Percentage Study Guide

📖 Core Concepts Percent: A dimension‑less number that means “out of 100”. 45 % = \(45/100 = 0.45\). Reading: “Forty‑five percent” → \(\frac{45}{100}\) → 0.45. Other parts‑per units: ‰ (per mille) = \(1/1000\). Percent change: Relative to the original amount. A 10 % rise makes the new value \(1.10\)× the original. Percentage point: The absolute difference between two percent values (e.g., 10 % → 15 % is a 5‑percentage‑point increase). 📌 Must Remember Convert percent ↔ decimal: divide or multiply by 100. Formula for a percent change: final = original × \((1 + 0.01x)\) where \(x\) is the percent increase (negative for a decrease). Reversing a change is not symmetric: to undo a +25 % increase you need a –20 % decrease because \((1.25)(0.80)=1\). Compound changes: after \(x\)% then \(y\)%, final = original × \((1+0.01x)(1+0.01y)\). Percent of a percent: multiply the two decimals (e.g., 50 % of 40 % → \(0.5 \times 0.4 = 0.20 = 20\%\)). 🔄 Key Processes Convert a ratio to percent Method A: compute decimal \(= \frac{\text{numerator}}{\text{denominator}}\), then \(\times 100\). Method B (shortcut): \(\frac{\text{numerator} \times 100}{\text{denominator}}\). Find a percent of a quantity Change percent to decimal (\(\%/100\)), multiply by the base amount. Apply a percent change New amount = original × \((1 + 0.01x)\). Compound multiple percent changes Multiply each factor: \(\prod (1 + 0.01\text{change})\). 🔍 Key Comparisons Percent vs. Percentage point Percent: relative change (10 % of 100 = 10). Percentage point: absolute difference between two percent values (10 % → 15 % = +5 pp). Increase then decrease vs. Decrease then increase Same factors multiply, order does not matter: \((1+0.01x)(1-0.01x) = 1-(0.01x)^2\). Decimal conversion vs. Double‑division mistake Correct: \(25\% = 25/100 = 0.25\). Incorrect: \(\frac{25\%}{100} = 0.0025\) (you divided twice). ⚠️ Common Misunderstandings Dividing a percent by 100 while keeping the “%” sign → treats the percent as a raw number twice. Assuming a 25 % increase can be undone by a 25 % decrease (needs 20 % decrease). Confusing “5 % increase” with “5‑percentage‑point increase”. 🧠 Mental Models / Intuition “Per‑hundred” picture: imagine 100 equally sized pieces; the percent tells how many of those pieces you have. Compound factor: treat each percent change as a multiplier (e.g., +10 % → ×1.10). Multiplying all multipliers gives the overall effect. Gradient as slope: road grade = \(100 \times \frac{\text{rise}}{\text{run}}\); a 5 % grade means a 5‑unit rise for every 100‑unit run. 🚩 Exceptions & Edge Cases Zero or negative bases: percent formulas still work mathematically, but interpretation (e.g., “percent increase” of a negative amount) may be non‑intuitive. Large percent changes: the approximation “\(x\)% ≈ 0.01 x” holds for any \(x\), but for very large \(x\) the product \((1+0.01x)(1-0.01x)\) can become negative, indicating a sign reversal. 📍 When to Use Which Simple percent of a number → use decimal conversion (multiply). Percent change of a known original → use \((1+0.01x)\) multiplier. Multiple sequential changes → multiply each \((1+0.01\text{change})\) factor (compound formula). Comparing rates (interest, grade, concentration) → express as percentage points to avoid relative‑change confusion. 👀 Patterns to Recognize Whenever a problem gives “increase by \(x\)% then decrease by \(x\)%”, expect the result to be \(1-(0.01x)^2\) of the original. If a question asks for the “overall change” after two percent changes, look for the product of two \((1+0.01\text{change})\) terms. Gradient problems always involve the ratio “rise/run” multiplied by 100. 🗂️ Exam Traps Distractor: Using \(\frac{x\%}{100}\) instead of converting \(x\%\) to a decimal. Distractor: Treating a 5‑percentage‑point rise as a 5 % rise (the latter would give only a 0.5 % relative increase on a 10 % base). Distractor: Assuming symmetric reversal of percent changes (e.g., thinking a 30 % increase is undone by a 30 % decrease). Distractor: Forgetting to multiply by 100 when converting a ratio to a percent, leaving the answer as a decimal.