Algebraic Fractions (A) Worksheet | Cazoom Maths Worksheets - Free Printable

Problem: Simplify the given algebraic fractions.

#### Section A: Simplify the following algebraic fractions.

1. Simplify:
\[
\frac{x(x + 3)}{x}
\]
- Solution: Cancel the common factor \( x \) in the numerator and denominator (assuming \( x \neq 0 \)):
\[
\frac{x(x + 3)}{x} = x + 3
\]
- Answer: \( x + 3 \)

2. Simplify:
\[
\frac{y(y - 8)}{y}
\]
- Solution: Cancel the common factor \( y \) in the numerator and denominator (assuming \( y \neq 0 \)):
\[
\frac{y(y - 8)}{y} = y - 8
\]
- Answer: \( y - 8 \)

3. Simplify:
\[
\frac{x(x + 5)}{(x + 5)}
\]
- Solution: Cancel the common factor \( x + 5 \) in the numerator and denominator (assuming \( x \neq -5 \)):
\[
\frac{x(x + 5)}{(x + 5)} = x
\]
- Answer: \( x \)

4. Simplify:
\[
\frac{8(y - 16)}{4}
\]
- Solution: Factor out the constant in the numerator and cancel with the denominator:
\[
\frac{8(y - 16)}{4} = 2(y - 16)
\]
- Answer: \( 2(y - 16) \)

5. Simplify:
\[
\frac{5(x - 7)}{10x(x - 7)}
\]
- Solution: Cancel the common factor \( x - 7 \) in the numerator and denominator (assuming \( x \neq 7 \)), and simplify the constants:
\[
\frac{5(x - 7)}{10x(x - 7)} = \frac{5}{10x} = \frac{1}{2x}
\]
- Answer: \( \frac{1}{2x} \)

6. Simplify:
\[
\frac{3x(3x - 4)}{18x^2(3x - 4)}
\]
- Solution: Cancel the common factors \( 3x \) and \( 3x - 4 \) in the numerator and denominator (assuming \( x \neq 0 \) and \( x \neq \frac{4}{3} \)):
\[
\frac{3x(3x - 4)}{18x^2(3x - 4)} = \frac{1}{6x}
\]
- Answer: \( \frac{1}{6x} \)

7. Simplify:
\[
\frac{x(x + 5)(x - 5)}{(x + 5)}
\]
- Solution: Cancel the common factor \( x + 5 \) in the numerator and denominator (assuming \( x \neq -5 \)):
\[
\frac{x(x + 5)(x - 5)}{(x + 5)} = x(x - 5)
\]
- Answer: \( x(x - 5) \)

8. Simplify:
\[
\frac{9y(2y - 1)(y - 1)}{27y^2(y - 1)}
\]
- Solution: Cancel the common factors \( y \) and \( y - 1 \) in the numerator and denominator (assuming \( y \neq 0 \) and \( y \neq 1 \)), and simplify the constants:
\[
\frac{9y(2y - 1)(y - 1)}{27y^2(y - 1)} = \frac{9(2y - 1)}{27y} = \frac{2y - 1}{3y}
\]
- Answer: \( \frac{2y - 1}{3y} \)

9. Simplify:
\[
\frac{x(x + 1)(x - 1)(x + 1)(x - 1)}{(x + 1)(x - 1)}
\]
- Solution: Cancel the common factors \( (x + 1) \) and \( (x - 1) \) in the numerator and denominator (assuming \( x \neq -1 \) and \( x \neq 1 \)):
\[
\frac{x(x + 1)(x - 1)(x + 1)(x - 1)}{(x + 1)(x - 1)} = x(x + 1)(x - 1)
\]
- Answer: \( x(x + 1)(x - 1) \)

10. Simplify:
\[
\frac{8y(y + 4)^2}{12y^2(y + 4)}
\]
- Solution: Cancel the common factors \( y \) and \( y + 4 \) in the numerator and denominator (assuming \( y \neq 0 \) and \( y \neq -4 \)), and simplify the constants:
\[
\frac{8y(y + 4)^2}{12y^2(y + 4)} = \frac{8(y + 4)}{12y} = \frac{2(y + 4)}{3y}
\]
- Answer: \( \frac{2(y + 4)}{3y} \)

11. Simplify:
\[
\frac{x(3x - 2)}{7x^3(3x - 2)^2}
\]
- Solution: Cancel the common factors \( x \) and \( 3x - 2 \) in the numerator and denominator (assuming \( x \neq 0 \) and \( x \neq \frac{2}{3} \)):
\[
\frac{x(3x - 2)}{7x^3(3x - 2)^2} = \frac{1}{7x^2(3x - 2)}
\]
- Answer: \( \frac{1}{7x^2(3x - 2)} \)

