Fractions & Rational Expressions | Free ACT Math Course

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<h1>ACT Math: Fractions & Rational Expressions</h1>

Fractions & Rational Expressions is a foundational ACT Math topic, and it covers both numeric fractions and their algebraic cousins, rational expressions built from variables. This free, complete lesson covers simplifying fractions and mixed numbers, adding, subtracting, multiplying, and dividing fractions, simplifying rational expressions, and operations with rational expressions, including complex fractions. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.

</p>

<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-fractions-rational-expressions-quiz-1">Practice Fractions & Rational Expressions Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#simplifying">Simplifying Fractions and Mixed Numbers</a></li>

<li><a href="#operations">Operations with Fractions</a></li>

<li><a href="#rational-simplify">Simplifying Rational Expressions</a></li>

<li><a href="#rational-operations">Operations with Rational Expressions</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

</nav>

<h2 id="simplifying">1. Simplifying Fractions and Mixed Numbers</h2>

<p>To simplify a fraction, divide the numerator and denominator by their greatest common factor (GCF). An <strong>improper fraction</strong> (numerator larger than denominator) can be rewritten as a <strong>mixed number</strong>, a whole number plus a proper fraction, and vice versa.</p>

<p><strong>Worked Example:</strong> Simplify <span class="frac"><span class="num">18</span><span class="den">24</span></span>.</p>

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<p class="prompt">1. Simplify: <span class="frac"><span class="num">24</span><span class="den">36</span></span></p>

</div>

<p><span class="label">Explanation:</span> The GCF of 24 and 36 is 12: <span class="frac"><span class="num">24 ÷ 12</span><span class="den">36 ÷ 12</span></span> = <span class="frac"><span class="num">2</span><span class="den">3</span></span>.</p>

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<p class="prompt">2. Convert <span class="frac"><span class="num">7</span><span class="den">2</span></span> to a mixed number.</p>

</div>

<p><span class="label">Explanation:</span> 7 ÷ 2 = 3 with a remainder of 1, so <span class="frac"><span class="num">7</span><span class="den">2</span></span> = 3<span class="frac"><span class="num">1</span><span class="den">2</span></span>.</p>

</div>

<p class="prompt">3. Convert 4<span class="frac"><span class="num">3</span><span class="den">5</span></span> to an improper fraction.</p>

</div>

<p><span class="label">Explanation:</span> Multiply the whole number by the denominator and add the numerator: 4 × 5 + 3 = 23, so the result is <span class="frac"><span class="num">23</span><span class="den">5</span></span>.</p>

</div>

<p class="prompt">4. Simplify: <span class="frac"><span class="num">45</span><span class="den">60</span></span></p>

</div>

<p><span class="label">Explanation:</span> The GCF of 45 and 60 is 15: <span class="frac"><span class="num">45 ÷ 15</span><span class="den">60 ÷ 15</span></span> = <span class="frac"><span class="num">3</span><span class="den">4</span></span>.</p>

</div>

<p class="prompt">5. Order the following fractions from least to greatest: <span class="frac"><span class="num">3</span><span class="den">5</span></span>, <span class="frac"><span class="num">7</span><span class="den">10</span></span>, <span class="frac"><span class="num">2</span><span class="den">3</span></span></p>

</div>

<p><span class="label">Explanation:</span> Rewrite with a common denominator of 30: <span class="frac"><span class="num">3</span><span class="den">5</span></span> = <span class="frac"><span class="num">18</span><span class="den">30</span></span>, <span class="frac"><span class="num">7</span><span class="den">10</span></span> = <span class="frac"><span class="num">21</span><span class="den">30</span></span>, and <span class="frac"><span class="num">2</span><span class="den">3</span></span> = <span class="frac"><span class="num">20</span><span class="den">30</span></span>. Ordering the numerators: 18 < 20 < 21, so <span class="frac"><span class="num">3</span><span class="den">5</span></span> < <span class="frac"><span class="num">2</span><span class="den">3</span></span> < <span class="frac"><span class="num">7</span><span class="den">10</span></span>.</p>

</div>

<h2 id="operations">2. Operations with Fractions</h2>

<p>To add or subtract fractions, first rewrite them with a common denominator. To multiply fractions, multiply straight across, numerator by numerator and denominator by denominator. To divide fractions, multiply by the reciprocal of the second fraction.</p>

<p><strong>Worked Example:</strong> Compute <span class="frac"><span class="num">2</span><span class="den">3</span></span> + <span class="frac"><span class="num">5</span><span class="den">6</span></span>.</p>

<p>The least common denominator is 6: <span class="frac"><span class="num">2</span><span class="den">3</span></span> = <span class="frac"><span class="num">4</span><span class="den">6</span></span>.</p>

</div>

<p class="prompt">1. Compute: <span class="frac"><span class="num">1</span><span class="den">4</span></span> + <span class="frac"><span class="num">3</span><span class="den">8</span></span></p>

