Free Printable Algebraic Fractions (Chapter 9)

1.

Factorise, then simplify: $\frac{8}{4x+8}$

a)

\frac{2}{x+2}

b)

\frac{1}{x+2}

c)

\frac{2}{x+4}

d)

\frac{1}{x+4}

2.

Factorise, then simplify: $\frac{2x+6}{x^2-9}$

a)

$\frac{2}{x+3}$

b)

$\frac{2}{x-3}$

c)

$\frac{2x}{x+3}$

d)

$\frac{2x}{x-3}$

3.

Factorise, then simplify: $\frac{16a+8b}{6a+3b}$

a)

$\frac{8}{3}$

b)

$\frac{8a+4b}{3a+b}$

c)

$\frac{8}{3} \cdot \frac{a+b}{a+b}$

d)

$\frac{8(2a+b)}{3(2a+b)}$

4.

Factorise, then simplify: $\frac{16-x^2}{2x-8}$

a)

$\frac{-(x+4)}{2}$

b)

$\frac{x+4}{2}$

c)

$\frac{-(x-4)}{2}$

d)

$\frac{x-4}{2}$

5.

What is the simplified form of $\frac{a^2+a-12}{a^2+10a+24}$ ?

a)

\frac{a-3}{a+4}

b)

\frac{a+3}{a+6}

c)

\frac{a-3}{a+6}

d)

\frac{a+3}{a+4}

6.

Factorise, then simplify: $\frac{3x^2-6x}{3x^2-5x-2}$

a)

$\frac{3(x-2)}{3(x+1)}$

b)

$\frac{3(x-2)}{(3x+1)(x-2)}$

c)

$\frac{3x(x-2)}{(3x+1)(x-2)}$

d)

$\frac{3(x-2)}{(3x-1)(x+2)}$

7.

Simplify: $\frac{m}{n} \times \frac{2}{n}$

a)

$\frac{2m}{n^2}$

b)

$\frac{m+2}{n^2}$

c)

$\frac{2m}{n}$

d)

$\frac{m}{2n}$

8.

What is the simplified form of $\frac{m}{n} \div \frac{2}{n}$ ?

a)

$\frac{m}{2}$

b)

$\frac{2m}{n^2}$

c)

$\frac{m}{2n}$

d)

$\frac{2n}{m}$

9.

Simplify: $\frac{3x}{2y^2} \times \frac{4y}{9xz^2}$

a)

$\frac{2}{3yz^2}$

b)

$\frac{2y}{3z^2}$

c)

$\frac{2}{3z^2}$

d)

$\frac{2}{3y^2z^2}$

10.

What is the simplified form of $\frac{y^2-5y}{y+2} \times \frac{3}{2y-10}$ ?

a)

$\frac{3y}{y+2}$

b)

$\frac{3(y-5)}{y+2}$

c)

$\frac{3y}{2(y-5)}$

d)

$\frac{3(y-5)}{2(y+2)}$

11.

Simplify: $\frac{9-3x}{4x+16} \div \frac{x^2-3x}{x+4}$

a)

$\frac{3}{x}$

b)

$\frac{3}{4}$

c)

$\frac{3}{x+4}$

d)

$\frac{3}{4x}$

12.

Simplify by writing as a single fraction: $\frac{2x}{3} + \frac{x}{4}$

a)

$\frac{11x}{12}$

b)

$\frac{5x}{7}$

c)

$\frac{7x}{12}$

d)

$\frac{8x}{12}$

13.

Simplify by writing as a single fraction: $2 + \frac{x}{7}$

a)

$\frac{2x + 7}{7}$

b)

$\frac{14 + x}{7}$

c)

$\frac{2x + 14}{7}$

d)

$\frac{14 + 2x}{7}$

14.

Simplify by writing as a single fraction: $\frac{x}{4} - 1$

a)

\frac{x-4}{4}

b)

\frac{x-1}{4}

c)

\frac{x-4}{3}

d)

\frac{x+4}{4}

15.

Simplify by writing as a single fraction: $\frac{x}{5} + \frac{2}{3}$

a)

$\frac{3x + 10}{15}$

b)

$\frac{3x + 2}{15}$

c)

$\frac{3x + 6}{15}$

d)

$\frac{5x + 10}{15}$

16.

Simplify by writing as a single fraction: $\frac{5}{a} + \frac{9}{b}$

a)

\frac{5b + 9a}{ab}

b)

\frac{5a + 9b}{ab}

c)

\frac{5 + 9}{ab}

d)

\frac{5b - 9a}{ab}

17.