12. Simplify:
\[
\frac{3x^3(5y - 3)(y + 3)}{18x^4(5y - 3)^3}
\]
- Solution: Cancel the common factors \( 3x^3 \) and \( 5y - 3 \) in the numerator and denominator (assuming \( x \neq 0 \) and \( y \neq \frac{3}{5} \)), and simplify the constants:
\[
\frac{3x^3(5y - 3)(y + 3)}{18x^4(5y - 3)^3} = \frac{(y + 3)}{6x(5y - 3)^2}
\]
- Answer: \( \frac{y + 3}{6x(5y - 3)^2} \)

---

Section B: Simplify the following algebraic fractions.

1. Simplify:
\[
\frac{8x + 4}{2}
\]
- Solution: Factor out the common factor in the numerator and cancel with the denominator:
\[
\frac{8x + 4}{2} = \frac{2(4x + 2)}{2} = 4x + 2
\]
- Answer: \( 4x + 2 \)

2. Simplify:
\[
\frac{2y + 6}{4}
\]
- Solution: Factor out the common factor in the numerator and cancel with the denominator:
\[
\frac{2y + 6}{4} = \frac{2(y + 3)}{4} = \frac{y + 3}{2}
\]
- Answer: \( \frac{y + 3}{2} \)

3. Simplify:
\[
\frac{7x}{14x - 21}
\]
- Solution: Factor the denominator and simplify:
\[
\frac{7x}{14x - 21} = \frac{7x}{7(2x - 3)} = \frac{x}{2x - 3}
\]
- Answer: \( \frac{x}{2x - 3} \)

4. Simplify:
\[
\frac{9y^2}{3y + 27y^2}
\]
- Solution: Factor the denominator and simplify:
\[
\frac{9y^2}{3y + 27y^2} = \frac{9y^2}{3y(1 + 9y)} = \frac{3y}{1 + 9y}
\]
- Answer: \( \frac{3y}{1 + 9y} \)

5. Simplify:
\[
\frac{x - 4}{5x - 20}
\]
- Solution: Factor the denominator and simplify:
\[
\frac{x - 4}{5x - 20} = \frac{x - 4}{5(x - 4)} = \frac{1}{5}
\]
- Answer: \( \frac{1}{5} \)

6. Simplify:
\[
\frac{6y - 30y^2}{24y^2}
\]
- Solution: Factor the numerator and simplify:
\[
\frac{6y - 30y^2}{24y^2} = \frac{6y(1 - 5y)}{24y^2} = \frac{1 - 5y}{4y}
\]
- Answer: \( \frac{1 - 5y}{4y} \)

7. Simplify:
\[
\frac{x - 4}{4 - x}
\]
- Solution: Rewrite the denominator as \( -(x - 4) \):
\[
\frac{x - 4}{4 - x} = \frac{x - 4}{-(x - 4)} = -1
\]
- Answer: \( -1 \)

8. Simplify:
\[
\frac{21 - 3x}{42x - 6x^2}
\]
- Solution: Factor the numerator and denominator, and simplify:
\[
\frac{21 - 3x}{42x - 6x^2} = \frac{3(7 - x)}{6x(7 - x)} = \frac{3}{6x} = \frac{1}{2x}
\]
- Answer: \( \frac{1}{2x} \)

9. Simplify:
\[
\frac{x + 2}{x^2 + 7x + 10}
\]
- Solution: Factor the denominator and simplify:
\[
\frac{x + 2}{x^2 + 7x + 10} = \frac{x + 2}{(x + 2)(x + 5)} = \frac{1}{x + 5}
\]
- Answer: \( \frac{1}{x + 5} \)

10. Simplify:
\[
\frac{x^2 - 13x + 36}{x - 4}
\]
- Solution: Factor the numerator and simplify:
\[
\frac{x^2 - 13x + 36}{x - 4} = \frac{(x - 4)(x - 9)}{x - 4} = x - 9
\]
- Answer: \( x - 9 \)

11. Simplify:
\[
\frac{x^2 - 8x - 20}{9x + 18}
\]
- Solution: Factor the numerator and denominator, and simplify:
\[
\frac{x^2 - 8x - 20}{9x + 18} = \frac{(x - 10)(x + 2)}{9(x + 2)} = \frac{x - 10}{9}
\]
- Answer: \( \frac{x - 10}{9} \)

