Percentages
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
A percentage means “per hundred” — 35% = 35/100 = 0.35. To convert between forms: percentage to decimal by dividing by 100; decimal to percentage by multiplying by 100; fraction to percentage by multiplying by 100 (and simplifying if possible). For example, 3/8 = 0.375 = 37.5%.
Essential Formulas:
- x% of y = (x/100) × y
- Express a as a percentage of b: (a/b) × 100%
- Percentage increase: ((new − original) / original) × 100%
- Percentage decrease: ((original − new) / original) × 100%
- Original value from percentage: if x% of original = value, original = value × (100/x)
- Percentage change can be negative (decrease) or positive (increase)
Key Facts:
- To increase a value by 15%, multiply by 1.15 (the growth factor = 1 + 15/100)
- To decrease by 15%, multiply by 0.85 (the reduction factor = 1 − 15/100)
- Successive percentage changes multiply: a 10% increase followed by a 20% increase = ×1.10 ×1.20 = ×1.32 (32% total increase, NOT 30%)
- Reverse percentage: if a price is reduced by 25% to Rp 150,000, original = 150,000 / 0.75 = Rp 200,000
⚡ Exam Tip: In the UI entrance test, percentage questions frequently appear in real-world contexts — shopping discounts, population growth, interest rates, and exam score improvements. The key trap is that successive percentage changes compound, not add. A 10% increase then 10% decrease does not return to the original value — it goes to 1.10 × 0.90 = 0.99 = 99% of the original, a 1% loss.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Basic Percentage Conversions
Converting between fractions, decimals, and percentages:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… | 33.33% |
| 2/3 | 0.666… | 66.67% |
These common conversions appear frequently — learn them.
Finding a Percentage of a Quantity
Example: A school has 800 students. 35% study Science. How many Science students? 35% of 800 = (35/100) × 800 = 0.35 × 800 = 280 students.
Example: A jacket costs Rp 450,000 with a 15% discount. What is the sale price? Discount amount = 15% of 450,000 = 0.15 × 450,000 = 67,500. Sale price = 450,000 − 67,500 = 382,500. OR directly: 450,000 × 0.85 = 382,500.
Finding What Percentage One Value is of Another
Formula: (value / total) × 100%
Example: In a survey of 1,200 people, 330 said they preferred tea. What percentage preferred tea? (330 / 1200) × 100% = 0.275 × 100% = 27.5%.
Percentage Increase and Decrease
Example (increase): A town’s population of 45,000 grows to 51,300. What is the percentage increase? Change = 51,300 − 45,000 = 6,300. Percentage increase = (6,300 / 45,000) × 100% = 0.14 × 100% = 14%.
Example (decrease): A car worth Rp 80,000,000 depreciates to Rp 68,000,000. What is the percentage depreciation? Change = 80,000,000 − 68,000,000 = 12,000,000. Percentage decrease = (12,000,000 / 80,000,000) × 100% = 15%.
Reverse Percentage Problems
If you know the result of a percentage change and want to find the original value:
Example: After a 20% discount, a laptop costs Rp 8,800,000. What was the original price? Let original = P. After 20% discount: P × 0.80 = 8,800,000. P = 8,800,000 / 0.80 = 11,000,000.
Successive Percentage Changes
When two or more percentage changes apply in sequence, multiply the growth/reduction factors.
Example: A product costs Rp 200,000. Its price increases by 10% in January, then another 15% in February. What is the final price? After January: 200,000 × 1.10 = 220,000. After February: 220,000 × 1.15 = 253,000. Or directly: 200,000 × 1.10 × 1.15 = 200,000 × 1.265 = 253,000. Overall increase: 26.5% (NOT 25%).
Compound Percentage — Repeated Change
Example: A bank’s savings account pays 8% interest per year, compounded annually. If you deposit Rp 5,000,000, how much do you have after 3 years? Year 1: 5,000,000 × 1.08 = 5,400,000 Year 2: 5,400,000 × 1.08 = 5,832,000 Year 3: 5,832,000 × 1.08 = 6,298,560 Or formula: 5,000,000 × (1.08)³ = 5,000,000 × 1.2597 = 6,298,560.
Percentage Points vs Percent
When comparing percentages, “percentage points” refers to the absolute difference between two percentages, while “percent” refers to the relative change.
Example: Interest rates rise from 10% to 12%. The increase is 2 percentage points, but the relative increase is (2/10) × 100% = 20%.
