Order of Operations, LCM, GCF & Fractions

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<h1>PSAT/NMSQT Math: Order of Operations, LCM, GCF & Fractions</h1>

Before tackling algebra, the PSAT/NMSQT tests whether you can reliably handle arithmetic fundamentals: the order of operations, least common multiple, greatest common factor, and fraction arithmetic. These skills quietly show up inside almost every other topic, so a small error here can throw off an otherwise correct approach to a much harder problem. This free, complete lesson covers PEMDAS, LCM and GCF, simplifying and combining fractions, and expanding expressions with the distributive property. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/order-of-operations-lcm-gcf-fractions-quiz-1">Practice Order of Operations Free</a>

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

<ol>

<li><a href="#pemdas">Order of Operations (PEMDAS)</a></li>

<li><a href="#lcm-gcf">Least Common Multiple and Greatest Common Factor</a></li>

<li><a href="#fractions">Simplifying and Combining Fractions</a></li>

<li><a href="#expanding">Expanding Expressions and Radicals of Fractions</a></li>

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

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

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<h2 id="pemdas">1. Order of Operations (PEMDAS)</h2>

<p>The <strong>order of operations</strong> tells you which part of an expression to compute first: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right), often remembered as PEMDAS. Multiplication and division are always performed at the same priority level as each other, as are addition and subtraction.</p>

<p><strong>Worked Example:</strong> Evaluate 4 + 3 × (8 − 5)².</p>

<p>Parentheses first: 8 − 5 = 3. Then the exponent: 3² = 9.</p>

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<p class="prompt">1. Evaluate: 6 + 2 × (9 − 4)</p>

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<p><span class="label">Explanation:</span> Parentheses first: 9 − 4 = 5. Then multiply: 2 × 5 = 10. Then add: 6 + 10 = 16.</p>

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<p class="prompt">2. Evaluate: 20 − 3² + 4 × 2</p>

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<p><span class="label">Explanation:</span> Exponent first: 3² = 9. Then multiplication: 4 × 2 = 8. Then left to right: 20 − 9 + 8 = 19.</p>

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<p class="prompt">3. Evaluate: <span class="frac"><span class="num">18</span><span class="den">3</span></span> + 2 × (5 − 3)²</p>

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<p><span class="label">Explanation:</span> 18/3 = 6. Parentheses: 5 − 3 = 2, then squared: 4. Multiply: 2 × 4 = 8. Add: 6 + 8 = 14.</p>

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<p class="prompt">4. Evaluate −√(36/9).</p>

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<p><span class="label">Explanation:</span> Inside the radical first: 36/9 = 4. Then √4 = 2. Applying the negative sign outside gives −2.</p>

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<h2 id="lcm-gcf">2. Least Common Multiple and Greatest Common Factor</h2>

<p>The <strong>least common multiple (LCM)</strong> of two numbers is the smallest number that both divide into evenly, used constantly when adding or subtracting fractions with different denominators. The <strong>greatest common factor (GCF)</strong> is the largest number that divides evenly into both, used to simplify fractions and factor expressions.</p>

<p><strong>Worked Example:</strong> Find the LCM and GCF of 12 and 18.</p>

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<p><span class="label">Explanation:</span> Multiples of 8: 8, 16, 24... Multiples of 12: 12, 24... The smallest shared multiple is 24.</p>

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<p><span class="label">Explanation:</span> 24 = 2³ × 3, and 36 = 2² × 3². The shared factors are 2² × 3 = 12.</p>

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<p class="prompt">3. Two joggers complete a loop, one every 6 minutes and the other every 8 minutes. If they start together, after how many minutes will they next be at the starting point at the same time?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p2-3',true)">A) 24 minutes</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p2-3',false)">B) 48 minutes</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p2-3',false)">C) 14 minutes</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p2-3',false)">D) 2 minutes</button>

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<p><span class="label">Explanation:</span> This asks for the LCM of 6 and 8, which is 24 minutes.</p>

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<h2 id="fractions">3. Simplifying and Combining Fractions</h2>

<p>To <strong>simplify</strong> a fraction, divide the numerator and denominator by their GCF. To <strong>add or subtract</strong> fractions, first rewrite them with a common denominator (their LCM), then combine the numerators. To <strong>multiply</strong> fractions, multiply straight across; to <strong>divide</strong>, multiply by the reciprocal of the second fraction.</p>

<p><strong>Worked Example:</strong> Simplify <span class="frac"><span class="num">2</span><span class="den">3</span></span> + <span class="frac"><span class="num">1</span><span class="den">4</span></span>.</p>

