Arithmetic
Percentages
How Bankers Invented Percentages
Italian bankers of the Middle Ages used the term **'per cento'** (per hundred) for calculating returns on loans. Gradually this was abbreviated to the symbol **%** - a stylized '/100' with two zeros.
Today percentages are the universal language of finance. Central banks manage economies by adjusting interest rates by fractions of a percent, and those changes affect millions of lives.
'50% off plus another 20%' - is that 70% off? No, only 60%! Percentages are everywhere: discounts, loans, statistics. Understanding percentages prevents manipulation and is the key to financial literacy.
- **Finance:** loans, deposits, investments
- **Commerce:** discounts, markups, taxes (VAT 20%)
- **Statistics:** ratings, polls, changes in indicators
What Is a Percentage
The word 'percent' comes from the Latin *per centum* - 'per hundred'. A **percent** is one hundredth of a number. It is a convenient way to compare proportions of different quantities.
A **percent (%)** is 1/100 of a number. 1% = 1/100 = 0.01 50% = 50/100 = 1/2 = 0.5 100% = 100/100 = 1 (the whole)
Percentages are convenient for comparison: 'a 20% discount' is clearer than '1/5 off' or 'a factor of 0.8'. Everyone speaks the same language - the language of hundredths.
What is 40% written as a fraction?
Three Types of Percentage Problems
All percentage problems reduce to three types. Mastering them is enough to solve any problem.
**Quick calculations:** • 10% of X = X / 10 (shift the decimal point) • 5% of X = half of 10% • 1% of X = X / 100 • 25% of X = X / 4 • 50% of X = X / 2
The key to solving: identify what is given (part? whole? percent?) and what needs to be found. Then substitute into the formula.
A class has 30 students, 18 of whom are girls. What percentage are girls?
Compound Interest
When interest is charged on interest - that's **compound interest**. It grows faster than simple interest.
**The magic of compound interest:** At 7% per year, an investment roughly doubles every ~10 years: • After 10 years: ×2 • After 20 years: ×4 • After 30 years: ×8 • After 40 years: ×16 The earlier the investing starts, the greater the effect!
Compound interest is the foundation of finance: bank deposits, loans, investments. Understanding this mechanism is critical for personal financial literacy.
A deposit of $1000 at 20% per year with compound interest. How much will there be after 2 years?
Percentage Traps
Percentages can be tricky! Many common mistakes come from misinterpreting them.
**Percentage points (pp):** Used when talking about a change in the percentages themselves: • Unemployment rose from 5% to 8% - rose by 3 pp. • But that's a 60% increase relative to the original value! Always clarify: percentage points or percent of the original.
In advertising and statistics, percentages are often used to mislead. Think critically: always ask 'percent of what?'
A rise of X% followed by a fall of X% returns to the original
Percentages are taken from DIFFERENT bases, so the result differs
+20% of 100 = 20, but -20% of 120 = 24. After rising 20% and falling 20%: 100 → 120 → 96. A 4% loss. This is the fundamental asymmetry of percentages.
A product's price rose 25%, then fell 25%. What is the final price relative to the original?
Key Ideas
- Percent = one hundredth: 1% = 0.01 = 1/100
- Three problem types: find the part, find the percent, find the whole
- Compound interest: interest on interest (exponential growth)
- Trap: +X% and -X% do NOT cancel each other out!
Related Topics
Percentages are a bridge between arithmetic and finance:
- Proportions — A percentage is a proportion with base 100
- Exponents — The compound interest formula
- Word Problems — Practical problems involving percentages
Вопросы для размышления
- Why do marketers love writing '30% off plus another 20%'?
- How to tell when percentages are being used incorrectly in the news?
- Why is compound interest called 'the eighth wonder of the world'?