Fractions, Decimals, and Percentages
Three representations of the same thing — parts of a whole — and how to move fluently between them.
The Same Idea, Three Representations
A fraction, a decimal, and a percentage are three ways of writing the same quantity — a part of a whole. Fluency means being able to move between them without thinking.
1/4 = 0.25 = 25%
The choice of representation is about context: fractions for exact values and algebra, decimals for calculation, percentages for communication.
Fractions
A fraction p/q means p parts of a whole divided into q equal parts.
- Numerator (p) — how many parts you have
- Denominator (q) — how many parts the whole is divided into
Equivalent fractions
Multiplying or dividing numerator and denominator by the same number gives an equivalent fraction:
1/2 = 2/4 = 3/6 = 50/100
Simplifying
Divide numerator and denominator by their GCD (greatest common divisor):
18/24 → GCD(18,24) = 6 → 3/4
Operations
Addition/Subtraction — common denominator required:
1/3 + 1/4 = 4/12 + 3/12 = 7/12
Multiplication — straight across:
2/3 × 3/5 = 6/15 = 2/5
Division — multiply by the reciprocal:
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Mixed numbers vs improper fractions
2¾ = 11/4 (multiply whole by denominator, add numerator)
11/4 = 2¾ (divide numerator by denominator: quotient + remainder/denominator)
Decimals
A decimal is a fraction with a power of 10 in the denominator, written in positional notation.
0.375 = 3/10 + 7/100 + 5/1000 = 375/1000 = 3/8
Decimal places
Each position to the right of the decimal point is a power of 10:
| Position | Value |
|---|---|
| Tenths | 1/10 |
| Hundredths | 1/100 |
| Thousandths | 1/1000 |
Terminating vs repeating
- Terminating: the decimal ends — only when the denominator’s prime factors are 2s and 5s only
- 1/4 = 0.25, 1/8 = 0.125, 3/20 = 0.15
- Repeating: a block of digits cycles forever — all other fractions
- 1/3 = 0.333…, 1/7 = 0.142857142857…, 1/11 = 0.090909…
Converting repeating decimal to fraction
x = 0.333...
10x = 3.333...
10x − x = 3
9x = 3
x = 1/3
Percentages
Percent means “per hundred.” A percentage is a fraction with denominator 100.
37% = 37/100 = 0.37
Converting
- Fraction → percentage: multiply by 100
- 3/8 = 0.375 = 37.5%
- Percentage → decimal: divide by 100
- 65% = 0.65
- Percentage → fraction: put over 100, simplify
- 40% = 40/100 = 2/5
Finding a percentage
“What is 35% of 80?”
0.35 × 80 = 28
Percentage of, percentage change
- What percent is A of B? → (A/B) × 100
- Percentage increase: ((new − old) / old) × 100
- Percentage decrease: ((old − new) / old) × 100
Price rises from 200 to 250:
((250 − 200) / 200) × 100 = 25% increase
Percentage points vs percent
A common mistake. If a rate rises from 10% to 15%, that’s:
- 5 percentage points increase (arithmetic difference)
- 50% increase (relative change: 5/10 × 100)
These are different things. Percentage points are absolute; percent change is relative.
The Key Conversions
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.1666… | 16.7% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 2/3 | 0.666… | 66.7% |
| 3/4 | 0.75 | 75% |
| 3/8 | 0.375 | 37.5% |
These are worth memorising. They’re anchors — knowing 1/8 = 12.5% lets you instantly compute 5/8 = 62.5%.
Compound Interest — Fractions and Percentages in Action
If you invest £P at annual rate r% for n years:
Final = P × (1 + r/100)ⁿ
At 10% for 7 years:
P × (1.1)⁷ ≈ P × 1.95
The rule of 72: divide 72 by the interest rate to get the approximate doubling time in years. At 10%, money doubles in ≈ 7.2 years.
This works because ln(2) ≈ 0.693, and for small r, (1 + r/100)ⁿ ≈ 2 when n ≈ 69.3/r — which 72 approximates (chosen because it has many divisors).