Simplify by writing as a single fraction: $\frac{6}{x} - \frac{7}{4x}$

a)

$\frac{17}{4x}$

b)

$\frac{17}{x}$

c)

$\frac{17}{-4x}$

d)

$\frac{17}{-x}$

18.

Write as a single fraction, and hence simplify: $\frac{2x+1}{5} - \frac{x-1}{10}$

a)

\frac{3x+1}{10}

b)

\frac{3x+3}{10}

c)

\frac{4x+1}{10}

d)

\frac{4x+3}{10}

19.

Write as a single fraction, and hence simplify: $\frac{1}{x+1} + \frac{2}{x-2}$

a)

\frac{3x-1}{(x+1)(x-2)}

b)

\frac{x+3}{(x+1)(x-2)}

c)

\frac{3x+1}{(x+1)(x-2)}

d)

\frac{x-3}{(x+1)(x-2)}

20.

Write as a single fraction, and hence simplify: $\frac{3}{(x+1)(x+2)} + \frac{4x}{x+2}$

a)

\frac{3 + 4x(x+1)}{(x+1)(x+2)}

b)

\frac{3 + 4x}{(x+1)(x+2)}

c)

\frac{3 + 4x^2 + 4x}{(x+1)(x+2)}

d)

\frac{3 + 4x^2}{(x+1)(x+2)}

21.

Solve for $x$ : $\frac{4x+5}{5} = \frac{x-6}{2}$

a)

x = -5

b)

x = 5

c)

x = 10

d)

x = -10

22.

Solve for $x$ : $\frac{4-3x}{5} = \frac{x-6}{11}$

a)

x = 2

b)

x = 3

c)

x = 4

d)

x = 5

23.

Solve for $x$ : $\frac{1}{3x} = 5$

a)

$\frac{1}{15}$

b)

$\frac{1}{5}$

c)

$\frac{1}{3}$

d)

$\frac{1}{10}$

24.

Solve for $x$ : $\frac{2}{x+1} = \frac{7}{x(x+1)}$

a)

x = 2

b)

x = 3

c)

x = 4

d)

x = 5

25.

Write a fraction for which the difference between the numerator and denominator is 24, and the fraction has value $\frac{3}{2}$.

a)

$\frac{36}{24}$

b)

$\frac{48}{32}$

c)

$\frac{30}{20}$

d)

$\frac{45}{30}$

26.

Ivan and Rico play in the same soccer team. Last season, Ivan scored half of his goals from penalties. Rico scored 8 more goals than Ivan, and scored one fifth of his goals from penalties. Suppose Ivan scored $x$ goals for the season. What is the total number of penalty goals scored by Ivan and Rico as a single fraction in terms of $x$ ?

a)

$ \frac{3x + 16}{10} $

b)

$ \frac{3x + 8}{10} $

c)

$ \frac{3x + 4}{10} $

d)

$ \frac{3x + 2}{10} $

27.

Ivan and Rico play in the same soccer team. Last season, Ivan scored half of his goals from penalties. Rico scored 8 more goals than Ivan, and scored one fifth of his goals from penalties. Suppose Ivan scored $x$ goals for the season. How many penalty goals did they score between them, if Ivan scored: (i) 12 goals for the season (ii) 22 goals for the season?

a)

10 penalty goals for (i) and 16 penalty goals for (ii)

b)

8 penalty goals for (i) and 14 penalty goals for (ii)

c)

6 penalty goals for (i) and 12 penalty goals for (ii)

d)

4 penalty goals for (i) and 10 penalty goals for (ii)

28.

If m = -4, n = 3, and p = 6, evaluate: \frac{p}{m+n}

a)

-6

b)

6

c)

-2

d)

2

29.

If $m = -4$ , $n = 3$ , and $p = 6$ , evaluate: $\frac{p-2n}{m+n}$

a)

-2

b)

2

c)

1

d)

-1

30.

If $m = -4$ , $n = 3$ , and $p = 6$ , evaluate: $\frac{p-m}{\sqrt{m^2+n^2}}$

a)

2

b)

5

c)

3

d)

4

31.

Factorise, then simplify: $\frac{a+b}{3b+3a}$

a)

\frac{1}{3}

b)

1

c)

\frac{a}{b}

d)

\frac{b}{a}

32.

