Rates & Ratios | Free ACT Math Course | The School of Mathematics

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<h1>ACT Math: Rates & Ratios</h1>

Rates & Ratios is one of the most frequently tested Integrating Essential Skills topics on the ACT, showing up in speed problems, recipe and mixture questions, map scales, and unit conversions. This free, complete lesson covers ratios and proportions, unit rates and unit conversion (including converting between derived units like feet per second and miles per hour), rate-distance-time problems, and multi-step rate and proportion problems, including classic work-rate questions. 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-rates-ratios-quiz-1">Practice Rates & Ratios 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="#ratios">Ratios and Proportions</a></li>

<li><a href="#unit-rates">Unit Rates and Unit Conversion</a></li>

<li><a href="#distance">Rate, Distance, and Time</a></li>

<li><a href="#multi-step">Multi-Step Rate and Proportion 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="ratios">1. Ratios and Proportions</h2>

<p>A <strong>ratio</strong> compares two quantities, and can be simplified the same way as a fraction. A <strong>proportion</strong> is an equation stating that two ratios are equal, solved by cross-multiplying.</p>

<p><strong>Worked Example:</strong> Solve the proportion <span class="frac"><span class="num">3</span><span class="den">4</span></span> = <span class="frac"><span class="num">x</span><span class="den">20</span></span>.</p>

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<p class="prompt">1. Simplify the ratio 18:24 to lowest terms.</p>

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<p><span class="label">Explanation:</span> The GCF of 18 and 24 is 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3:4.</p>

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<p class="prompt">2. Solve the proportion: <span class="frac"><span class="num">5</span><span class="den">8</span></span> = <span class="frac"><span class="num">x</span><span class="den">40</span></span></p>

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<p><span class="label">Explanation:</span> Cross-multiply: 8x = 5 × 40 = 200, so x = 25.</p>

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<p class="prompt">3. A recipe uses a ratio of 2 cups flour to 3 cups sugar. If you use 9 cups of sugar, how many cups of flour are needed?</p>

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<p><span class="label">Explanation:</span> <span class="frac"><span class="num">2</span><span class="den">3</span></span> = <span class="frac"><span class="num">x</span><span class="den">9</span></span>. Cross-multiply: 3x = 18, so x = 6.</p>

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<p class="prompt">4. The ratio of boys to girls in a class is 3:5. If there are 24 students total, how many are boys?</p>

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<p><span class="label">Explanation:</span> The ratio parts add to 3 + 5 = 8. Boys make up <span class="frac"><span class="num">3</span><span class="den">8</span></span> of the total: <span class="frac"><span class="num">3</span><span class="den">8</span></span> × 24 = 9.</p>

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

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<p><span class="label">Explanation:</span> Cross-multiply: 3(x + 2) = 6 × 5 = 30, so 3x + 6 = 30, 3x = 24, and x = 8.</p>

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<h2 id="unit-rates">2. Unit Rates and Unit Conversion</h2>

<p>A <strong>unit rate</strong> compares a quantity to a single unit of another (like miles per hour). To convert between units, multiply by conversion factors arranged so the unwanted units cancel out.</p>

<p><strong>Worked Example:</strong> Convert 60 miles per hour to feet per second. (1 mile = 5,280 feet; 1 hour = 3,600 seconds)</p>

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<p class="prompt">1. A car travels at 45 miles per hour. What is this speed in feet per second? (1 mile = 5,280 feet)</p>

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<p><span class="label">Explanation:</span> 45 × 5,280 ÷ 3,600 = 237,600 ÷ 3,600 = 66 ft/s.</p>

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<p class="prompt">2. A runner completes a 400-meter race in 50 seconds. What is the runner's unit rate in meters per second?</p>

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<p><span class="label">Explanation:</span> 400 ÷ 50 = 8 meters per second.</p>

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<p class="prompt">3. A printer prints 15 pages in 3 minutes. At this rate, how many pages can it print in 8 minutes?</p>

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<p><span class="label">Explanation:</span> Unit rate: 15 ÷ 3 = 5 pages per minute. In 8 minutes: 5 × 8 = 40 pages.</p>

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<p class="prompt">4. Convert 2 hours 15 minutes into minutes.</p>

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<p><span class="label">Explanation:</span> 2 hours = 2 × 60 = 120 minutes. 120 + 15 = 135 minutes.</p>

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<p class="prompt">5. A recipe requires 1.5 cups of sugar per batch. How many cups are needed for 6 batches?</p>

</div>

<p><span class="label">Explanation:</span> 1.5 × 6 = 9 cups.</p>

</div>

<h2 id="distance">3. Rate, Distance, and Time</h2>

<p>Distance, rate, and time are linked by the formula <strong>d = rt</strong>. Rearrange it as needed: r = d ÷ t, or t = d ÷ r.</p>

<p><strong>Worked Example:</strong> A car travels at 60 mph for 3.5 hours. How far does it travel?</p>

<p class="step-math">d = rt = 60 × 3.5 = 210 miles</p>

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<p class="prompt">1. A cyclist rides 45 miles in 3 hours. What is the cyclist's average speed?</p>

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<p><span class="label">Explanation:</span> r = d ÷ t = 45 ÷ 3 = 15 mph.</p>

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<p class="prompt">2. A train travels at 80 mph. How long does it take to travel 320 miles?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-2',true)">A) 4 hours</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-2',false)">B) 40 hours</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-2',false)">C) 2.5 hours</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-2',false)">D) 400 hours</button>

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<p><span class="label">Explanation:</span> t = d ÷ r = 320 ÷ 80 = 4 hours.</p>

