Percentages
Part of Mathematics
Using percentages to express proportions, calculate shares, understand growth, and communicate clearly about quantities.
Why This Matters
Percentages are the language of proportion. When a healer says a treatment succeeds 70% of the time, when a miller keeps 5% of grain as a milling fee, when a population has grown 12% in a year — these statements use percentages because they allow direct comparison regardless of the underlying quantities. You can immediately compare a 70% success rate with a 65% success rate without knowing the absolute numbers.
In rebuilding, percentages appear constantly: taxation rates, survival statistics, crop yield improvements, material waste factors, nutritional content of foods, interest rates for loans, and error tolerance in construction. A community whose leaders and craftspeople understand percentages makes better decisions and is harder to deceive with misleading statistics.
Percentages are also the bridge between fractions and decimal notation. Understanding that 25% = 1/4 = 0.25 — all three representations of the same proportion — gives you flexibility in calculation and prevents the mental blocks that come from treating these as three separate concepts.
What a Percentage Is
“Percent” means “per hundred” — from the Latin per centum. A percentage is a ratio expressed with 100 as the denominator.
- 50% means 50 out of every 100, or 50/100 = 1/2
- 25% means 25 out of every 100, or 25/100 = 1/4
- 1% means 1 out of every 100, or 1/100
Converting between forms:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/3 | 0.333… | 33.3% |
| 2/3 | 0.667… | 66.7% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/20 | 0.05 | 5% |
| 1/100 | 0.01 | 1% |
Memorize the highlighted ones. For fractions not in the table: divide the numerator by the denominator to get the decimal, then multiply by 100.
Example: 3/8 as a percentage: 3 ÷ 8 = 0.375; 0.375 × 100 = 37.5%
Core Calculations
There are three fundamental percentage calculations. Every other percentage problem is a variation on these.
1. Finding a Percentage of a Number
“What is X% of N?”
Formula: Result = (X ÷ 100) × N
Shortcut: Move the decimal point two places left to get 1%, then multiply.
Example: What is 15% of 240?
- 1% of 240 = 2.4
- 15% = 2.4 × 15 = 36
Faster for common percentages:
- 10%: divide by 10 → 240 ÷ 10 = 24
- 5%: half of 10% → 24 ÷ 2 = 12
- 15%: 10% + 5% → 24 + 12 = 36 ✓
- 25%: divide by 4 → 240 ÷ 4 = 60
- 50%: divide by 2 → 240 ÷ 2 = 120
2. Finding What Percentage One Number Is of Another
“N is what percentage of M?”
Formula: Percentage = (N ÷ M) × 100
Example: 36 is what percentage of 240?
- 36 ÷ 240 = 0.15
- 0.15 × 100 = 15%
Example (harvest waste): You planted 80 kg of seed and harvested 520 kg of grain. What percentage return is that?
- 520 ÷ 80 = 6.5
- 6.5 × 100 = 650%
- The harvest returned 650% of the seed invested (a 6.5× return, or 550% gain)
3. Finding the Original Number from a Percentage
“X is P% of what number?”
Formula: Original = X ÷ (P ÷ 100) = X × (100 ÷ P)
Example: 36 is 15% of what number?
- Original = 36 ÷ 0.15 = 240
Practical use: You have received 42 kg of grain as your 7% milling fee. How much grain passed through your mill?
- Original = 42 ÷ 0.07 = 600 kg
Percentage Change
One of the most important applications is measuring change over time: population growth, harvest improvement, price change.
Percentage increase formula: Increase % = ((New − Old) ÷ Old) × 100
Percentage decrease formula: Decrease % = ((Old − New) ÷ Old) × 100
Example: Last year’s harvest was 1,200 kg. This year it is 1,380 kg. What is the increase?
- Increase = 1,380 − 1,200 = 180 kg
- Increase % = (180 ÷ 1,200) × 100 = 15%
Example: Population was 340 last year, now 323. What is the decrease?
- Decrease = 340 − 323 = 17
- Decrease % = (17 ÷ 340) × 100 = 5%
Percentage Change is Not Symmetric
A 50% decrease followed by a 50% increase does NOT return you to the original value. If you start with 100, lose 50% (→50), then gain 50% (→75), you end at 75, not 100. This asymmetry matters for understanding population recovery and resource depletion.
Applying Percentages to Multiple Steps
Finding the final value after a percentage change:
Final = Original × (1 + Rate) for increase Final = Original × (1 − Rate) for decrease
Where rate is the percentage expressed as a decimal.
Example: Grain supply of 800 kg decreases by 20% due to spoilage.
- Final = 800 × (1 − 0.20) = 800 × 0.80 = 640 kg
Chaining percentage changes:
- Start: 800 kg
- After 20% loss: 640 kg
- After 10% gain from new harvest: 640 × 1.10 = 704 kg
Practical Community Applications
Taxation and Tithes
A common system is a percentage tithe: 10% of harvest to community store.
Example: Three households harvest 320 kg, 185 kg, and 410 kg. What is each tithe?
- 10% of 320 = 32 kg
- 10% of 185 = 18.5 kg → round to 18 or 19 by convention
- 10% of 410 = 41 kg
- Total community store addition: 91 kg
Waste and Yield Factors
No process is 100% efficient. Standard waste factors help with planning:
| Process | Typical Waste |
|---|---|
| Grain milling (stone mill) | 10–15% |
| Timber cutting (saw waste) | 20–25% |
| Clay brick firing (cracking) | 5–10% |
| Tanning hides | 10–15% |
| Rendering lard | 15–20% |
Example: You need 500 usable bricks. Expecting 8% cracking, how many do you fire?
- You need 100% of final bricks to be 92% of what you fire (100% − 8% = 92%)
- Bricks to fire = 500 ÷ 0.92 = 543 (round up to 545)
Loan Interest
If your community uses simple interest on loans of grain or tools:
Simple interest: Interest = Principal × Rate × Time
Example: You lend 200 kg of grain for 1 season at 15% interest.
- Interest = 200 × 0.15 × 1 = 30 kg
- Borrower repays 230 kg at harvest
Checking Your Work
Always sanity-check percentage results:
- Order of magnitude: 15% of 240 should be between 10% (24) and 20% (48). Answer 36 ✓
- Direction of change: An increase should give a higher number; a decrease a lower one
- Reverse check: If X is P% of N, then N × P/100 should equal X
Common errors:
- Confusing percentage of with percentage change: “prices rose 10%” means multiply by 1.10, not by 0.10
- Applying percentages to the wrong base: “20% discount off the sale price” vs. “20% off the original price” can differ significantly
- Adding percentage changes directly: a 30% gain does not cancel a 30% loss (see asymmetry warning above)
Fluency with percentages is one of the most practical mathematical skills for everyday community life. It enables fair taxation, accurate planning, clear communication about risk and yield, and resistance to manipulation by misleading statistics.