Fractions, Decimals, and Percentages
Fractions, decimals, and percentages are three ways to express the same thing: a part of a whole. Being fluent in all three — and in moving between them — is one of the most practical maths skills you can develop.
What Each Notation Means
A fraction expresses a ratio directly: 3/4 means "3 out of every 4 equal parts". The number on top is the numerator; the number on the bottom is the denominator. Fractions are exact and preserve the relationship between part and whole clearly.
A decimal writes the same quantity using the base-10 place-value system extended beyond the ones column. The decimal 0.75 means 7 tenths and 5 hundredths — the same as 3/4.
A percentage is a special fraction whose denominator is always 100. The symbol % literally means "per hundred". So 75% = 75/100 = 0.75 = 3/4. Percentages are useful for comparisons because they standardise everything to a scale of 100.
Converting Fractions to Decimals
Divide the numerator by the denominator. For 3/4: 3 ÷ 4 = 0.75. For 5/8: 5 ÷ 8 = 0.625. Some fractions produce terminating decimals (the division ends). Others produce repeating decimals — for example, 1/3 = 0.333… (the 3 repeats forever) and 1/7 = 0.142857142857…
Fractions whose denominators have only 2 and 5 as prime factors always terminate (because 10 = 2 × 5). Fractions with any other prime factors in the denominator produce repeating decimals.
Converting Decimals to Fractions
Write the decimal digits as the numerator and the appropriate power of 10 as the denominator, then simplify:
- 0.6 = 6/10 = 3/5
- 0.75 = 75/100 = 3/4
- 0.125 = 125/1000 = 1/8
- 0.04 = 4/100 = 1/25
For a repeating decimal, an algebraic trick works: let x = 0.333…, then 10x = 3.333…, subtract to get 9x = 3, so x = 3/9 = 1/3.
Converting Between Fractions and Percentages
To convert a fraction to a percentage, multiply by 100: 3/4 × 100 = 75%. To convert a percentage to a fraction, divide by 100 and simplify: 45% = 45/100 = 9/20.
Alternatively, work through the decimal form: fraction → decimal → percentage (multiply by 100) or percentage → decimal (divide by 100) → fraction.
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 ≈ 33.3% | 2/3 ≈ 0.667 ≈ 66.7%
Adding and Subtracting Fractions
To add or subtract fractions, they must share a common denominator. Find the lowest common denominator (LCD), convert each fraction, then add or subtract the numerators:
1/3 + 1/4: LCD = 12. Convert: 4/12 + 3/12 = 7/12.
5/6 − 1/4: LCD = 12. Convert: 10/12 − 3/12 = 7/12.
With decimals or percentages, addition and subtraction are straightforward: just align decimal points (for decimals) or add the numbers and keep the % sign.
Multiplying and Dividing Fractions
Multiplying fractions is simple: multiply numerators together, multiply denominators together, and simplify. 2/3 × 3/5 = 6/15 = 2/5. You can also cancel common factors before multiplying ("cross-cancel") to keep the numbers small.
Dividing by a fraction means multiplying by its reciprocal (flip it). 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
Percentage Problems: Three Types
Almost every percentage problem is one of three types:
- Find the percentage of a number: What is 35% of 80? → 0.35 × 80 = 28.
- Find what percentage one number is of another: 21 is what % of 60? → (21/60) × 100 = 35%.
- Find the original value: 45 is 90% of what? → 45 ÷ 0.90 = 50.
Percentage change is calculated as: ((new value − old value) / old value) × 100. If a price rises from $40 to $46, the percentage increase is ((46 − 40) / 40) × 100 = 15%.
To find 10% of any number, just move the decimal point one place left. 10% of 340 = 34. To find 5%, halve that: 17. To find 15%, add them: 34 + 17 = 51. To find 20%, double the 10%: 68. This approach handles most percentage calculations in your head.
Simplifying Fractions
A fraction is in its simplest (lowest) form when the numerator and denominator share no common factor other than 1. To simplify, divide both by their greatest common factor (GCF). For 18/24: GCF is 6, so 18/24 = 3/4. You can find the GCF by prime factorisation or by the Euclidean algorithm (repeatedly applying division with remainder until the remainder is 0).
Summary
Fractions, decimals, and percentages represent the same quantities in different forms. Converting between them relies on two basic operations: division (fraction to decimal) and multiplication or division by 100 (decimal to/from percentage). Memorising the key benchmarks speeds up mental arithmetic considerably. The three types of percentage problem — finding a percentage, finding what percentage, and finding the whole from a percentage — cover the vast majority of real-world applications, from discounts and tax rates to statistics and science data.