Learn about how to perform sum and difference of two or more algebraic fractions on this page. We are giving a detailed step by step procedure that helps to calculate the addition, subtraction of algebraic fractions easily. Have a look at some example questions and answers for a better understanding of the concept.

## How to Add and Subtract Algebraic Fractions?

You may feel that performing addition or subtraction of algebraic fractions is a bit difficult. To help you out in solving those questions, we are providing the step by step procedure in the below sections of this page. Follow these steps while solving the questions.

- If the denominator of fractions is the same, then just add or subtract the numerators and keep the denominator as it is.
- If the denominator of the algebraic fractions is different, then find the lowest common multiple of those denominators.
- Express all fractions in terms of the lowest common denominator.
- Perform the required operation among the numerators to obtain the result.

### Solved Example Questions on Sum & Difference of Algebraic Fractions

**Example 1.**

Find the sum of a / (a – b) + b / (a² – b²)?

**Solution:**

We can observe that the denominators of the fractions are different. Those are (a-b) and (a² – b²).

The factors of denominators are (a – b), and (a + b) (a – b).

L.C.M of (a-b), (a² – b²) is (a – b) (a + b)

To make the two fractions having common denominator both the numerator and denominator of these are to be multiplied by (a * (a + b)) / ((a + b) (a – b)) in case of a / (a – b), (b * 1) / ((a – b) (a + b)) in case of b / (a² – b²).

Therefore, a / (a – b) + b / (a² – b²)

= (a * (a + b)) / ((a + b) (a – b)) + b / ((a + b) (a – b))

= (a (a + b) + b) / ((a + b) (a – b))

= (a² + ab + b) / (a² – b²).

**Example 2:**

Find the difference of (x² + 5x + 6) / (7x + 7y) – (y² – 8y + 16) / (x² – xy)?

**Solution:**

We can observe that the denominators of the fractions are different. Those are (7x + 7y), (x² – xy).

The factors of denominators are 7 (x + y), x (x – y).

Least common multiple of denominators (7x + 7y), (x² – xy) is 7x (x + y) (x – y)

To make the two fractions having common denominator both the numerator and denominator of these are to be multiplied by [(x² + 5x + 6) * x (x – y)] / [7x (x + y) (x – y)] in case of (x² + 5x + 6) / (7x + 7y), [(y² – 8y + 16) * 7 (x + y)] / [7x (x + y) (x – y)] in case of (y² – 8y + 16) / (x² – xy).

Therefore, (x² + 5x + 6) / (7x + 7y) – (y² – 8y + 16) / (x² – xy)

= [(x² + 5x + 6) * x (x – y)] / [7x (x + y) (x – y)] – [(y² – 8y + 16) * 7 (x + y)] / [7x (x + y) (x – y)]

= [x (x – y) (x² + 3x + 2x + 6)] / [7x (x + y) (x – y)] – [7 (x + y) (y² – 4y – 4y + 16)] / [7x (x + y) (x – y)]

= [x (x – y) (x (x + 3) + 2 (x + 3))] / [7x (x + y) (x – y)] – [7 (x + y) (y (y – 4) – 4 (y – 4))] / [7x (x + y) (x – y)]

= [x (x – y) (x + 3) (x + 2)] / [7x (x + y) (x – y)] – [7 (x + y) (y – 4) (y – 4)] / [7x (x + y) (x – y)]

= [x (x – y) (x + 3) (x + 2) – 7 (x + y) (y – 4) (y – 4)] / [7x (x + y) (x – y)]

= [x (x – y) (x + 3) (x + 2) – 7 (x + y) (y – 4)²] / [7x (x + y) (x – y)].

**Example 3.**

Simplify the algebraic fractions 1 / (m – n) – 1 / (m + n) + 2n / (m² – n²)?

**Solution:**

We can say that all the denominators are different, those are (m – n), (m + n), (m² – n²)

The factors of denominators are (m – n), (m + n), (m + n) (m – n)

L.C.M of denominators is (m + n) (m – n)

To make the fractions having common denominator both the numerator and denominator of these are to be multiplied by (m + n) / [(m + n) (m – n)] in case of 1 / (m – n), (m – n) / [(m + n) (m – n)] in case of 1 / (m + n), 2n / [(m + n) (m – n)] in case of 2n / (m² – n²).

Therefore, 1 / (m – n) – 1 / (m + n) + 2n / (m² – n²)

= (m + n) / [(m + n) (m – n)] – (m – n) / [(m + n) (m – n)] + 2n / [(m + n) (m – n)]

= [m + n – (m – n) + 2n] / [(m + n) (m – n)]

= [m + n – m + n + 2n] / [(m + n) (m – n)]

= 4n / [(m + n) (m – n)].

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