SAT Math: Percents

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<title>Percents: Complete Free SAT Math Lesson & Practice | The School of Mathematics</title>

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<h1>Percents: Complete Free SAT Math Lesson</h1>

"Percentages" is the second skill listed in College Board's official Problem-Solving and Data Analysis domain, right after ratios and rates, and it's tested constantly, from straightforward conversions to multi-step discount, interest, and mixture problems. This free lesson covers everything: converting between percents, decimals, and fractions, the percent increase and percent decrease formulas, successive percent changes, finding the part, the percent, or the whole using an equation or a proportion, and percent word problems covering mixtures, interest, and combined discount-and-tax scenarios. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.

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

<ol>

<li><a href="#converting">Converting Between Percents, Decimals, and Fractions</a></li>

<li><a href="#percent-of-change">Percent of Change</a></li>

<li><a href="#successive">Successive Percent Changes</a></li>

<li><a href="#equations">Finding the Part, the Percent, or the Whole</a></li>

<li><a href="#word-problems">Percent Word 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="converting">1. Converting Between Percents, Decimals, and Fractions</h2>

<p>The word <strong>percent</strong> means "hundredth" or "out of every hundred." To write a decimal or fraction as a percent, multiply by 100 and add the % sign. To write a percent as a decimal or fraction, multiply by <span class="frac"><span class="num">1</span><span class="den">100</span></span> and drop the % sign, then simplify.</p>

<p><strong>Worked Example:</strong> Write 175% as a decimal and as a fraction.</p>

<p class="step-math">175% = <span class="frac"><span class="num">175</span><span class="den">100</span></span> = <span class="frac"><span class="num">7</span><span class="den">4</span></span>   (Simplify the fraction)</p>

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<p class="prompt">1. Write 0.07 as a percent.</p>

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<p><span class="label">Explanation:</span> Multiply by 100 and add the % sign: 0.07 × 100 = 7, so 0.07 = 7%.</p>

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<p class="prompt">2. Write <span class="frac"><span class="num">3</span><span class="den">5</span></span> as a percent.</p>

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<p><span class="label">Explanation:</span> <span class="frac"><span class="num">3</span><span class="den">5</span></span> × 100% = 60%.</p>

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<p class="prompt">3. Write 125% as a decimal.</p>

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<p><span class="label">Explanation:</span> Divide 125 by 100 (multiply by <span class="frac"><span class="num">1</span><span class="den">100</span></span>) and drop the % sign: 125% = 1.25.</p>

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<p class="prompt">4. Write 8% as a fraction in simplest form.</p>

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<p><span class="label">Explanation:</span> 8% = <span class="frac"><span class="num">8</span><span class="den">100</span></span>. The greatest common factor of 8 and 100 is 4, so simplify to <span class="frac"><span class="num">2</span><span class="den">25</span></span>.</p>

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<p class="prompt">5. Which decimal is equivalent to 0.5%?</p>

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<p><span class="label">Explanation:</span> 0.5% = 0.5 × <span class="frac"><span class="num">1</span><span class="den">100</span></span> = 0.005. It's easy to lose track of a zero here since the percent itself already contains a decimal.</p>

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<h2 id="percent-of-change">2. Percent of Change</h2>

<p>The percent a quantity increases or decreases from its original amount is the <strong>percent of change</strong>.</p>

<tr><th>Percent increase</th><td><span class="frac"><span class="num">amount of increase</span><span class="den">original amount</span></span></td></tr>

<tr><th>Percent decrease</th><td><span class="frac"><span class="num">amount of decrease</span><span class="den">original amount</span></span></td></tr>

</table>

<p>The denominator is always the <strong>original</strong> amount, never the new amount, this is the detail that trips students up most often.</p>

<p><strong>Worked Example:</strong> A $450 laptop is on sale for $342. What is the percent discount?</p>

<p class="step-math">percent discount = <span class="frac"><span class="num">450 − 342</span><span class="den">450</span></span> = <span class="frac"><span class="num">108</span><span class="den">450</span></span> = 0.24 = 24%</p>

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<p class="prompt">1. A shirt's price increased from $25 to $32. What is the percent increase?</p>

