ASVAB Study Guide
Fractions & Decimals · Lesson 4 of 5

Algebraic Fractions (Rational Expressions)

Key rules and formulas, worked examples and 12 practice problems with step-by-step solutions.

1. Simplifying Expressions

Reducing algebraic fractions to lowest terms

$$\frac{a \times c}{b \times c} = \frac{a}{b}$$

Factor both the numerator and denominator completely, then cancel common factors.

Example

Simplify $$\frac{(x^2 - 9)}{(x^2 + 5x + 6)}$$

Show solution

Step 1: Factor numerator: $$x^2 - 9 = (x - 3)(x + 3)$$.

Step 2: Factor denominator: $$x^2 + 5x + 6 = (x + 2)(x + 3)$$.

Step 3: Cancel common factor $$(x + 3)$$:

  • $$[(x - 3)(x + 3)] / [(x + 2)(x + 3)] = \frac{(x - 3)}{(x + 2)}$$.

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

Practice problems

  1. Problem 1

    Simplify $$\frac{(6x^2)}{(3x)}$$

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    Answer: $$2x$$

    Step 1: Divide numbers: $$\frac{6}{3} = 2$$.

    Step 2: Simplify variables: $$\frac{x^2}{x} = x$$.

    => Result: $$2x$$.

    => Correct Answer: A ($$2x$$)

  2. Problem 2

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

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    Answer: $$\frac{(x - 2)}{2}$$

    Step 1: Factor numerator (difference of squares): $$x^2 - 4 = (x - 2)(x + 2)$$.

    Step 2: Factor denominator: $$2x + 4 = 2(x + 2)$$.

    Step 3: Cancel common factor $$(x + 2)$$:

    • $$[(x - 2)(x + 2)] / [2(x + 2)] = \frac{(x - 2)}{2}$$.

    => Correct Answer: B ($$\frac{(x - 2)}{2}$$)

  3. Problem 3

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

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    Answer: $$\frac{(x + 2)}{(x - 2)}$$

    Step 1: Factor numerator: $$x^2 + 4x + 4 = (x + 2)^2 = (x + 2)(x + 2)$$.

    Step 2: Factor denominator: $$x^2 - 4 = (x - 2)(x + 2)$$.

    Step 3: Cancel common factor $$(x + 2)$$:

    • $$[(x + 2)(x + 2)] / [(x - 2)(x + 2)] = \frac{(x + 2)}{(x - 2)}$$.

    => Correct Answer: A ($$\frac{(x + 2)}{(x - 2)}$$)

2. Multiplying Expressions

Finding the product of two rational expressions

$$\frac{a}{b} \times \frac{c}{d} = a \times \frac{c}{b} \times d$$

Factor all parts first. Multiply across, then simplify by canceling common factors.

Example

Multiply $$(\frac{x}{3}) \times (\frac{6}{(x + 2)})$$

Show solution

Step 1: Multiply numerators and denominators: $$(x \times 6) / [3 \times (x + 2)]$$.

Step 2: Simplify $$\frac{6}{3} = 2$$.

Result: $$2x / (x + 2)$$

Practice problems

  1. Problem 4

    Multiply $$(2x / 5) \times (\frac{10}{x})$$

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    Answer: 4

    Step 1: Multiply numerators: $$2x \times 10 = 20x$$.

    Step 2: Multiply denominators: $$5 \times x = 5x$$.

    Step 3: Simplify $$\frac{(20x)}{(5x)} = 4$$.

    => Correct Answer: A (4).

  2. Problem 5

    Multiply $$[\frac{(x - 1)}{4}] \times [\frac{8}{(x^2 - 1)}]$$

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    Answer: $$\frac{2}{(x + 1)}$$

    Step 1: Factor $$(x^2 - 1) = (x - 1)(x + 1)$$.

    Step 2: Expression becomes $$[(x - 1) \times 8] / [4 \times (x - 1)(x + 1)]$$.

    Step 3: Cancel $$(x - 1)$$ and simplify $$\frac{8}{4} = 2$$.

    • Result: $$\frac{2}{(x + 1)}$$.

    => Correct Answer: A ($$\frac{2}{(x + 1)}$$)

  3. Problem 6

    Multiply $$[\frac{(x^2 - 9)}{(2x)}] \times [\frac{4}{(x + 3)}]$$

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    Answer: $$2(x - 3) / x$$

    Step 1: Factor numerator: $$x^2 - 9 = (x - 3)(x + 3)$$.

    Step 2: Expression becomes $$[(x - 3)(x + 3) \times 4] / [2x \times (x + 3)]$$.

    Step 3: Cancel $$(x + 3)$$ and simplify $$\frac{4}{2} = 2$$.

    • Result: $$2(x - 3) / x$$.

    => Correct Answer: B ($$2(x - 3) / x$$)

3. Dividing Expressions

Dividing one algebraic fraction by another

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Keep, Change, Flip: Flip the second fraction and then follow multiplication rules.

