Ratios, Proportions, and Percents: Comprehensive Study Guide

Foundations of Ratios and Proportions

• Ratios and Proportions Overview     * Definition of Ratios: A ratio represents a relationship between two related items or quantities.         * Example: In a classroom setting, the ratio of women to men might be expressed as 15 to 1 ($15:1$).     * Definition of Proportions: A proportion is defined as two ratios that are equal to one another.     * Solving Proportions: Proportions are solved through cross-multiplication.         * Mechanism: If a ratio is equal to another ratio ($\frac{a}{b} = \frac{c}{d}$), the cross products are equal ($ad = bc$).

• Conceptual Distinction: Fractions vs. Proportions     * Multiplying Fractions: When multiplying two fractions (e.g., $\frac{a}{b} \times \frac{c}{d}$), one should cross-cancel if possible and then multiply across the numerators and the denominators.     * Solving Proportions: In a proportion, where one ratio equals another, you set the cross products equal to each other to solve for a variable.

• Mathematical Example: Solving for $x$     * Given the proportion:         $\frac{x}{48} = \frac{47}{21}$     * Step 1: Set cross products equal:         $21x = 48 \times 47$     * Step 2: Simplify the product:         $21x = 2256$     * Step 3: Solve for $x$ by dividing both sides by 21:         $x = 107.4285...$     * Rounding Rule: If the instruction is to round to the nearest tenth, check the hundredths place. Because the hundredths digit is 2, the tenths digit stays the same. The final answer is $107.4$.

Algebraic Applications and Complex Proportions

• Incorporating Algebra     * Proportions serve as a tool for practicing linear equations.     * Distributive Property Warning: When a proportion contains a binomial (more than one term in a numerator or denominator), it is vital to use parentheses. This ensures that the multiplier is distributed to 18 to each term within the binomial.

• Algebraic Exercise 1: Solving for $z$     * Equation:         $\frac{-z}{6} = \frac{26}{-12}$     * Calculation:         $(-z) \times (-12) = 6 \times 26$         $12z = 156$         $z = 13$

• Algebraic Exercise 2: Solving with Binomials     * Equation:         $\frac{18}{90} = \frac{4}{w + 2}$     * Step 1: Cross multiply:         $18(w + 2) = 90 \times 4$     * Step 2: Distribute the 18:         $18w + 36 = 360$     * Step 3: Subtract 36 from both sides:         $18w = 324$     * Step 4: Divide by 18:         $w = 18$

• Algebraic Exercise 3: Simplified Variables     * Equation:         $\frac{3t}{10} = \frac{60}{50}$     * Step 1: Cross multiply:         $3t \times 50 = 10 \times 60$         $150t = 600$     * Step 2: Divide by 150:         $t = 4$

• Algebraic Exercise 4: Variables on Both Sides     * Equation:         $\frac{-9}{-3} = \frac{5y - 10}{2y - 4}$     * Step 1: Set up cross products with parentheses:         $-9(2y - 4) = -3(5y - 10)$     * Step 2: Distribute carefully (mind the signs):         $-18y + 36 = -15y + 30$     * Step 3: Strategy for variables—Bring the smallest variable to the other side to avoid sign errors and ensure a positive coefficient. Since $-18$ is smaller than $-15$, add $18y$ to both sides:         $36 = 3y + 30$     * Step 4: Isolate the term with $y$:         $6 = 3y$     * Step 5: Solve:         $y = 2$     * Pedagogical Note: For health care and medical scenarios involving inequalities, keeping variables positive prevents the need to flip the inequality sign when dividing.

Real-World Scenarios and Case Studies

• Diversity of Applications     * Proportions are universally applicable across disciplines including healthcare, business, geometry (angles), and finance.     * Anecdote: Proportions in Standardized Testing: The speaker recounts helping their eighth-grade child study for the PSAT. While complex trigonometric methods existed for an angle problem, the child solved it in seconds using only the principles of proportion.

