Percents | Free ACT Math Course | The School of Mathematics

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<h1>ACT Math: Percentages</h1>

Percentages appear throughout the ACT Math test, from quick single-step calculations to multi-part word problems about discounts, tax, and repeated changes. This free, complete lesson covers percent basics (finding a percent of a number, and reverse percent problems), percent increase and decrease, discounts, markups, and tax, and multi-step successive percent problems, where more than one percent change is applied in sequence. 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-percentages-quiz-1">Practice Percentages 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="#basics">Percent Basics</a></li>

<li><a href="#increase-decrease">Percent Increase and Decrease</a></li>

<li><a href="#discounts">Discounts, Markups, and Tax</a></li>

<li><a href="#multi-step">Multi-Step and Successive Percent Problems</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="basics">1. Percent Basics</h2>

<p>"Percent" means "per hundred." To find a percent of a number, convert the percent to a decimal and multiply. To find what percent one number is of another, divide and convert to a percent. To find the original number when you know a percent of it, divide by the decimal form of that percent.</p>

<p><strong>Worked Example:</strong> What is 35% of 80?</p>

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<p><span class="label">Explanation:</span> 0.40 × 150 = 60.</p>

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<p class="prompt">2. 24 is what percent of 80?</p>

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<p><span class="label">Explanation:</span> 24 ÷ 80 = 0.30 = 30%.</p>

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<p class="prompt">3. 45 is 15% of what number?</p>

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<p><span class="label">Explanation:</span> 45 ÷ 0.15 = 300.</p>

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<p class="prompt">4. Convert <span class="frac">3/8</span> to a percent.</p>

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<p><span class="label">Explanation:</span> 3 ÷ 8 = 0.375 = 37.5%.</p>

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<p><span class="label">Explanation:</span> 1.25 × 40 = 50. A percent greater than 100% simply means the result is larger than the original number.</p>

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<h2 id="increase-decrease">2. Percent Increase and Decrease</h2>

<p class="step-math" style="text-align:center;">percent change = <span class="frac">(new value − original value)/(original value)</span> × 100%</p>

<p>The denominator is always the <strong>original</strong> value, not the new one, this matters more than it might seem.</p>

<p><strong>Worked Example:</strong> A shirt's price increased from $40 to $52. What is the percent increase?</p>

<p>Increase: 52 − 40 = 12.</p>

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<p class="prompt">1. A population grows from 200 to 250. What is the percent increase?</p>

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<p><span class="label">Explanation:</span> Increase: 250 − 200 = 50. Percent increase: 50 ÷ 200 = 0.25 = 25%.</p>

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<p class="prompt">2. A stock price falls from $80 to $68. What is the percent decrease?</p>

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<p><span class="label">Explanation:</span> Decrease: 80 − 68 = 12. Percent decrease: 12 ÷ 80 = 0.15 = 15%.</p>

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<p class="prompt">3. If a value increases by 20% and the new value is 96, what was the original value?</p>

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<p><span class="label">Explanation:</span> A 20% increase means the new value is 120% of the original: original × 1.20 = 96, so original = 96 ÷ 1.20 = 80.</p>

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<p class="prompt">4. A price decreases by 25% to $60. What was the original price?</p>

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<p><span class="label">Explanation:</span> A 25% decrease means the new price is 75% of the original: original × 0.75 = 60, so original = 60 ÷ 0.75 = 80.</p>

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<p class="prompt">5. A quantity increases from 50 to 70, then later decreases back from 70 to 50. Are the percent increase and percent decrease equal?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pb-5',true)">A) No, the percent decrease is smaller because it's based on a larger starting value</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">D) No, the percent increase is smaller</button>

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<p><span class="label">Explanation:</span> The percent increase is 20 ÷ 50 = 40%. The percent decrease is 20 ÷ 70 ≈ 28.6%. Since percent change always uses the original (starting) value as the denominator, and 70 is larger than 50, the decrease percentage comes out smaller.</p>

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<h2 id="discounts">3. Discounts, Markups, and Tax</h2>

<p>A <strong>discount</strong> is a percent decrease from a listed price. A <strong>markup</strong> is a percent increase from a cost price. <strong>Sales tax</strong> is a percent added on top of a price. All three use the same percent increase/decrease relationships from the previous section.</p>

<p><strong>Worked Example:</strong> A $60 jacket is on sale for 30% off. What is the sale price?</p>

<p class="step-math">discount = 0.30 × 60 = 18</p>

<p class="step-math">sale price = 60 − 18 = 42</p>

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<p class="prompt">1. A $150 television is discounted by 20%. What is the sale price?</p>

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<p><span class="label">Explanation:</span> Discount: 0.20 × 150 = 30. Sale price: 150 − 30 = 120.</p>

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<p class="prompt">2. A store marks up the wholesale price of an item by 40%. If the wholesale price is $25, what is the retail price?</p>

