# Worksheet on Algebraic Fractions | Simplifying Algebraic Fractions Worksheet with Answers

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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]