What is the simplified form of $\frac{2a-2b}{b-a}$ ?

a)

-2

b)

2

c)

1

d)

-1

33.

Factorise, then simplify: $\frac{2x^2-8}{x+2}$

a)

2x - 4

b)

2x + 4

c)

2x - 2

d)

2x + 2

34.

Factorise, then simplify: $\frac{5x-15}{3x-x^2}$

a)

$\frac{5}{x-3}$

b)

$\frac{5}{x+3}$

c)

$\frac{5}{3-x}$

d)

$\frac{5}{x}$

35.

Factorise, then simplify: $\frac{x^2+4x+4}{x^2+2x}$

a)

\frac{x+2}{x}

b)

\frac{x+2}{2x}

c)

\frac{x+4}{x}

d)

\frac{x+4}{2x}

36.

Factorise, then simplify: $\frac{x^2-6x+9}{2x^2-5x-3}$

a)

$\frac{x-3}{2x+1}$

b)

$\frac{x-3}{2x-3}$

c)

$\frac{x-3}{x+1}$

d)

$\frac{x-3}{2x+3}$

37.

Simplify: $\frac{4}{k} \times \frac{k^2}{12}$

a)

$\frac{k}{3}$

b)

$\frac{4k}{3}$

c)

$\frac{1}{3}$

d)

$\frac{4}{3}$

38.

What is the simplified form of $\frac{9a}{b^2} \times \frac{ab}{3}$ ?

a)

\frac{3a^2}{b}

b)

\frac{3a^2}{b^3}

c)

\frac{3a^2b}{b^2}

d)

\frac{9a^2}{b^3}

39.

What is the simplified form of $\frac{t^2 - 3t}{6t + 6} \times \frac{t + 1}{4t - 12}$ ?

a)

$\frac{t}{12(t-3)}$

b)

$\frac{t}{6(t-3)}$

c)

$\frac{t}{24(t-3)}$

d)

$\frac{t}{3(t-3)}$

40.

What is the simplified form of $\frac{9}{n} \div 6$ ?

a)

$\frac{3}{2n}$

b)

$\frac{9}{6n}$

c)

$\frac{9}{n} \times \frac{1}{6}$

d)

$\frac{3}{n}$

41.

What is the simplified form of $\frac{x^2}{4y} \div \frac{3}{x}$ ?

a)

\frac{x^3}{12y}

b)

\frac{x^3}{4y}

c)

\frac{x^3}{3y}

d)

\frac{x^3}{12}

42.

What is the simplified form of $\frac{7}{3x - 6} \div \frac{x + 5}{x^2 - 2x}$ ?

a)

$\frac{7x}{3(x + 5)}$

b)

$\frac{7(x + 5)}{3x}$

c)

$\frac{7x}{3(x - 2)}$

d)

$\frac{7(x - 2)}{3x}$

43.

Simplify by writing as a single fraction: $\frac{3}{4} + \frac{5}{2x}$

a)

$\frac{3x + 10}{4x}$

b)

$\frac{6x + 5}{8x}$

c)

$\frac{3x + 5}{4x}$

d)

$\frac{3 + 10x}{4x}$

44.

Simplify by writing as a single fraction: $\frac{6}{y} = \frac{a}{x}$

a)

\frac{6x}{ay}

b)

\frac{6a}{xy}

c)

\frac{6y}{ax}

d)

\frac{ax}{6y}

45.

Simplify by writing as a single fraction: $\frac{3x}{7} - \frac{x}{14}$

a)

$\frac{5x}{14}$

b)

$\frac{6x}{14}$

c)

$\frac{5x}{28}$

d)

$\frac{6x}{28}$

46.

Simplify by writing as a single fraction: $\frac{4}{3x} + \frac{3}{x^2}$

a)

$\frac{4x + 9}{3x^2}$

b)

$\frac{4x + 3}{3x^2}$

c)

$\frac{4x + 9}{x^2}$

d)

$\frac{4x + 3}{x^2}$

47.

Simplify by writing as a single fraction: $3 - \frac{y}{x}$

a)

$\frac{3x - y}{x}$

b)

$\frac{3 - y}{x}$

c)

$\frac{3x + y}{x}$

d)

$\frac{3 + y}{x}$

48.

Simplify by writing as a single fraction: $1 + \frac{x}{2} + \frac{y}{3}$

a)

$\frac{6 + 3x + 2y}{6}$

b)

$\frac{6 + x + y}{6}$

c)

$\frac{3 + x + y}{6}$

d)

$\frac{6 + 2x + 3y}{6}$

49.