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<p class="prompt">1. Simplify <span class="frac"><span class="num">18</span><span class="den">24</span></span> to lowest terms.</p>

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<p><span class="label">Explanation:</span> The GCF of 18 and 24 is 6. Dividing both by 6 gives 3/4.</p>

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<p class="prompt">2. Evaluate <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">4</span></span>.</p>

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<p><span class="label">Explanation:</span> The LCM of 6 and 4 is 12. 5/6 = 10/12 and 1/4 = 3/12, so 10/12 − 3/12 = 7/12.</p>

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<p class="prompt">3. Evaluate <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">9</span></span>.</p>

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<p><span class="label">Explanation:</span> Multiply by the reciprocal: (2/3) × (9/4) = 18/12 = 3/2.</p>

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<p class="prompt">4. Evaluate the complex fraction: <span class="frac"><span class="num">1/2 + 1/3</span><span class="den">1/6</span></span></p>

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<p><span class="label">Explanation:</span> 1/2 + 1/3 = 5/6 (common denominator 6). Then (5/6) ÷ (1/6) = (5/6) × 6 = 5.</p>

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<h2 id="expanding">4. Expanding Expressions and Radicals of Fractions</h2>

<p>Expanding an expression like (x + a)(x + b) uses the <strong>distributive property</strong> (often remembered as FOIL for two binomials: First, Outer, Inner, Last), multiplying every term in the first factor by every term in the second. A <strong>radical of a fraction</strong> can be evaluated by taking the square root of the numerator and denominator separately, since √(a/b) = √a / √b.</p>

<p><strong>Worked Example:</strong> Expand (x + 3)(x − 5).</p>

<p>First: x · x = x². Outer: x · (−5) = −5x. Inner: 3 · x = 3x. Last: 3 · (−5) = −15.</p>

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<p class="prompt">1. Expand (x + 4)(x + 7).</p>

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<p><span class="label">Explanation:</span> FOIL: x² + 7x + 4x + 28 = x² + 11x + 28.</p>

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<p class="prompt">2. Expand (2x − 3)(x + 5).</p>

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<p><span class="label">Explanation:</span> FOIL: 2x² + 10x − 3x − 15 = 2x² + 7x − 15.</p>

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<p class="prompt">3. Evaluate √(25/16).</p>

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<p><span class="label">Explanation:</span> √(25/16) = √25 / √16 = 5/4.</p>

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<p class="prompt">4. Evaluate −√(49/4).</p>

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<p><span class="label">Explanation:</span> √(49/4) = 7/2, and applying the negative sign outside the radical gives −7/2.</p>

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<p class="prompt">5. A worker can paint a room in 4 hours; a second worker can paint the same room in 6 hours. Working together, what fraction of the room do they complete in 1 hour?</p>

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<p><span class="label">Explanation:</span> The first worker completes 1/4 per hour and the second completes 1/6 per hour. Adding with a common denominator of 12: 3/12 + 2/12 = 5/12.</p>

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

<li><strong>Adding or subtracting before multiplying or dividing,</strong> ignoring PEMDAS priority. Multiplication and division always come before addition and subtraction unless parentheses say otherwise.</li>

<li><strong>Confusing LCM with GCF.</strong> LCM is the smallest shared multiple (used for common denominators); GCF is the largest shared factor (used for simplifying).</li>

<li><strong>Adding or subtracting fractions without a common denominator.</strong> You must find the LCM of the denominators first before combining numerators.</li>

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

<li><strong>Only multiplying the first terms when expanding two binomials.</strong> Every term in the first factor must be multiplied by every term in the second (all four FOIL products).</li>

</ul>

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

<h3>What does PEMDAS actually stand for?</h3>

<p>Parentheses, Exponents, Multiplication and Division (performed together, left to right), Addition and Subtraction (performed together, left to right). It's a priority order, not a strict left-to-right sequence of six separate steps.</p>

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<h3>Is there a shortcut for finding the LCM of two numbers?</h3>

<p>One reliable method: LCM(a, b) = (a × b) / GCF(a, b). For small numbers, listing out multiples until you find a match also works well and avoids extra calculation.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/order-of-operations-lcm-gcf-fractions-quiz-1">Order of Operations, LCM, GCF & Fractions quizzes</a> in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the <a href="https://theschoolofmathematics.com/quiz/course/PSAT-NMSQT-QBank/practice">full PSAT/NMSQT Math Question Bank</a>.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/order-of-operations-lcm-gcf-fractions-quiz-1">Practice Order of Operations Free</a>

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