Reduce Fractions

An important step that we do while we solve fractions is to reduce them to the simplest form. Though we reduce them to simplify, the value of the fraction remains unchanged. The reduced fraction is equivalent to the original fraction. In fact, the original fraction and the reduced fractions form a pair of equivalent fractions. In this lesson, we will learn how to reduce fractions using three different ways.

Reducing fractions means simplifying a fraction, wherein we divide the numerator and denominator by a common divisor until the common factor becomes 1. In other words, a fraction cannot be divided anymore by the same whole number other than 1. For example, consider the fraction 8/24. Here is the step by step process to reduce the fraction.

• Step 1: Write the factors of the numerator and the denominator. The factors of 8 are 1, 2, 4, and 8, and the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24
• Step 2: Determine the common factors of the numerator and the denominator. The common factors of 8 and 24 are 1, 2, 4, and 8
• Step 3: Divide the numerator and denominator by the common factors until they have no common factor except 1. The fraction thus obtained is in the reduced form.

Let's start dividing by 2: (8 ÷ 2) / (24 ÷ 2) = 4/12. We will continue to divide by 2 until we can't go any further. So, we have (4 ÷ 2) / (12 ÷ 2) = 2/6 = (2 ÷ 2) / (6 ÷ 2) = 1/3. Therefore, the reduced form of 8/24 is 1/3

Let us take another example.

Example: Reduce the fraction, 10/20. We will find out a common factor of the numerator and the denominator. Repeat the process until there are no more common factors. 5 is a common factor of both 10 and 20. On dividing the numerator and denominator by 5, we get, 10/20 = (10 ÷ 5) / (20 ÷ 5) = 2/4. The fraction reduces to 2/4 in the first step. Now, 2 is a common factor of 2 and 4. Reducing the fraction further, (2 ÷ 2) / (4 ÷ 2) = 1/2. Therefore, the reduced form of 10/20 is 1/2.

Let's have a look at the figure given below. The first circle has 2 shaded parts out of 8 total parts, whereas, the second circle has only one shaded part out of 4 total parts. It is to be noted that the shaded portion is the same in both circles. So we can conclude that 2 equal parts out of 8 equal parts are the same as 1 equal part out of 4 equal parts.

Reducing a fraction means making a fraction as simple as possible. In order to find the reduced forms of fractions we just simplify the fraction to its lowest form. Let us look at three easy methods of reducing fractions.

Equivalent Fractions Method

Equivalent fractions have the same value irrespective of their numerators and denominators. Given below are the steps to reduce fractions by the equivalent fractions method.

• Step 1: Find any common factor of the numerator and the denominator.
• Step 2: Divide the numerator and denominator by the common factor.
• Step 3: Repeat the same step in the resulting fraction until there are no more common factors other than 1.

GCF Method

The GCF (Greatest Common Factor) of two or more numbers is the greatest number among all the common factors of the given numbers. Given below are the steps to reduce fractions by the GCF method.

• Step 1: Find the greatest common factor (GCF) of the numerator and the denominator.
• Step 2: Divide the numerator and denominator by the GCF. The fraction so obtained is the reduced fraction.

Prime Factorization Method

Prime factorization is a way of expressing a number as a product of its prime factors. Given below are the steps to reduce fractions by the prime factorization method.

• Step 1: Find the prime factorization of the numerator and the denominator.
• Step 2: Cancel out the common factors of the numerator and denominator.
• Step 3: Take away the remaining numbers in the numerator and denominator to find the reduced fraction.

We already know how to represent whole numbers on a number line. We can also show fractions on a number line and identify equivalent fractions on a number line with the help of the following example and steps.

• Step 1: Draw 6 lines with two whole numbers marked at their ends.
• Step 2: Divide each number line into equal parts as shown in the figure. For example, starting from the first number line, we can see that it is divided into two equal parts and the division is marked by the fraction 1/2. Similarly, the second number line is divided into three equal parts and the divisions are marked by the fractions 1/3 and 2/3. In the same way, for all the number lines the divisions are marked by fractions.
• Step 3: After this step, we can easily identify the equivalent fractions by checking their length from zero. For example, we can identify the fractions 1/2, 2/4, 3/6 and 4/8 as equivalent fractions because if we observe their length (distance) from 0, we find that they are of the same length. Similarly, 1/3 and 2/6 are equivalent fractions because they represent the same distance on the number line.
• Step 4: Thus, equivalent fractions can easily be marked and identified on a number line using this method.