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<p><span class="label">Explanation:</span> percent increase = <span class="frac"><span class="num">32 − 25</span><span class="den">25</span></span> = <span class="frac"><span class="num">7</span><span class="den">25</span></span> = 0.28 = 28%.</p>

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<p class="prompt">2. A town's population decreased from 8,400 to 7,140. What is the percent decrease?</p>

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<p><span class="label">Explanation:</span> percent decrease = <span class="frac"><span class="num">8,400 − 7,140</span><span class="den">8,400</span></span> = <span class="frac"><span class="num">1,260</span><span class="den">8,400</span></span> = 0.15 = 15%.</p>

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<p class="prompt">3. A $60 jacket is marked down to $45. What is the percent discount?</p>

</div>

<p><span class="label">Explanation:</span> percent discount = <span class="frac"><span class="num">60 − 45</span><span class="den">60</span></span> = <span class="frac"><span class="num">15</span><span class="den">60</span></span> = 0.25 = 25%.</p>

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<p class="prompt">4. A stock's value grew from $50 to $68 in one year. What is the percent increase?</p>

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<p><span class="label">Explanation:</span> percent increase = <span class="frac"><span class="num">68 − 50</span><span class="den">50</span></span> = <span class="frac"><span class="num">18</span><span class="den">50</span></span> = 0.36 = 36%.</p>

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<p class="prompt">5. A company's revenue fell by 20% from $500,000. What is the new revenue?</p>

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<p><span class="label">Explanation:</span> A 20% decrease means the new revenue is 80% of the original: 0.80 × $500,000 = $400,000.</p>

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<p class="prompt">6. If a price is increased by 15% and then the new price is decreased by 15%, is the final price the same as the original price?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-6',false)">A) Yes, it returns to the original price</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pb-6',true)">B) No, the final price is lower than the original</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-6',false)">C) No, the final price is higher than the original</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-6',false)">D) Cannot be determined</button>

</div>

<p><span class="label">Explanation:</span> Starting at 100: a 15% increase gives 115. A 15% decrease of 115 is 17.25, leaving 115 − 17.25 = 97.75. The final value is lower than the original 100, because the 15% decrease was taken from a larger number than the 15% increase was added to.</p>

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<h2 id="successive">3. Successive Percent Changes</h2>

<p>When a quantity changes by one percent and then by another, each change applies to the <strong>result of the previous step</strong>, not to the original amount. This means you can't simply add or subtract the two percents to find the combined effect, you have to apply them one at a time, or multiply the corresponding decimal factors together.</p>

<p><strong>Worked Example:</strong> A price of $200 is increased by 10%, and then the new price is increased by another 10%. What is the final price?</p>

<p class="step-math">$200 × 1.10 = $220   (First increase)</p>

<p class="step-math">$220 × 1.10 = $242   (Second increase)</p>

<p>Two successive 10% increases combine to a 21% increase overall, not 20%, since the second 10% was taken from the already-larger $220.</p>

</div>

<p class="prompt">1. A price of $200 is increased by 10%, and then the new price is increased by another 10%. What is the final price?</p>

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<p><span class="label">Explanation:</span> This is the worked example above: $200 × 1.10 = $220, then $220 × 1.10 = $242.</p>

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<p class="prompt">2. A 10% increase followed by another 10% increase is equivalent to a single percent increase of:</p>

</div>

<p><span class="label">Explanation:</span> Multiply the decimal factors: 1.10 × 1.10 = 1.21, which represents a 21% total increase.</p>

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<p class="prompt">3. A price is increased by 25% and then the new price is decreased by 20%. Is the final price the same as, greater than, or less than the original price?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-3',true)">A) The same as the original</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">B) Greater than the original</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">C) Less than the original</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">D) Cannot be determined</button>

</div>

<p><span class="label">Explanation:</span> Starting at 100: a 25% increase gives 125. A 20% decrease of 125 is 25, leaving 125 − 25 = 100. In this specific case, the two changes exactly cancel out, since 1.25 × 0.80 = 1.00. This won't happen with just any pair of percents, it's a coincidence of these particular numbers.</p>

</div>

<p class="prompt">4. An $80 item is discounted 30%, and then an additional 10% is taken off the sale price. What is the final price?</p>

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<p><span class="label">Explanation:</span> $80 × 0.70 = $56 after the first discount. Then $56 × 0.90 = $50.40 after the second discount.</p>