</div>

<p><span class="label">Explanation:</span> <span class="frac"><span class="num">1</span><span class="den">4</span></span> = <span class="frac"><span class="num">2</span><span class="den">8</span></span>, and <span class="frac"><span class="num">2</span><span class="den">8</span></span> + <span class="frac"><span class="num">3</span><span class="den">8</span></span> = <span class="frac"><span class="num">5</span><span class="den">8</span></span>.</p>

</div>

<p class="prompt">2. Compute: <span class="frac"><span class="num">5</span><span class="den">6</span></span> − <span class="frac"><span class="num">1</span><span class="den">3</span></span></p>

</div>

<p><span class="label">Explanation:</span> <span class="frac"><span class="num">1</span><span class="den">3</span></span> = <span class="frac"><span class="num">2</span><span class="den">6</span></span>, and <span class="frac"><span class="num">5</span><span class="den">6</span></span> − <span class="frac"><span class="num">2</span><span class="den">6</span></span> = <span class="frac"><span class="num">3</span><span class="den">6</span></span> = <span class="frac"><span class="num">1</span><span class="den">2</span></span>.</p>

</div>

<p class="prompt">3. Compute: <span class="frac"><span class="num">2</span><span class="den">3</span></span> × <span class="frac"><span class="num">3</span><span class="den">4</span></span></p>

</div>

<p><span class="label">Explanation:</span> Multiply straight across: <span class="frac"><span class="num">2 × 3</span><span class="den">3 × 4</span></span> = <span class="frac"><span class="num">6</span><span class="den">12</span></span> = <span class="frac"><span class="num">1</span><span class="den">2</span></span>.</p>

</div>

<p class="prompt">4. Compute: <span class="frac"><span class="num">3</span><span class="den">5</span></span> ÷ <span class="frac"><span class="num">2</span><span class="den">7</span></span></p>

</div>

<p><span class="label">Explanation:</span> Multiply by the reciprocal of the second fraction: <span class="frac"><span class="num">3</span><span class="den">5</span></span> × <span class="frac"><span class="num">7</span><span class="den">2</span></span> = <span class="frac"><span class="num">21</span><span class="den">10</span></span>.</p>

</div>

<p class="prompt">5. Compute: 1<span class="frac"><span class="num">1</span><span class="den">2</span></span> + 2<span class="frac"><span class="num">1</span><span class="den">3</span></span></p>

</div>

<p><span class="label">Explanation:</span> As improper fractions: <span class="frac"><span class="num">3</span><span class="den">2</span></span> + <span class="frac"><span class="num">7</span><span class="den">3</span></span> = <span class="frac"><span class="num">9</span><span class="den">6</span></span> + <span class="frac"><span class="num">14</span><span class="den">6</span></span> = <span class="frac"><span class="num">23</span><span class="den">6</span></span> = 3<span class="frac"><span class="num">5</span><span class="den">6</span></span>.</p>

</div>

<h2 id="rational-simplify">3. Simplifying Rational Expressions</h2>

<p>A <strong>rational expression</strong> is a fraction with a polynomial in the numerator, denominator, or both. Simplify one by factoring the numerator and denominator completely, then canceling any factors they share.</p>

<p><strong>Worked Example:</strong> Simplify <span class="frac"><span class="num">x² − 9</span><span class="den">x + 3</span></span>.</p>

</div>

<p class="prompt">1. Simplify: <span class="frac"><span class="num">6x²</span><span class="den">9x</span></span></p>

</div>

<p><span class="label">Explanation:</span> The GCF of 6x² and 9x is 3x: <span class="frac"><span class="num">6x² ÷ 3x</span><span class="den">9x ÷ 3x</span></span> = <span class="frac"><span class="num">2x</span><span class="den">3</span></span>.</p>

</div>

<p class="prompt">2. Simplify: <span class="frac"><span class="num">x² − 16</span><span class="den">x − 4</span></span></p>

</div>

<p><span class="label">Explanation:</span> <span class="frac"><span class="num">(x − 4)(x + 4)</span><span class="den">x − 4</span></span> = x + 4.</p>

</div>

<p class="prompt">3. Simplify: <span class="frac"><span class="num">x² + 5x + 6</span><span class="den">x + 2</span></span></p>

</div>

<p><span class="label">Explanation:</span> Factor the numerator: <span class="frac"><span class="num">(x + 2)(x + 3)</span><span class="den">x + 2</span></span> = x + 3.</p>

</div>

<p class="prompt">4. Simplify: <span class="frac"><span class="num">4x + 8</span><span class="den">x + 2</span></span></p>

</div>

<p><span class="label">Explanation:</span> Factor the numerator: <span class="frac"><span class="num">4(x + 2)</span><span class="den">x + 2</span></span> = 4.</p>

