<h3>The Main Idea</h3> <p><strong>Compare the same things in the same way.</strong></p> <p>This is the single most important concept for solving proportion problems. When you set up a proportion, both sides must compare the same quantities in the same order.</p> <h3>What is a Part-to-Part Proportion?</h3> <p>A part-to-part proportion compares two parts of a whole to each other. When you see "the ratio of A to B is 3:5", you're comparing parts to parts, not parts to the total.</p> <h3>Setting Up the Proportion</h3> <ol> <li><strong>Read the question sentence first</strong> — Know what you're solving for before reading the details.</li> <li><strong>Identify the comparison</strong> — What two things are being compared?</li> <li><strong>Write the known ratio</strong> — Put the given ratio as one fraction.</li> <li><strong>Write the unknown ratio</strong> — Use the same comparison order with your variable.</li> <li><strong>Solve</strong> — Cross-multiply and divide, or look for a convenient multiplier.</li> </ol> <h3>Finding the Total from Parts</h3> <p>If a problem gives you a part-to-part ratio (like 3:5) but asks for the total:</p> <ul> <li>Add the ratio parts: 3 + 5 = 8 total parts</li> <li>Now you can compare any part to the total (e.g., 3:8 or 5:8)</li> </ul>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Compare the same things in the same way.</strong></p> <p>This is the same core principle from part-to-part proportions. The difference? Now one side of your comparison is the <strong>total</strong>, not another part.</p> <h3>What is a Part-to-Total Proportion?</h3> <p>A part-to-total proportion compares one part of a group to the whole group. When you see "the ratio of A to total is 3:20" or "6 out of every 20 are fiction," you're comparing a part to the total.</p> <h3>The Two Scenarios</h3> <p>Part-to-total problems come in two forms:</p> <ol> <li><strong>Total is already in the ratio</strong> — "The ratio of honor students to total students is 3:20" <ul> <li>The 20 IS the total — use it directly</li> <li>Don't add anything — you're already comparing part to total</li> </ul> </li> <li><strong>Total is NOT in the ratio</strong> — "The ratio of boys to girls is 4:7" <ul> <li>You have part-to-part, but the question asks for total</li> <li>Add the ratio parts: 4 + 7 = 11 total parts</li> <li>Now you can compare any part to the total (4:11 or 7:11)</li> </ul> </li> </ol> <h3>The "X out of Y" Format</h3> <p>When a problem says "6 out of every 20 books are fiction":</p> <ul> <li>6 = the part (fiction)</li> <li>20 = the total</li> <li>This IS a part-to-total ratio (6:20)</li> </ul> <h3>Finding the Complement</h3> <p>Sometimes you need the "other" part. If 6 out of 20 are fiction and you need non-fiction:</p> <ul> <li>Non-fiction = Total − Fiction = 20 − 6 = 14</li> <li>Now compare non-fiction to total: 14:20</li> </ul> <h3>Common Pitfall</h3> <p>The biggest mistake: using numbers from the ratio without checking if they match what you're comparing. If you're comparing "boys to total" but the ratio gives "boys to girls," you'll get the wrong answer. Always verify your setup matches your goal.</p>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Compare the same things in the same way — even with three parts.</strong></p> <p>Triple ratios look intimidating, but the core principle doesn't change. You still set up a proportion comparing just two things — you just need to identify which two you actually need.</p> <h3>What is a Triple Ratio?</h3> <p>A triple ratio compares three quantities at once, written as A:B:C. For example, "the ratio of juniors to seniors to veterans is 2:3:1" means for every 2 juniors, there are 3 seniors and 1 veteran.</p> <h3>The Two Types of Triple Ratio Questions</h3> <p><strong>Type 1: Find one part from another part</strong></p> <ul> <li>You're given one part and asked for a different part</li> <li>The third ratio value is <em>extra information</em> — ignore it</li> <li>Set up a proportion using only the two parts you need</li> </ul> <p><strong>Type 2: Find the total</strong></p> <ul> <li>You're given one part and asked for the total</li> <li>First, add all ratio parts to get "total parts"</li> <li>Then set up a proportion comparing that part to the total</li> </ul> <h3>The Approach</h3> <ol> <li><strong>Read the question sentence first</strong> — What are you looking for? What are you given?</li> <li><strong>Label the ratio parts</strong> — Write out what each number represents (e.g., 5:8:6 = brushes:markers:crayons)</li> <li><strong>Identify what you need</strong> — Do you need two parts, or a part and the total?</li> <li><strong>For totals:</strong> Add all ratio parts first (2+3+1 = 6 total parts)</li> <li><strong>Set up the proportion</strong> — Compare the same things in the same way</li> <li><strong>Solve</strong> — Scale factor or cross-multiply</li> </ol> <h3>Common Pitfall</h3> <p>Don't try to use all three ratio numbers in one proportion. A proportion only compares <em>two</em> things. With triple ratios, you either:</p> <ul> <li>Use two of the three parts (ignoring the extra one), OR</li> <li>Use one part and the sum of all parts (for total questions)</li> </ul>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Find the middleman, make it match, then connect the endpoints.</strong></p> <p>Chained ratio problems give you two separate ratios that share a common term (the "middleman"). Your job is to connect the endpoints by using that middleman as a bridge.</p> <h3>What is a Chained Ratio?</h3> <p>A chained ratio connects two ratios through a shared term. For example:</p> <ul> <li>A:B = 4:6</li> <li>B:C = 6:5</li> </ul> <p>Here, B is the middleman connecting A to C. You want to find A:C.</p> <h3>The Two Cases</h3> <p><strong>Case 1: Middleman Already Matches</strong></p> <p>If the middleman value is the same in both ratios (like B=6 in both), just combine them directly:</p> <ul> <li>A:B = 4:6 and B:C = 6:5</li> <li>Since B=6 in both → A:B:C = 4:6:5</li> <li>Answer: A:C = 4:5</li> </ul> <p><strong>Case 2: Middleman Doesn't Match</strong></p> <p>If the middleman values differ, you must scale the ratios to make them match:</p> <ul> <li>A:B = 5:6 and B:C = 9:4</li> <li>B is 6 in one ratio, 9 in the other — they don't match!</li> <li>Find LCM of 6 and 9 = 18</li> <li>Scale first ratio by 3: A:B = 15:18</li> <li>Scale second ratio by 2: B:C = 18:8</li> <li>Now combine: A:B:C = 15:18:8</li> <li>Answer: A:C = 15:8</li> </ul> <h3>The Approach</h3> <ol> <li><strong>Read the question sentence</strong> — What comparison are you looking for? (e.g., A:C)</li> <li><strong>Identify the middleman</strong> — Which term appears in both ratios?</li> <li><strong>Check if the middleman matches</strong> — Are the numbers the same?</li> <li><strong>If different, find the LCM</strong> — Scale both ratios so the middleman is equal</li> <li><strong>Combine the ratios</strong> — Write out A:B:C (or whatever the three terms are)</li> <li><strong>Extract the endpoints</strong> — Pull out just the two terms you need</li> </ol> <h3>Common Pitfall</h3> <p>Don't assume you can directly combine ratios when the middleman values are different! This is the #1 mistake. Always check if the shared term has the same value in both ratios before combining.</p>
Practice This Goal