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<h2 id="equations">4. Finding the Part, the Percent, or the Whole</h2>

<p>Every percent problem asks for one of three missing quantities: the <strong>part</strong>, the <strong>percent</strong>, or the <strong>whole</strong>. Each type can be solved either by writing a direct equation or by writing a proportion.</p>

<tr><th>Type</th><th>Example question</th><th>Equation</th><th>Proportion</th></tr>

<tr><td>Finding the part</td><td>What is 15% of 72?</td><td>n = 0.15 × 72</td><td><span class="frac"><span class="num">15</span><span class="den">100</span></span> = <span class="frac"><span class="num">n</span><span class="den">72</span></span></td></tr>

<tr><td>Finding the percent</td><td>What percent of 20 is 6?</td><td><span class="frac"><span class="num">n</span><span class="den">100</span></span> × 20 = 6</td><td><span class="frac"><span class="num">n</span><span class="den">100</span></span> = <span class="frac"><span class="num">6</span><span class="den">20</span></span></td></tr>

<tr><td>Finding the whole</td><td>17 is 25% of what number?</td><td>17 = 0.25 × n</td><td><span class="frac"><span class="num">25</span><span class="den">100</span></span> = <span class="frac"><span class="num">17</span><span class="den">n</span></span></td></tr>

</table>

<p><strong>Worked Example:</strong> Translate each verbal phrase into an equation, then solve.</p>

<p>b. What percent of 40 is 16?</p>

<p>c. 18 is 30% of what number?</p>

</div>

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<p><span class="label">Explanation:</span> This is the worked example above: n = 0.40 × 85 = 34.</p>

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<p class="prompt">2. What percent of 40 is 16?</p>

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<p><span class="label">Explanation:</span> <span class="frac"><span class="num">n</span><span class="den">100</span></span> × 40 = 16, so n = <span class="frac"><span class="num">1,600</span><span class="den">40</span></span> = 40. The answer is 40%.</p>

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<p class="prompt">3. 18 is 30% of what number?</p>

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<p><span class="label">Explanation:</span> This is the worked example above: 18 = 0.30n, so n = 60.</p>

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<p><span class="label">Explanation:</span> n = 0.0075 × 400 = 3. Remember that 0.75% converts to the decimal 0.0075, not 0.075.</p>

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<p class="prompt">5. 45 is what percent of 180?</p>

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<p><span class="label">Explanation:</span> <span class="frac"><span class="num">45</span><span class="den">180</span></span> = 0.25 = 25%.</p>

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<p class="prompt">6. 72 is 120% of what number?</p>

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<p><span class="label">Explanation:</span> 72 = 1.20n, so n = <span class="frac"><span class="num">72</span><span class="den">1.20</span></span> = 60. A percent over 100%, like 120%, is completely valid, it just means "more than the whole."</p>

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<h2 id="word-problems">5. Percent Word Problems</h2>

<p>Three types of word problems account for most of the multi-step percent questions on the SAT: mixing two solutions of different concentrations, splitting an investment between two interest rates, and combining a discount with a tax. Each one is solved by writing an equation that balances a total quantity before and after the percent is applied.</p>

<h3>Mixtures of Two Solutions</h3>

<p><strong>Worked Example:</strong> How many milliliters of 70% acid solution must be added to 50 milliliters of a 30% acid solution to make a 50% acid solution?</p>

<p>Let x = the amount of 70% solution added.</p>

<p>50 milliliters of the 70% solution must be added.</p>

</div>

<p class="prompt">1. How many liters of a 60% saline solution must be added to 20 liters of a 20% saline solution to make a 40% saline solution?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pe-1',true)">A) 20 liters</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-1',false)">B) 10 liters</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-1',false)">C) 40 liters</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pe-1',false)">D) 15 liters</button>

</div>

<p><span class="label">Explanation:</span> Let x = liters of 60% solution added. 0.20(20) + 0.60x = 0.40(20 + x), so 4 + 0.6x = 8 + 0.4x. Then 0.2x = 4, and x = 20.</p>

</div>

<h3>Interest and Investments</h3>

<p><strong>Worked Example:</strong> Maria invested $9,000 in stocks and bonds. The stocks pay 4% interest a year and the bonds pay 7% interest a year. Her interest income is $480 this year. How much money was invested in stocks?</p>