• Healthcare Application: Medication Mixing     * Scenario: A nurse must mix Medication A with Medication B in constant proportions.     * Data: Last week, the doctor ordered $20\,mg$ of Medication A with $18\,mg$ of Medication B.     * Problem: This week, the order is only $9\,mg$ of Medication B. How much Medication A is required?     * Setup:         $\frac{\text{Medication A}}{\text{Medication B}} = \frac{20}{18} = \frac{x}{9}$     * Visual Shortcut: Since 18 divided by 2 is 9, you can simply divide 20 by 2 to get the answer.     * Standard Method:         $18x = 20 \times 9$         $18x = 180$         $x = 10\,mg$

• Business Application: Property Tax     * Scenario: In Chicago, a property assessed at $\$410,000$ has a tax of $\$8,200$. Tacoma uses a proportional rate.     * Problem: Calculate the tax for a property assessed at $\$690,000$.     * Setup:         $\frac{\text{Property Value}}{\text{Tax Amount}} = \frac{410,000}{8,200} = \frac{690,000}{x}$     * Calculation:         $410,000x = 8,200 \times 690,000$         $410,000x = 5,658,000,000$         $x = 13,800$     * Reasonableness Check: $\$13,800$ is larger than $\$8,200$, which makes sense as the property value increased from $\$410,000$ to $\$690,000$.

Proportions and Percentage Calculations

• Percentage Formula via Proportion     * Instead of moving decimals manually, one can use the proportion formula:         $\frac{\text{Percent}}{100} = \frac{\text{Part}}{\text{Whole}}$     * Advantage: Using this method eliminates the risk of decimal placement errors because the value is solved directly as a percentage.

• Conversion Rule: To convert a decimal to a percent manually, move the decimal point two places to the right (equivalent to multiplying by 100). Note that 'D' comes before 'P' in the alphabet (Decimal $\rightarrow$ Percent).

• Application: Percentage Change (Increase and Decrease)     * When calculating percentage increase or decrease, the formula is modified:         $\frac{\text{Percent}}{100} = \frac{\text{Amount of Change (Increase or Decrease)}}{\text{Original Amount}}$     * Important: The denominator (the "whole") is always the original value, not necessarily the larger or smaller number.

• Example: Calculating Percent Increase     * Scenario: A value increases from 200 to 340.     * Step 1: Find the amount of change:         $340 - 200 = 140$     * Step 2: Set up the proportion:         $\frac{x}{100} = \frac{140}{200}$     * Step 3: Solve:         $200x = 14000$         $x = 70\%$

• Example: Calculating Percent Decrease (Vehicle Depreciation)     * Story Context: The speaker mentions their personal experience with car depreciation due to accidents on Highway 1 and collisions with deer and uninsured students at Newlands Tech.     * Scenario (Eva): A car bought for $\$26,000$ is worth $\$14,000$ one year later. Find the percent decrease.     * Step 1: Calculate the decrease amount (the "part"):         $26,000 - 14,000 = 12,000$     * Step 2: Set up the proportion against the original cost (the "whole"):         $\frac{x}{100} = \frac{12,000}{26,000}$     * Step 3: Solve:         $26,000x = 1,200,000$         $x \approx 46.2\%$

• Example: Computer Depreciation     * Original cost: $\$800$. Current value: $\$650$.     * Step 1: Find the decrease amount:         $800 - 650 = 150$     * Step 2: Set up the proportion:         $\frac{x}{100} = \frac{150}{800}$     * Step 3: Solve:         $800x = 15,000$         $x = 18.75\%$

Questions & Discussion

• Interpreting Survey Data on Procrastination:     * Question: What proportion of male students responded that procrastination negatively impacted their performance in class?     * Data Provided:         1. Did not affect academic performance.         2. Negatively impacted performance in class: 1,138 males.         3. Total number of male students: 2,281.     * Interpretation Debate: There was a discussion about whether the denominator should be the absolute total of male students ($2,281$) or only those who provided valid responses. The class concluded to use the provided total, noting that students who didn't respond might have been procrastinating themselves.     * Result calculation:         $\text{Decimal} = \frac{1138}{2281} = 0.4989...$         $\text{Percent} = 49.9\%$

• Rounding for People:     * Question: How do we round when the answer represents human beings (e.g., predicted number of men at an event based on a ratio)?     * Response: In the women-to-men ratio problem where the result was $27.3$, the instructor noted you cannot have a tenth of a person. In such cases, round to the nearest whole number. The 3 tells the 7 to stay the same; therefore, the answer is 27 people.

• Rounding on Tests/Homework:     * Question: How many decimal places are required?     * Response: If the instruction is vague, rounding to the nearest tenth is usually acceptable for tests, and the instructor manually reviews scratch work to ensure fairness. For certain homework systems, entering the full decimal might be necessary to avoid being marked incorrect.