Simplification of algebraic fractions is nothing but finding the factors of both numerator and denominator and canceling the like terms. A fraction is a real number that represents the part of a single object from a group of objects. It is a combination of the numerator, denominator, and separator (/). The denominator of a fraction can not be zero. We can represent fractions as multiple fractions without changing the original one simply by altering its numerator, denominator values. Get the two simple methods to simply the fractions and solved examples in the below sections.

Algebraic Fractions

Fractions which has polynomial expressions in the numerator, denominator are called algebraic fractions. The algebraic fractions denominator can never be a zero. Every polynomial may be represented as an algebraic fraction with a denominator.

Adding Fractions:

To add two or more fractions, you must have a common denominator. To make a common denominator, just find the lcm of denominators or find the product of denominators. After making the denominator common, just add the numerators to get the addition of fractions.

Example: Solve 5 / 6 + 1 / 12?

LCM of 6, 12 is 12.

(5 * 2) / (6 * 2) + 1 / 12 = 10 / 12 + 1 / 12

= (10 + 1) / 12 = 11 / 12

Subtracting Fractions:

Addition and subtrcation of fractions is same. Here also, make a common denominator and subtract the numerator values.

Example: Solve [(x + 3) / (x – 3)] – (3 / (x – 3))

(x + 3 – 3) / (x – 3) = x / x – 3.

Multiplying Fractions:

Multiply numerators of the fractions and denominators of the fractions separately. Write the obtained numerator and denominator as a fraction to get the multiplication of fractions.

Example: Evaluate 1 / 2 * 23 / 25?

(1 * 23) / (2 * 25) = 23 / 50.

Dividing Fractions:

When you divide two fractions (a/b) / (c/d), you will get the answer as (ad / bc).

Example: Find 1/2 / 2/1?

(1 * 1) / (2 * 2) = 1 / 4.

Equivalent Fractions:

We can say that two or more fractions are equivalent when their numerator, denominators are equal. To make the fractions equal, you can divide or multiply the fractions by the same value.

Example: Check whether 12/42 and 2/7 are equivalent or not?

To make the numerator, the denominator of a second fraction equal to the first fraction, multiply the second fraction with 6.

(2 * 6) / (7 * 6) = 12 / 42

So, both are equivalent fractions.

What is Meant by Simplification of Algebraic Fractions?

Simplification of algebraic fractions means reducing a fraction to its lowest value. Both the numerator and denominator of a fraction are reduced so that it should have no common factor between them. The original value of the fraction will never change after the simplification. Therefore, it also called the equivalent fractions where numerator, denominator have no common factor except 1. An example is 25 / 15 = (5 * 5) / (5 * 3) = 5 / 3

Methods to Simplify Algebraic Fractions

Actually, we have two simple methods to simplify algebraic fractions. The one and only most important rule in the simplification of algebraic fractions is you must divide numerator, denominator with the same number at a time. Learn those steps in the below sections.

Method 1:

Simply, divide numerator and denominator by 2, 3, 4, 5… so on until you cannot find the common factor for those numbers.

Method 2:

• Find the highest common factor (HCF) for both the numerator, denominator.
• Divide both by the HCF number.
• The obtained result is a simplified fraction.

Whenever a polynomial expression is given in a fraction, then you can use this better and shorter way to simplify the fractions.

• Find the factors of both the numerator, denominator.
• Cancel the common factors in the numerator, denominator.
• Multiply the remaining values to get the result.

Example Questions on Simplification of Algebraic Fractions

Example 1.

Simplify the algebraic fraction: (10x³y³z²) / (2xy²z)

Solution:

Given fraction is (10x³y³z) / (2xy²z²)

The factors of the fraction are

(5 * 2 * x * x * x * y * y * y * z * z) / (2 * x * y * y * z)

We can see that ‘2’, ‘x’, ‘y * y’, ‘z’ are the common factors in the numerator and denominator. So, we cancel the common factors from the numerator and denominator.

