Are you looking for the Worksheet on Algebraic Fractions? Then you have reached the correct place where you will find the solved example questions on algebraic fractions. Find the step by step explanation for each question that helps you solve different types of questions on algebraic fractions. Practice all the questions from Algebraic Fractions Worksheet to increase your knowledge and problem-solving skills.

1. Simplify the algebraic fraction: 1 / (a² – 5a + 6) – 1 / (a² – 4a + 3)

## Solution:

1 / (a² – 5a + 6) – 1 / (a² – 4a + 3)

Get the factors of the denominators (a² – 5a + 6), (a² – 4a + 3)

= (a² – 3a – 2a + 6), (a² – 3a – a + 3)

= (a(a – 3) – 2(a – 3)), (a(a – 3) -1(a – 3))

=(a – 3) (a – 2), (a – 1)(a – 3)

The L.C.M of denominators is (a – 3) (a – 2) (a – 1).

= (a – 1) / (a – 3) (a – 2) (a – 1) – (a – 2) / (a – 3) (a – 2) (a – 1)

= [(a – 1) – (a – 2)] / (a – 3) (a – 2) (a – 1)

= (a – 1 – a + 2) / (a – 3) (a – 2) (a – 1)

= 1 / (a – 3) (a – 2) (a – 1)

2. Reduce the algebraic fraction to the lowest form: a – (a – 1) / 2 + (a – 2) / 6

## Solution:

a – (a – 1) / 2 + (a – 2) / 6

The lowest common multiple of 2, 6 is 6.

= a/6 – (a-1)/6 + (a-2)/6

=[a – (a – 1) + a – 2] / 6

= [a – a + 1 + a – 2] / 6

= (a – 1) / 6.

3. Multiply the algebraic expressions: [(25x² – 16a²) / (ax + 2a²) x [2a / (x – 2a)] x [(x + 2a) / (5x – 4a)]

## Solution:

[(25x² – 16a²) / (ax + 2a²) x [2a / (x – 2a)] x [(x + 2a) / (5x – 4a)]

Multiply the numerators and denominators together.

= [((5x)² – (4a)²) / a(x + 2a)] x [2a / (x – 2a)] x [(x + 2a) / (5x – 4a)]

= [(5x – 4a) (5x + 4a) * 2a * (x + 2a)] / [a(x + 2a) (x – 2a) (5x – 4a)]

Cancel the common terms.

= [2(5x + 4a)] / [(x – 2a) (5x – 4a)]

4. Find the sum of the algebraic fractions: [5x / (x² – 25)] + [(x² + x – 20) / (x² + 2x – 15)]

## Solution:

[5x / (x² – 25)] + [(x² + x – 20) / (x² + 2x – 15)]

Denominators of the two algebraic fractions are (x² – 25), (x² + 2x – 15)

The factors of denominators are (x² – 5²), (x² + 5x – 3x – 15)

= (x – 5) (x + 5), (x(x + 5) -3(x + 5))

= (x – 5) (x + 5), (x + 5) (x – 3)

L.C.M of denominators are (x – 5) (x + 5) (x – 3).

= [5x(x – 3)/ (x – 5) (x + 5) (x – 3)] + [(x² + 5x – 4x – 20) / (x – 5) (x + 5) (x – 3)]

= [(5x² – 3x) / (x – 5) (x + 5) (x – 3)] + [x² + x – 20 / (x – 5) (x + 5) (x – 3)]

= [5x² – 3x + x² + x – 20] / (x – 5) (x + 5) (x – 3)

= [6x² – 2x – 20] / (x – 5) (x + 5) (x – 3)

= 2(3x² – x – 10] / (x – 5) (x + 5) (x – 3)

= 2(3x² – 6x +5x – 10) / (x – 5) (x + 5) (x – 3)

= 2(3x(x – 2) + 2(x – 2)) / (x – 5) (x + 5) (x – 3)

= 2(3x + 2) (x – 2) / (x – 5) (x + 5) (x – 3)

5. Subtract the algebraic fractions: [3x / 4a²b] – [7 / 6ab⁵] – [5x / 2ab²]

## Solution:

[3x / 4a²b] – [7 / 6ab⁵] – [5x / 2ab²]

L.C.M of 4a²b, 6ab⁵, and 2ab² is 12a²b⁵.

= [3x . 3 . b⁴ / 12a²b⁵] – [7 . 2. a / 12a²b⁵] – [5x . 6ab³ / 12a²b⁵]

= (9xb⁴ – 14a – 30ab³x) / 12a²b⁵

6. Simplify the algebraic fractions: [(2a – b) / 10a] – [b / 2a] + [(2b – a) / 15a]

## Solution:

[(2a – b) / 10a] – [b / 2a] + [(2b – a) / 15a]

The least common multiple of 10a, 2a, 15a is 30a.

= [3(2a – b) / 30a] – [15b / 30a] + [2(2b – a) / 30a]

= [6a – b – 15b + 4b – a] / 30a

= (5a – 12b) / 30a.