Example

Divide $$(\frac{x^2}{4}) \div (\frac{x}{2})$$

Show solution

Step 1: Keep first fraction, change ÷ to *, flip second fraction:

  • $$(\frac{x^2}{4}) \times (\frac{2}{x})$$.

Step 2: Multiply and simplify:

  • $$\frac{(2x^2)}{(4x)} = \frac{x}{2}$$.

Result: $$\frac{x}{2}$$

Practice problems

  1. Problem 7

    Divide $$(3x / 4) \div (\frac{x}{8})$$

    Show solution

    Answer: 6

    Step 1: Flip second fraction: $$(3x / 4) \times (\frac{8}{x})$$.

    Step 2: Multiply: $$\frac{(24x)}{(4x)}$$.

    Step 3: Simplify: $$\frac{24}{4} = 6$$.

    => Correct Answer: A (6).

  2. Problem 8

    Divide $$[\frac{(x + 2)}{3}] \div [\frac{(x^2 - 4)}{9}]$$

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    Answer: $$\frac{3}{(x - 2)}$$

    Step 1: Flip second fraction: $$[\frac{(x + 2)}{3}] \times [\frac{9}{(x^2 - 4)}]$$.

    Step 2: Factor $$x^2 - 4 = (x - 2)(x + 2)$$.

    Step 3: Expression $$= [(x + 2) \times 9] / [3 \times (x - 2)(x + 2)]$$.

    Step 4: Cancel $$(x + 2)$$ and simplify $$\frac{9}{3} = 3$$-> $$\frac{3}{(x - 2)}$$.

    => Correct Answer: B ($$\frac{3}{(x - 2)}$$)

  3. Problem 9

    Divide $$[\frac{(2x - 6)}{x^2}] \div [\frac{(x - 3)}{(5x)}]$$

    Show solution

    Answer: $$\frac{10}{x}$$

    Step 1: Flip second fraction: $$[\frac{(2x - 6)}{x^2}] \times [\frac{(5x)}{(x - 3)}]$$.

    Step 2: Factor $$2x - 6 = 2(x - 3)$$.

    Step 3: Expression $$= [2(x - 3) \times 5x] / [x^2 \times (x - 3)]$$.

    Step 4: Cancel $$(x - 3)$$ and simplify $$\frac{(10x)}{x^2} = \frac{10}{x}$$.

    => Correct Answer: A ($$\frac{10}{x}$$)

4. Adding/ Subtracting

Combining expressions with common/unlike denominators

$$\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$$

Find the Least Common Denominator (LCD). Rewrite terms, combine numerators, and simplify.

Example

Add $$(\frac{2}{x}) + (\frac{3}{(2x)})$$

Show solution

Step 1: Find LCD = $$2x$$.

Step 2: Rewrite first fraction: $$\frac{(2 \times 2)}{(x \times 2)} = \frac{4}{(2x)}$$.

Step 3: Combine numerators: $$\frac{(4 + 3)}{(2x)} = \frac{7}{(2x)}$$.

Result: $$\frac{7}{(2x)}$$

Practice problems

  1. Problem 10

    Subtract $$(5x / 7) - (2x / 7)$$

    Show solution

    Answer: $$3x / 7$$

    Step 1: Denominators are the same (7).

    Step 2: Subtract numerators: $$5x - 2x = 3x$$.

    => Result: $$3x / 7$$.

    => Correct Answer: A ($$3x / 7$$)

  2. Problem 11

    Add $$(\frac{1}{x}) + (\frac{2}{(x + 1)})$$

    Show solution

    Answer: $$(3x + 1) / [x(x + 1)]$$

    Step 1: Find LCD = $$x(x + 1)$$.

    Step 2: Convert first fraction: $$1(x + 1) / [x(x + 1)]$$.

    Step 3: Convert second fraction: $$2x / [x(x + 1)]$$.

    Step 4: Combine numerators: $$(x + 1 + 2x) / [x(x + 1)] = (3x + 1) / [x(x + 1)]$$.

    => Correct Answer: A ($$(3x + 1) / [x(x + 1)]$$)

  3. Problem 12

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

    Show solution

    Answer: $$\frac{(2x + 8)}{(x^2 - 4)}$$

    Step 1: Find LCD = $$(x - 2)(x + 2) = x^2 - 4$$.

    Step 2: First fraction $$= 3(x + 2) / (x^2 - 4) = \frac{(3x + 6)}{(x^2 - 4)}$$.

    Step 3: Second fraction $$= 1(x - 2) / (x^2 - 4) = \frac{(x - 2)}{(x^2 - 4)}$$.

    Step 4: Subtract: $$[(3x + 6) - (x - 2)] / (x^2 - 4) = \frac{(3x + 6 - x + 2)}{(x^2 - 4)} = \frac{(2x + 8)}{(x^2 - 4)}$$.

    => Correct Answer: A ($$\frac{(2x + 8)}{(x^2 - 4)}$$)