= 5x²y / z

Example 2.

Simplify (2x – 10)⁶ / (x – 5)⁷.

Solution:

Given fraction is (2x – 10)⁶ / (x – 5)⁷

The common factors are (2 * (x – 5))⁶ / (x-5)⁷

= [2⁶ * (x-5) * (x-5) * (x-5) * (x-5) * (x-5) * (x-5)] / [(x-5) * (x-5) * (x-5) * (x-5) * (x-5) * (x-5) * (x-5)]

We can see that (x-5) is the common factor in the numerator, denominator. So, we cancel the common factors.

= 2⁶ / (x-5) = 64 / (x-5)

Example 3.

Reduce the algebraic fraction to its lowest term:

(x² + 2x – 35) / (x² – 25)

Solution:

Given fraction is (x² + 2x – 35) / (x² – 25)

x² + 2x – 35 = x² + 7x – 5x – 35 = x (x + 7) – 5(x + 7)

= (x – 5) (x + 7)

x² – 25 = (x – 5) (x + 5)

The common factors of the fraction is

[(x – 5) (x + 7)] / [(x – 5) (x + 5)]

Cancel the like terms

(x + 7) / (x + 5)

Example 4.

Simplify by adding and subtracting algebraic fractions:

[(2x – 1) / 3] – [(x – 5) / 6] + [(x – 4) / 2]

Solution:

Given that,

[(2x – 1) / 3] – [(x – 5) / 6] + [(x – 4) / 2]

The lcm of 3 , 6, 2 is 6.

Make the denominator as 6 for all parts of the expression.

[(2x – 1) / 3] – [(x – 5) / 6] + [(x – 4) / 2] = [2(2x – 1) / 6] – [(x – 5) / 6] + [3(x – 4) / 6]

= [(4x – 2) / 6] – [(x – 5) / 6] + [(3x – 12) / 6]

As, all denominators are equal perform arithmetic operations on numerator.

= [(4x – 2) – (x – 5) + (3x – 12)] / 6

= [4x – 2 – x + 5 + 3x – 12] / 6

= [6x – 9] / 6

= 3(x – 3) / 6

= 3(x – 3) / (3 * 2) = (x – 3) / 2

Example 5.

Simplify the following:

(i) [(x²y² + 3xy) / (4x² – 1)] / [(xy + 3) / (2x + 4)]

(ii) [(a² – 4b²) / (ab + 2b²)] * [(2b) / (a – 2b)]

Solution:

(i) Given that,

[(x²y² + 3xy) / (4x² – 16)] / [(xy + 3) / (2x + 4)]

(a/b) / (c/d) = (ad / bc)

[(x²y² + 3xy) / (4x² – 16)] / [(xy + 3) / (2x + 4)] = [(x²y² + 3xy) * (2x + 4)] / [(4x² – 16) * (xy + 3)]

The common factors are

= [(xy (xy + 3) * (2x + 4)] / [(2x -4) * (2x + 4) * (xy + 3)]

Cancel the common factors in both numerator, denominator.

= (xy) / (2x + 4)

= (xy) / [2(x + 2)]

(ii) Given that,

[(a² – 4b²) / (ab + 2b²)] * [(2b) / (a – 2b)]

Multiply numerator, denominator

= [(a² – 4b²) * (2b)] / [(ab + 2b²) * (a – 2b)]

The common factors are

= [(a + 2b) * (a – 2b) * 2b] / [(b(a + 2b) * (a – 2b)]

Cancel the like terms in both numerator, denominator

= (2b) / b = 2

FAQs on Simplification of Algebraic Fractions

1. How do you simplify an algebraic fraction?

In order to simplify an algebraic fraction, find the common factors for the numerator, denominator. Cancel the like terms in the numerator and denominator to get the simplified form.

2. How do you simplify algebraic fractions with quadratics?

Get the factors of polynomial expressions by using factorization. Find the common factors and cancel them. Perform the required operations to get the simplified fraction.

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