</div>

<p class="prompt">5. For what value of x is the expression <span class="frac"><span class="num">x + 5</span><span class="den">x − 3</span></span> undefined?</p>

</div>

<p><span class="label">Explanation:</span> A rational expression is undefined wherever its denominator equals 0: x − 3 = 0, so x = 3.</p>

</div>

<h2 id="rational-operations">4. Operations with Rational Expressions</h2>

<p>Rational expressions follow the exact same rules as numeric fractions: multiply straight across (then simplify), divide by multiplying by the reciprocal, and add or subtract only after finding a common denominator. A <strong>complex fraction</strong>, a fraction within a fraction, is simplified the same way as dividing two fractions.</p>

<p><strong>Worked Example:</strong> Simplify <span class="frac"><span class="num">3</span><span class="den">x</span></span> × <span class="frac"><span class="num">x²</span><span class="den">6</span></span>.</p>

</div>

<p class="prompt">1. Simplify: <span class="frac"><span class="num">x</span><span class="den">4</span></span> ÷ <span class="frac"><span class="num">x²</span><span class="den">8</span></span></p>

</div>

<p><span class="label">Explanation:</span> Multiply by the reciprocal: <span class="frac"><span class="num">x</span><span class="den">4</span></span> × <span class="frac"><span class="num">8</span><span class="den">x²</span></span> = <span class="frac"><span class="num">8x</span><span class="den">4x²</span></span> = <span class="frac"><span class="num">2</span><span class="den">x</span></span>.</p>

</div>

</div>

<p><span class="label">Explanation:</span> These fractions already share a denominator, so just add the numerators: <span class="frac"><span class="num">3 + 2</span><span class="den">x</span></span> = <span class="frac"><span class="num">5</span><span class="den">x</span></span>.</p>

</div>

</div>

<p><span class="label">Explanation:</span> The common denominator is xy: <span class="frac"><span class="num">y</span><span class="den">xy</span></span> + <span class="frac"><span class="num">x</span><span class="den">xy</span></span> = <span class="frac"><span class="num">x + y</span><span class="den">xy</span></span>.</p>

</div>

<p class="prompt">4. Simplify the complex fraction: <span class="frac"><span class="num">1/2</span><span class="den">3/4</span></span></p>

</div>

<p><span class="label">Explanation:</span> A complex fraction is division: <span class="frac"><span class="num">1</span><span class="den">2</span></span> ÷ <span class="frac"><span class="num">3</span><span class="den">4</span></span> = <span class="frac"><span class="num">1</span><span class="den">2</span></span> × <span class="frac"><span class="num">4</span><span class="den">3</span></span> = <span class="frac"><span class="num">4</span><span class="den">6</span></span> = <span class="frac"><span class="num">2</span><span class="den">3</span></span>.</p>

</div>

<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Adding or subtracting fractions without a common denominator.</strong> Unlike multiplication, addition and subtraction of fractions require the same denominator first. Never add numerators and denominators separately.</li>

<li><strong>Forgetting to flip the second fraction when dividing.</strong> Dividing by a fraction means multiplying by its reciprocal, flipping only the second fraction, not the first.</li>

<li><strong>Canceling terms instead of factors in a rational expression.</strong> You can only cancel a factor that multiplies the entire numerator and entire denominator, never a term that's only added or subtracted within a larger expression.</li>

<li><strong>Forgetting to factor before canceling.</strong> An expression like <span class="frac"><span class="num">4x + 8</span><span class="den">x + 2</span></span> can't be simplified by canceling the x + 2 pieces directly, it must be factored first, into <span class="frac"><span class="num">4(x + 2)</span><span class="den">x + 2</span></span>, before anything cancels.</li>

<li><strong>Treating a complex fraction like a sum instead of a division.</strong> A fraction stacked over another fraction represents division between the two, not addition, so multiply by the reciprocal of the bottom fraction.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>Do I need to find the least common denominator, or will any common denominator work?</h3>

<p>Any common denominator technically works, but the least common denominator keeps the numbers smaller and the arithmetic simpler, which matters on a timed test like the ACT.</p>

</div>

<h3>Are rational expressions harder than numeric fractions?</h3>

<p>The rules are identical, add/subtract with a common denominator, multiply straight across, divide by the reciprocal. The extra step with rational expressions is factoring first, since the "common factors" you're canceling are algebraic expressions rather than plain numbers.</p>

</div>

<h3>Why does a rational expression have restricted values?</h3>

<p>Division by zero is undefined, so any x-value that makes the denominator of a rational expression equal zero must be excluded from its domain, even after the expression has been simplified.</p>

</div>

<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-fractions-rational-expressions-quiz-1">Fractions & Rational Expressions quizzes</a> in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-fractions-rational-expressions-quiz-1">Practice Fractions & Rational Expressions Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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