12. Simplify:
\[
\frac{5x + 40}{x^2 + 6x - 16}
\]
- Solution: Factor the numerator and denominator, and simplify:
\[
\frac{5x + 40}{x^2 + 6x - 16} = \frac{5(x + 8)}{(x + 8)(x - 2)} = \frac{5}{x - 2}
\]
- Answer: \( \frac{5}{x - 2} \)

13. Simplify:
\[
\frac{12x + 20}{9x^2 + 9x - 10}
\]
- Solution: Factor the numerator and denominator, and simplify:
\[
\frac{12x + 20}{9x^2 + 9x - 10} = \frac{4(3x + 5)}{(3x + 5)(3x - 2)} = \frac{4}{3x - 2}
\]
- Answer: \( \frac{4}{3x - 2} \)

14. Simplify:
\[
\frac{x^2 + 5x + 6}{x^2 + 14x + 24}
\]
- Solution: Factor both the numerator and denominator, and simplify:
\[
\frac{x^2 + 5x + 6}{x^2 + 14x + 24} = \frac{(x + 2)(x + 3)}{(x + 2)(x + 12)} = \frac{x + 3}{x + 12}
\]
- Answer: \( \frac{x + 3}{x + 12} \)

15. Simplify:
\[
\frac{x^2 - 7x - 44}{x^2 - 17x + 66}
\]
- Solution: Factor both the numerator and denominator, and simplify:
\[
\frac{x^2 - 7x - 44}{x^2 - 17x + 66} = \frac{(x - 11)(x + 4)}{(x - 11)(x - 6)} = \frac{x + 4}{x - 6}
\]
- Answer: \( \frac{x + 4}{x - 6} \)

16. Simplify:
\[
\frac{6x^2 - x - 1}{15x^2 + 8x + 1}
\]
- Solution: Factor both the numerator and denominator, and simplify:
\[
\frac{6x^2 - x - 1}{15x^2 + 8x + 1} = \frac{(3x + 1)(2x - 1)}{(3x + 1)(5x + 1)} = \frac{2x - 1}{5x + 1}
\]
- Answer: \( \frac{2x - 1}{5x + 1} \)

17. Simplify:
\[
\frac{x^2 - y^2}{(x + y)^2}
\]
- Solution: Factor the numerator using the difference of squares, and simplify:
\[
\frac{x^2 - y^2}{(x + y)^2} = \frac{(x - y)(x + y)}{(x + y)^2} = \frac{x - y}{x + y}
\]
- Answer: \( \frac{x - y}{x + y} \)

18. Simplify:
\[
\frac{4y^2 - 9x^2}{4x^2y + 6x^3}
\]
- Solution: Factor both the numerator and denominator, and simplify:
\[
\frac{4y^2 - 9x^2}{4x^2y + 6x^3} = \frac{(2y - 3x)(2y + 3x)}{2x^2(2y + 3x)} = \frac{2y - 3x}{2x^2}
\]
- Answer: \( \frac{2y - 3x}{2x^2} \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Section A:} & 1. x + 3, \quad 2. y - 8, \quad 3. x, \quad 4. 2(y - 16), \quad 5. \frac{1}{2x}, \quad 6. \frac{1}{6x}, \\
& 7. x(x - 5), \quad 8. \frac{2y - 1}{3y}, \quad 9. x(x + 1)(x - 1), \quad 10. \frac{2(y + 4)}{3y}, \\
& 11. \frac{1}{7x^2(3x - 2)}, \quad 12. \frac{y + 3}{6x(5y - 3)^2}.
\end{array}
}
\]

\[
\boxed{
\begin{array}{ll}
\text{Section B:} & 1. 4x + 2, \quad 2. \frac{y + 3}{2}, \quad 3. \frac{x}{2x - 3}, \quad 4. \frac{3y}{1 + 9y}, \quad 5. \frac{1}{5}, \quad 6. \frac{1 - 5y}{4y}, \\
& 7. -1, \quad 8. \frac{1}{2x}, \quad 9. \frac{1}{x + 5}, \quad 10. x - 9, \quad 11. \frac{x - 10}{9}, \quad 12. \frac{5}{x - 2}, \\
& 13. \frac{4}{3x - 2}, \quad 14. \frac{x + 3}{x + 12}, \quad 15. \frac{x + 4}{x - 6}, \quad 16. \frac{2x - 1}{5x + 1}, \\
& 17. \frac{x - y}{x + y}, \quad 18. \frac{2y - 3x}{2x^2}.
\end{array}
}
\]