Write as a single fraction, and hence simplify: $\frac{x}{4} - \frac{2 - x}{8}$

a)

$\frac{2x - 2}{8}$

b)

$\frac{2x + 2}{8}$

c)

$\frac{x - 2}{8}$

d)

$\frac{x + 2}{8}$

50.

Write as a single fraction, and hence simplify: $\frac{2}{x - 1} - \frac{3}{x + 2}$

a)

\frac{5}{(x - 1)(x + 2)}

b)

\frac{-x - 7}{(x - 1)(x + 2)}

c)

\frac{x + 7}{(x - 1)(x + 2)}

d)

\frac{-5}{(x - 1)(x + 2)}

51.

Write as a single fraction, and hence simplify: $\frac{x}{x + 2} - \frac{2}{x} + \frac{x}{x + 3}$

a)

\frac{x^3 + 3x^2 - 2x - 12}{x(x + 2)(x + 3)}

b)

\frac{x^3 + 3x^2 - 2x}{x(x + 2)(x + 3)}

c)

\frac{x^3 + 3x^2 - 2x - 6}{x(x + 2)(x + 3)}

d)

\frac{x^3 + 3x^2 - 2x - 6}{(x + 2)(x + 3)}

52.

If $m$ is an even integer and $m \neq 10$ , what is the value of $\frac{m^2 - 3m - 70}{m - 10}$ ?

a)

An odd integer

b)

An even integer

c)

A prime number

d)

A negative integer

53.

Solve for $x$ : $\frac{1 - 3x}{4} = \frac{x - 2}{2}$

a)

x = 1

b)

x = 2

c)

x = 3

d)

x = 4

54.

Solve for $x$ : $\frac{2x + 1}{3} - \frac{4 - x}{6} = -2$

a)

-5

b)

-4

c)

-3

d)

-2

55.

Solve for $x$ : $\frac{18}{x} = 3$

a)

x = 6

b)

x = 3

c)

x = 9

d)

x = 12

56.

Solve for $x$ : $\frac{3}{x + 4} = \frac{8}{x - 1}$

a)

x = -2

b)

x = 2

c)

x = 3

d)

x = -3

57.

Solve for $x$ : $\frac{x + 2}{4x} = \frac{2}{3}$

a)

x = 1

b)

x = 2

c)

x = 3

d)

x = 4

58.

Solve for $x$ : $\frac{x + 3}{x + 3} + \frac{x}{x + 1} = 2$

a)

x = 1

b)

x = 2

c)

x = 3

d)

x = 4

59.

How much water must be added to 250 mL of 20% cordial mixture so that the resulting mixture is 16% cordial?

a)

50 mL

b)

62.5 mL

c)

75 mL

d)

100 mL

60.

Determine which sum is larger: $\frac{1}{3} + \frac{1}{3}$ or $\frac{1}{2} + \frac{1}{4}$

a)

$\frac{1}{3} + \frac{1}{3}$

b)

$\frac{1}{2} + \frac{1}{4}$

61.

Determine which sum is larger: $\frac{1}{5} + \frac{1}{5}$ or $\frac{1}{4} + \frac{1}{6}$

a)

$\frac{1}{5} + \frac{1}{5}$

b)

$\frac{1}{4} + \frac{1}{6}$

62.

Determine which sum is larger: $\frac{1}{9} + \frac{1}{9}$ or $\frac{1}{8} + \frac{1}{10}$

a)

$\frac{1}{9} + \frac{1}{9}$

b)

$\frac{1}{8} + \frac{1}{10}$

63.

For any positive integer $x > 1$ , which is larger: $\frac{1}{x} + \frac{1}{x}$ or $\frac{1}{x - 1} + \frac{1}{x + 1}$ ?

a)

$\frac{1}{x} + \frac{1}{x}$

b)

$\frac{1}{x - 1} + \frac{1}{x + 1}$

64.

Prove your answer to b by simplifying $\frac{1}{x - 1} + \frac{1}{x + 1} = \frac{2}{x}$ .

a)

The equation is simplified correctly.

b)

The equation is not simplified correctly.

c)

The equation is partially simplified.

d)

The equation cannot be simplified.

Algebraic Fractions (Chapter 9)

Total questions: 64

Worksheet time: 32mins

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