<p>Let x = the amount invested in stocks, so 9,000 − x is invested in bonds.</p>

<p>$5,000 was invested in stocks.</p>

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<p class="prompt">2. James invested $6,000 in two accounts. One pays 3% interest and the other pays 6% interest. His total interest income for the year was $270. How much was invested at 6%?</p>

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<p><span class="label">Explanation:</span> Let y = the amount invested at 6%, so 6,000 − y is invested at 3%. 0.03(6,000 − y) + 0.06y = 270, so 180 − 0.03y + 0.06y = 270. Then 0.03y = 90, and y = 3,000.</p>

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<h3>Discounts and Tax</h3>

<p><strong>Worked Example:</strong> The sale price of a bicycle is $246.24 after a 20% discount and 8% additional tax. What was the original price of the bicycle before discount and tax?</p>

<p>Let x = the original price.</p>

<p class="step-math">x − 0.20x = 0.80x   (Price after the 20% discount)</p>

<p class="step-math">0.80x(1.08) = 0.864x   (Price after 8% tax)</p>

<p>The original price of the bicycle was $285.</p>

</div>

<p class="prompt">3. The sale price of a television is $481.50 after a 25% discount and 7% additional tax. What was the original price before discount and tax?</p>

</div>

<p><span class="label">Explanation:</span> Let x = the original price. x(0.75)(1.07) = 481.50, so 0.8025x = 481.50, and x = 600. Check: $600 × 0.75 = $450 after the discount, and $450 × 1.07 = $481.50 after tax.</p>

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<p class="prompt">4. A jacket costs $84 before an 8% sales tax. What is the total cost including tax?</p>

</div>

<p><span class="label">Explanation:</span> $84 × 1.08 = $90.72.</p>

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

<li><strong>Dividing by the new amount instead of the original amount.</strong> Percent of change always uses the original, starting amount in the denominator, never the final amount.</li>

<li><strong>Adding or subtracting successive percents directly.</strong> A 10% increase followed by a 10% decrease is not a net 0% change. Each percent applies to a different base amount, so you must apply them one step at a time or multiply the decimal factors.</li>

<li><strong>Losing track of a decimal place when converting a small percent.</strong> Percents under 1%, like 0.5% or 0.75%, are easy to convert incorrectly. Write out the multiplication by <span class="frac"><span class="num">1</span><span class="den">100</span></span> carefully rather than estimating.</li>

<li><strong>Confusing "percent of" with "percent more/less than."</strong> "20% of 80" means 0.20 × 80 = 16. "20% more than 80" means 80 + 0.20(80) = 96. These are very different calculations that share similar wording.</li>

<li><strong>Setting up a mixture or investment equation with the wrong total.</strong> The two amounts on the left side of a mixture or interest equation must add up to the actual total described in the problem, double-check that the "60 + x" or "9,000 − x" style term matches what the problem states.</li>

<li><strong>Applying tax before discount, or forgetting to apply one of them entirely.</strong> When a problem combines a discount and a tax, apply the discount first, then apply the tax to the discounted price, not the original price, unless the problem specifically states otherwise.</li>

</ul>

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

<h3>Should I use the equation method or the proportion method for percent problems?</h3>

<p>Both give the same answer, it's a matter of preference. The equation method (like n = 0.15 × 72) is usually faster for simple problems. The proportion method can feel more intuitive for "what percent" questions, since it directly compares part to whole.</p>

</div>

<h3>Why doesn't a percent increase and an equal percent decrease cancel out?</h3>

<p>Because the two percents are calculated from different base amounts. The increase is a percent of the smaller, original amount, while the decrease is the same percent of the larger, already-increased amount, so the decrease removes more than the increase added.</p>

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<h3>What's the fastest way to set up a mixture or investment word problem?</h3>

<p>Build a small table with three columns: the amount, the percent (or rate), and the resulting part. Fill in what you know, represent the unknown amount with a variable, and set the sum of the individual parts equal to the total part described in the problem.</p>

</div>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/sat-percentages-quiz-1">Percentages quizzes</a> in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/sat-percentages-quiz-1">Practice Percentages Free</a>

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