Variables are letters like a, b, c, x, y, z, etc., that appear in a mathematical expression and they represent unknown values. Fractions can have variables along with numbers. To reduce a fraction with variables, follow the steps given below:

• Step 1: Group the like terms together. For example, in the fraction (8a - a + 2a) / (12a). We group the like terms of a. On simplifying the numerator, we get 9a. The fraction now reduces to 9a /12a
• Step 2: Find the common factors and cancel them. 9a / 12a = (3 × 3 × a) / (3 × 4 × a). Canceling the common factors and simplifying, we get the fraction reduced to 3/4

Tips & Tricks on Reducing Fractions

So, now you know the three methods to reduce a fraction to its simplest form. Here are some tricks for you that will help you to reduce fractions quickly. Follow these tips and tricks while reducing fractions to their simplest form.

• If either numerator or denominator of a fraction is a prime number then the fraction cannot be simplified further.
• A fraction that has 1 in the numerator cannot be reduced further.
• To reduce an improper fraction, first, write it as a mixed fraction and follow the same method of simplifying a proper fraction.

Topics Related to Reduce Fractions

• Fractions
• Equivalent Fractions
• Addition and Subtraction of Fractions
• Comparing Fractions
• Simplifying Fractions
• Divisor
• Greatest Common Factor
• Long Division
• Improper Fractions

How do you Reduce Large Fractions?

In order to reduce large fractions, we divide the numerator and denominator of the large fraction by the common prime factors to reduce it to the simplest form. Another easy way to reduce large fractions is to divide the numerator and the denominator by their GCF. This makes the calculation faster and easier.

How do you Reduce Mixed Fractions?

Mixed fractions can be reduced after they are converted to an improper fraction. This can be done by using the formula: \(\dfrac{(\text{Whole}\times\text{Denominator})+\text{Numerator}}{\text{Denominator}}\). After the mixed fraction is converted to an improper fraction, it can be reduced, if needed. For example, \(5\dfrac{3}{7}=\dfrac{(5\times 7)+3}{7}=\dfrac{38}{7}\). Another way in which mixed fractions can be reduced is by keeping the whole number separate and reduce only the fractional part of the mixed fraction. For example, to reduce \(3\dfrac{4}{8}\), we will keep 3 separate, and reduce 4/8 to 1/2, so the final reduced fraction is \(3\dfrac{1}{2}\)

Why is the GCF used in Reducing Fractions?

While reducing fractions, we use the GCF to divide the numerator and the denominator because the GCF (Greatest Common Factor) is the largest number that divides the numerator and the denominator, so, it becomes easier to reduce the fractions. The other common factors of the numerator and the denominator are smaller and thus, they take more time and steps to reduce the fraction. For example, let us reduce the fraction: 12/18. The GCF of 12 and 18 is 6. So, we can use 6 to divide the numerator and the denominator in just one step. (12 ÷ 6)/(18 ÷ 6) = 2/3

How are Fractions Reduced to their Lowest Terms?

To reduce a fraction into its simplest form, divide the numerator and denominator by the highest common factor. For example, let us reduce 16/64. The highest common factor of 16 and 64 is 16. So, we will divide the numerator and the denominator by 16. (16 ÷ 16) / (64 ÷ 16) = 1/4. Therefore, the reduced form of 16/64 is 1/4.

What are the Steps to Reduce Fractions?

The following steps are used to reduce a fraction to its simplest form:

• Find the highest common factor of the numerator and the denominator.
• Divide the numerator and denominator by the highest common factor. The fraction thus obtained is in the simplest form.

What is the Easiest Way to Reduce a Fraction?

One of the easiest ways to reduce a fraction to its simplest form is to divide the numerator and denominator of the fraction by their (GCF) greatest common factor.

How do you Reduce Fractions with Exponents?

To reduce fractions with exponents, apply the exponent rules to the numerator and the denominator. For example, (a/b)ⁿ = aⁿ/bⁿ, where 'a' and 'b' are the numerator and denominator respectively and 'n' is the exponent of the fraction. After evaluating the fraction using the exponent rules, reduce the fraction to its simplest form. For example, let us reduce (2/4)³. This can be reduced and written as (1/2)³ and then using the exponent rules, this can be written as 1³/2³ = 1/8.

What is an Improper Fraction?

A fraction in which the numerator is greater than its denominator is called an improper fraction. For example, 7/4

How to Convert an Improper Fraction to a Mixed Fraction?

To convert an improper fraction to a mixed number, we divide the numerator by the denominator. Then, we write the quotient as the whole number, the remainder as the new numerator and the denominator remains the same. For example, to convert 26/7 to a mixed fraction, we will divide 26 by 7. This will give 3 as the quotient and 5 as the remainder. So, the mixed fraction will be \(3\dfrac{5}{7}\)