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<p class="prompt">3. Two cities are 210 miles apart. A car travels at a constant speed and arrives in 3.5 hours. What is the car's speed?</p>

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<p><span class="label">Explanation:</span> r = d ÷ t = 210 ÷ 3.5 = 60 mph.</p>

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<p class="prompt">4. A boat travels downstream at 20 mph and upstream (against the current) at 12 mph, covering the same distance in each direction. If the total round trip takes 8 hours, what is the one-way distance?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-4',true)">A) 60 miles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">B) 96 miles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">C) 40 miles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">D) 120 miles</button>

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<p><span class="label">Explanation:</span> Let d be the one-way distance. Total time: <span class="frac"><span class="num">d</span><span class="den">20</span></span> + <span class="frac"><span class="num">d</span><span class="den">12</span></span> = 8. Using a common denominator of 60: <span class="frac"><span class="num">3d</span><span class="den">60</span></span> + <span class="frac"><span class="num">5d</span><span class="den">60</span></span> = <span class="frac"><span class="num">8d</span><span class="den">60</span></span> = 8, so 8d = 480 and d = 60.</p>

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<p class="prompt">5. Two trains start from the same station at the same time, traveling in opposite directions. One travels at 50 mph and the other at 70 mph. How far apart are they after 2.5 hours?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-5',true)">A) 300 miles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">B) 250 miles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">C) 175 miles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">D) 60 miles</button>

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<p><span class="label">Explanation:</span> Traveling in opposite directions, their speeds combine: 50 + 70 = 120 mph. Distance apart: 120 × 2.5 = 300 miles.</p>

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

<p>Some rate problems require finding an intermediate unit rate first, like a scale factor or a per-worker/per-machine rate, then applying it to a new situation.</p>

<p><strong>Worked Example:</strong> A map has a scale of 1 inch = 25 miles. If two cities are 4.5 inches apart on the map, how far apart are they in reality?</p>

<p class="step-math">x = 4.5 × 25 = 112.5 miles</p>

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<p class="prompt">1. A map has a scale of 2 cm = 15 km. If two towns are 7 cm apart on the map, what is the actual distance between them?</p>

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<p><span class="label">Explanation:</span> <span class="frac"><span class="num">2</span><span class="den">15</span></span> = <span class="frac"><span class="num">7</span><span class="den">x</span></span>. Cross-multiply: 2x = 105, so x = 52.5 km.</p>

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<p class="prompt">2. A factory produces 240 units in 6 hours using 4 machines. At the same rate, how many units would 6 machines produce in 6 hours?</p>

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<p><span class="label">Explanation:</span> Rate per machine per hour: 240 ÷ (4 × 6) = 240 ÷ 24 = 10 units. With 6 machines for 6 hours: 6 × 6 × 10 = 360 units.</p>

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<p class="prompt">3. It takes 5 workers 8 days to complete a job. Working at the same rate, how many days would it take 10 workers to complete the same job?</p>

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<p><span class="label">Explanation:</span> The total work is 5 × 8 = 40 worker-days. With 10 workers: 40 ÷ 10 = 4 days.</p>

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<p class="prompt">4. A car uses 8 gallons of gas to travel 240 miles. At this rate, how many gallons are needed to travel 420 miles?</p>

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<p><span class="label">Explanation:</span> Unit rate: 240 ÷ 8 = 30 miles per gallon. Gallons needed: 420 ÷ 30 = 14.</p>

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

<li><strong>Setting up a proportion with mismatched units in the numerators or denominators.</strong> Keep the same quantity type in both numerators and the same quantity type in both denominators, mixing them up flips the answer.</li>

<li><strong>Averaging two speeds instead of using total distance over total time.</strong> If a trip has two different speeds over unequal times, the overall average speed is total distance ÷ total time, not a simple average of the two rates.</li>

<li><strong>Adding rates when they should be combined by distance or subtracted, or vice versa.</strong> Objects moving toward or away from each other add their speeds; objects moving in the same direction subtract them. Read the direction described carefully.</li>

<li><strong>Forgetting to convert units before comparing or combining quantities.</strong> Mixing hours with minutes, or miles with feet, without converting first, is one of the most common sources of errors on rate problems.</li>

<li><strong>Assuming a work-rate problem scales in a straight line without checking the relationship.</strong> More workers means less time (an inverse relationship), while more time at a fixed rate means more work completed (a direct relationship). Confusing which applies leads to the reciprocal of the correct answer.</li>

</ul>

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

<h3>What's the best way to keep unit conversions organized?</h3>

<p>Write out each conversion factor as a fraction, arranged so the unit you want to cancel appears once on top and once on the bottom. Multiplying several of these fractions in a row, and canceling as you go, avoids errors far more reliably than trying to do the conversion in your head.</p>

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<h3>How do I know whether a rate problem needs addition or subtraction of the two rates?</h3>

<p>If two objects move toward each other or apart from each other, their rates add, since the distance between them changes at the combined rate. If they move in the same direction, subtract the rates to find how quickly one catches up to (or falls behind) the other.</p>

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<h3>Why does adding more workers reduce completion time instead of dividing evenly?</h3>

<p>Work problems rely on total "worker-days" (or a similar combined unit) staying constant for a fixed job: workers × days = a constant amount of work. As the number of workers goes up, the number of days needed goes down proportionally, an inverse relationship, not a simple even split.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/act-rates-ratios-quiz-1">Rates & Ratios 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-rates-ratios-quiz-1">Practice Rates & Ratios 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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