7. Divide the algebraic expressions: [(3m² – 9m) / (16m² – 1)] ÷ [(4m² – m) / (m – 2)]

## Solution:

[(3m² – 9m) / (16m² – 1)] ÷ [(4m² – m) / (m – 2)]

Reverse the second algebraic expression and multiply it with the first one.

= [(3m² – 9m) / (16m² – 1)] x [(m – 2) / (4m² – m)]

= [(3m² – 9m) / ((4m)² – 1²) / (3m² – 9m)] x [(m – 2) / (4m² – m)]

= [(3m² – 9m) (m – 2)] / [(4m – 1) (4m + 1) (4m² – m)]

= [m(3m – 9) (m – 2)] / [(4m – 1) (4m + 1) m (4m – 1)]

Cancel the common term m in both the numerator and denominator.

= [(3m – 9) (m – 2)] / [(4m – 1)² (4m + 1)]

8. Reduce the following algenraic fractions to the lowest terms:

(a) [3x + 7] / [6x² – 25x – 91]

(b) [5a³ + 8a²b + ab² – 2b³] / [2a³ + 9a²b- 8ab² – 15b³]

## Solution:

(a) [3x + 7] / [6x² – 25x – 91]

Get the factors of the denominator.

= (3x + 7) / (6x² + 14x – 39x – 91)

= (3x + 7) / (2x(3x + 7) – 13(3x + 7))

= (3x + 7) / (2x – 13) (3x + 7)

Cancel the common factor (3x + 7) in the numerator and denominator to get the reduced form,

= 1 / (2x – 13)

(b) [5a³ + 8a²b + ab² – 2b³] / [2a³ + 9a²b- 8ab² – 15b³]

Find the factors of numerator and denominator of the algebraic expression.

= [(5a – b) (a² + 2ab + b²)] / [(2a – 3b) (a² + 5b² + 6ab)]

= [(5a – b) (a² + ab + ab + b²)] / [(2a – 3b) (a² + 5ab + ab + 5b²)]

= [(5a – b) (a(a + b) +b(a + b)] / [(2a – 3b) (a(a + 5b) + b(a + 5b)]

= [(5a – b) (a + b) (a + b)] / [(2a – 3b) (a + b) (a + 5b)]

Cancel the common factor (a + b) in both the numerator and denominator of the fraction.

= [(5a – b) (a + b)] / [(2a – 3b) (a + 5b)]

9. Divide the algebraic fractions: [(c² – 10c + 25) / (4c² + 12c + 5)] ÷ [(4c² + 8c + 3) / (c² + 4c + 3)]

## Solution:

Get the factors of each polynomial.

= [(c² – 5c – 5c + 25) / (4c² + 10c + 2c + 5)] ÷ [4c² + 6c + 2c + 3) / (c² + 3c + c+ 3)]

= [(c(c – 5) – 5(c – 5)) / (2c(c + 5) + 1(2c + 5)] ÷ [2c(2c + 3) +1(2c + 3)) / (c(c + 3) +1(c + 3))]

= [(c – 5) (c – 5) / (2c + 5) (c + 5)] ÷ [(2c + 1)(2c + 3) / (c + 1)(c + 3)]

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

= [(c – 5)² (c + 1)(c + 3) / (2c + 5) (c + 5) (2c + 1)(2c + 3)]

10. Perform the required arithmetic operation to the following algebraic fractions:

(a) [1/(6a-2)] – [1/2(a-⅓)] + [1/(1-3a)]

(b) [(a⁴ – b⁴) / a²b²] : [(1 + b²/a²)(1 – 2a/b + a²/b²]

## Solution:

(a) [1/(6a-2)] – [1/2(a-⅓)] + [1/(1-3a)]

= [1/(6a-2)] – [1/2(3a-1) / 3)] + [1/(1-3a)]

= [1/(6a-2)] – [3/2(3a – 1)] + [1/(1-3a)]

= [1/(6a-2)] – [3/2(3a – 1)] – [1/(3a)-1]

L.C.M of (6a – 2), (3a – 1), (3a – 1) is 2(3a – 1).

= [1 / 2(3a – 1)] – [3 / 2(3a – 1)] – [2/ 2(3a – 1)]

= [1 – 3 – 2] / 2(3a – 1)

= -4/2(3a – 1)

= -1/ (3a – 1)

(b) [(a⁴ – b⁴) / a²b²] : [(1 + b²/a²)(1 – 2a/b + a²/b²)]

(1 – 2a/b + a²/b²)

L.C.M of b, b² is b².

(b² – 2ab + a²) / b²

= (a – b)² / b²

(1 + b²/a²)

= (a² + b²) / a²

(1 + b²/a²)(1 – 2a/b + a²/b²) = ((a² + b²) / a²) ((a – b)² / b²)

Multiply numerators and denominators.

= (a² + b²) (a – b)² / a²b²

[(a⁴ – b⁴) / a²b²] : [(1 + b²/a²)(1 – 2a/b + a²/b²)]

= [(a⁴ – b⁴) / a²b²] : [(a² + b²) (a – b)² / a²b²]

= [(a⁴ – b⁴) (a² + b²) (a – b)²] / [a²b² a²b²]

= [(a⁴ – b⁴) (a² + b²) (a – b)²] / [a⁴b⁴]