It means the fraction is in its simplest form.

To simply the fraction, follow the steps given below.

• List all the factors of the numerator and the denominator.
• Write all the common factors.
• Find the greatest common factor.
• Divide both the numerator and the denominator by the GCF.
• Write the reduced fraction.

#### Note: Every number is divisible by 1 and itself. So, 1 and the number itself are the two factors for each number.

Let's see some examples.

Example 4: Simplify the fraction .

Solution:

Step 1: Factors of the numerator (12): 1, 2, 3, 4, 6, 12

Factors of the denominator (18): 1, 2, 3, 6, 9, 18

Step 2: Common Factors: 1, 2, 3, 6

Step 3: Greatest Common Factor: 6

Step 4: We will divide both the numerator and the denominator by the greatest common factor, i.e., 6. We cannot further simplify the fraction because both the numerator and the denominator are divisible by itself and having GCF 1.

Step 5: Hence, the simplified fraction is .

Example 5: Reduce the fraction .

Solution:

Step 1: Factors of the numerator (25): 1, 5, 25

Factors of the denominator (75): 1, 3, 5, 15, 25, 75

Step 2: Common Factors: 1, 5, 25

Step 3: Greatest Common Factor: 25

Step 4: We will divide both the numerator and the denominator by the greatest common factor, i.e., 25. We cannot further simplify the fraction because both the numerator and the denominator are divisible by itself and having GCF 1.

Step 5: Hence, the simplified fraction is .

Example 6: Reduce the fraction .

Solution:

Step 1: Factors of the numerator (8): 1, 2, 4, 8

Factors of the denominator (10): 1, 2, 5, 10

Step 2: Common Factors: 1, 2

Step 3: Greatest Common Factor: 2

Step 4: We will divide both the numerator and the denominator by the greatest common factor, i.e., 2. We cannot further simplify the fraction because both the numerator and the denominator are divisible by itself and having GCF 1.

Step 5: Hence, the simplified fraction is .

Example 7: Simplify the fraction .

Solution:

Step 1: Factors of the numerator (30): 1, 2, 3, 5, 6, 10, 15, 30

Factors of the denominator (36): 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 2: Common Factors: 1, 2, 3, 6

Step 3: Greatest Common Factor: 6

Step 4: We will divide both the numerator and the denominator by the greatest common factor, i.e., 6. We cannot further simplify the fraction because both the numerator and the denominator are divisible by itself and having GCF 1.

Step 5: Hence, the simplified fraction is .

### Using Prime Factor Tree

In this method, we find the prime factors of the numerator and the denominator and cancel out the common factors.

Prime Factor: Prime factor is the prime number that is divisible by itself. For example, 2, 3, 5, 7, 11, etc. To find the prime factor, branch off the given number into two numbers in which one must be prime. Repeat it until we do not get both numbers as prime.

To simplify the fraction using the prime factor tree, follow the steps given below.

• Determine the prime factors of the numerator and the denominator.
• Write the prime factors of each number with the multiplication sign, such as (2×2×3).
• Cancel out the common factors that are common in both.

Let's understand it through examples.

Example 8: Reduce the fraction Solution:

Let's find the prime factors of 24 and 60. Write the prime factors with the multiplication sign. Cancel out the common factors, we get: Hence, the simplified fraction is .

Example 9: Simplify the fraction .

Solution:

Let's find the prime factors of 820 and 240. Write the prime factors with the multiplication sign. Cancel out the common factors, we get: Hence, the simplified fraction is .

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