Problem-Solving Strategies:
- For discount problems, either subtract the discount amount or multiply by the remaining percentage — both should give the same answer
- When asked for “the original price” after a percentage change, divide by the factor (e.g., 0.80 if it was reduced by 20%)
- For successive changes, calculate step by step to avoid confusion
- In mixed problems (increase then decrease), the order matters — a 20% increase then 20% decrease on Rp 100,000 gives 100,000 × 1.20 × 0.80 = 96,000 (a net loss)
Common Mistakes:
- Adding percentage changes instead of multiplying factors: 10% + 20% ≠ 30% when applied successively
- Confusing “percentage of” with “what percentage”: “35% of 200” = 70, but “70 is what percentage of 200” = 35%
- Forgetting that a percentage decrease followed by the same percentage increase does not return to the original value
- In reverse percentage, dividing by (1 − percentage/100) instead of the correct factor
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Percentage Change Derivation
Percentage increase from original O to new N: Change = N − O. Percentage increase = ((N − O) / O) × 100%. If N = O × (1 + r), then percentage increase = (O × r / O) × 100% = r × 100%.
This generalises: N = O × (1 + r₁) × (1 + r₂) × … for successive changes, where r can be positive (increase) or negative (decrease).
Solving Complex Percentage Problems
Example: The price of a product first increases by 20%, then decreases by 20%. The final price is Rp 96,000. Find the original price. Let original = P. After +20%: P × 1.20. After −20%: P × 1.20 × 0.80 = P × 0.96. 0.96P = 96,000 → P = 96,000 / 0.96 = 100,000.
Example: A town’s population decreases by 10% per year. After 3 years, the population is 72,900. What was the original population? Each year: multiply by 0.90. After 3 years: P × (0.90)³ = 72,900. (0.90)³ = 0.729. So P × 0.729 = 72,900 → P = 72,900 / 0.729 = 100,000.
Profit and Loss Percentages
Selling Price (SP) and Cost Price (CP):
- Profit = SP − CP; Profit% = (Profit/CP) × 100%
- Loss = CP − SP; Loss% = (Loss/CP) × 100%
- If profit is x% on cost: SP = CP × (1 + x/100)
- If selling at a loss of x%: SP = CP × (1 − x/100)
Example: A trader buys goods for Rp 600,000 and sells them to make a 25% profit. What is the selling price? SP = 600,000 × 1.25 = 750,000.
Markup and Markdown
Markup: adding a percentage to the cost price to determine the selling price (retail context). Markdown: reducing the marked price to determine the sale price.
Example: A retailer marks up a product by 40% of cost (Rp 200,000 cost → Rp 280,000 marked price). During a sale, it is marked down by 20%. Sale price = 280,000 × 0.80 = 224,000. Overall profit = (224,000 − 200,000) / 200,000 × 100% = 12%.
Percentage in Data Interpretation
In charts and tables, percentages are often used to show composition:
- A segment in a pie chart representing 25% of the total: angle = 90°.
- If a bar chart shows values with percentages written on it, read both the value and percentage.
Example: A budget of Rp 500,000,000 is allocated: Education 35%, Health 25%, Infrastructure 30%, Administration 10%. Education allocation = 35% of 500M = 0.35 × 500M = 175M. If the Education budget increases by 20%: new Education budget = 175M × 1.20 = 210M. New total budget (if only Education increases): 500M − 175M + 210M = 535M. New Education percentage of new total: 210M / 535M × 100% = 39.25%.
Interest Calculations
Simple Interest: I = P × r × t (P = principal, r = rate per year as decimal, t = years). Example: Rp 1,000,000 at 6% simple interest for 3 years: I = 1,000,000 × 0.06 × 3 = 180,000. Total = 1,180,000.
Compound Interest (compounded annually): A = P(1 + r)^t. Example: Rp 1,000,000 at 6% compound interest for 3 years: A = 1,000,000 × (1.06)³ = 1,000,000 × 1.191016 = 1,191,016.
UI Entrance Exam Patterns
Percentage questions in the UI Academic Potential test often involve:
- Direct percentage calculation (finding a percentage of a number)
- Expressing one quantity as a percentage of another
- Percentage increase/decrease in context
- Reverse percentages (finding original value)
- Successive percentage changes (often involving multiple steps)
- Percentage in data presentation (reading charts)
⚡ Advanced Exam Tip: When a question involves multiple successive percentage changes, resist the urge to add or average the percentages. Always multiply the individual factors. If an answer choice matches the result of adding percentages (a common trap), it is likely wrong.
Content adapted based on your selected roadmap duration. Switch tiers using the selector above.