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<p><span class="label">Explanation:</span> Markup: 0.40 × 25 = 10. Retail price: 25 + 10 = 35.</p>

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<p class="prompt">3. A meal costs $45 before an 8% sales tax. What is the total cost including tax?</p>

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<p><span class="label">Explanation:</span> Tax: 0.08 × 45 = 3.60. Total: 45 + 3.60 = 48.60.</p>

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<p class="prompt">4. A jacket originally priced at $80 is on sale for $56. What percent discount was applied?</p>

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<p><span class="label">Explanation:</span> Discount amount: 80 − 56 = 24. Percent discount: 24 ÷ 80 = 0.30 = 30%.</p>

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<p class="prompt">5. An item's price after a 15% discount is $85. What was the original price?</p>

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<p><span class="label">Explanation:</span> A 15% discount leaves 85% of the original price: original × 0.85 = 85, so original = 85 ÷ 0.85 = 100.</p>

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<h2 id="multi-step">4. Multi-Step and Successive Percent Problems</h2>

<p>When more than one percent change is applied in sequence, apply each one to the <strong>result of the previous step</strong>, never add the percents together and apply them all at once, since that gives the wrong answer.</p>

<p><strong>Worked Example:</strong> A $100 item is discounted 20%, then an additional 10% is taken off the new (already discounted) price. What is the final price?</p>

<p class="step-math">After first discount: 100 × 0.80 = 80</p>

<p class="step-math">After second discount: 80 × 0.90 = 72</p>

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<p class="prompt">1. A $200 item is discounted 25%, then an additional 20% is taken off the new price. What is the final price?</p>

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<p><span class="label">Explanation:</span> After the first discount: 200 × 0.75 = 150. After the second discount: 150 × 0.80 = 120.</p>

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<p class="prompt">2. A price increases by 10%, then decreases by 10%. Is the final price equal to the original price?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pd-2',true)">A) No, the final price is slightly less than the original</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">B) Yes, the two changes cancel out exactly</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">C) No, the final price is slightly more than the original</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">D) Cannot be determined</button>

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<p><span class="label">Explanation:</span> Starting from x: x × 1.10 × 0.90 = 0.99x, which is 99% of the original, slightly less. The 10% decrease is applied to a larger number than the original, so it removes more than the 10% increase added.</p>

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<p class="prompt">3. A population increases by 20% one year and then by another 20% the next year. If the initial population was 500, what is the population after two years?</p>

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<p><span class="label">Explanation:</span> After year one: 500 × 1.20 = 600. After year two: 600 × 1.20 = 720.</p>

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<p class="prompt">4. A worker's salary increases by 10%, and then the new salary is decreased by 10%. If the original salary was $50,000, what is the final salary?</p>

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<p><span class="label">Explanation:</span> After the raise: 50,000 × 1.10 = 55,000. After the decrease: 55,000 × 0.90 = 49,500.</p>

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

<li><strong>Using the wrong value as the base of a percent change.</strong> Percent increase or decrease is always calculated relative to the original (starting) value, not the new value.</li>

<li><strong>Adding successive percent changes together.</strong> A 20% discount followed by a 10% discount is not a 30% discount overall, since the second discount applies to an already-reduced price. Apply each step in sequence instead.</li>

<li><strong>Assuming an increase and an equal decrease cancel out.</strong> Increasing by 10% and then decreasing by 10% does not return you to the original value, because the second percent is applied to a different base amount.</li>

<li><strong>Confusing "percent of" with "percent increase/decrease from."</strong> "What is 30% of 80" is a different calculation from "80 increased by 30%," even though both involve the same numbers.</li>

<li><strong>Forgetting to divide (not multiply) when solving a reverse percent problem.</strong> If you know a percent of a number and want the original number back, divide by the decimal form of the percent, don't multiply.</li>

</ul>

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

<h3>What's the fastest way to handle a percent increase or decrease?</h3>

<p>Convert directly to a single multiplier: a 30% increase means multiplying by 1.30, and a 30% decrease means multiplying by 0.70. This skips the extra step of calculating the change amount separately and adding or subtracting it.</p>

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<h3>How do I know whether a percent problem needs multiple steps?</h3>

<p>Watch for language like "then," "an additional," or a second discount or markup applied after the first. Any time a second percent change is based on a new, already-changed value, you're dealing with a successive percent problem rather than a single calculation.</p>

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<h3>Why doesn't a percent increase followed by an equal percent decrease return the original value?</h3>

<p>Because the base amount changes between the two steps. The increase is calculated on the smaller, original number, while the decrease is calculated on the larger, already-increased number, so the decrease removes more in absolute terms than the increase added.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/act-percentages-quiz-1">Percentages 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-percentages-quiz-1">Practice Percentages 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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