Fractions, decimals, and percentages are three different ways of expressing exactly the same thing: a part of a whole. One quarter, 0.25, and 25% all describe the same ratio. Yet these three forms appear in different contexts — fractions in recipes and probability, decimals in calculations and measurements, percentages in finance and statistics — and switching fluently between them is one of the most useful everyday mathematical skills there is. Once you see the conversion rules clearly, the forms stop feeling like separate topics and start feeling like dialects of the same language.
Fractions: Numerator Over Denominator
A fraction expresses a ratio of two whole numbers: the numerator (top) divided by the denominator (bottom). Three-quarters (¾) means 3 divided by 4. The denominator tells you how many equal parts the whole is divided into; the numerator tells you how many of those parts you're expressing.
Fractions can be proper (numerator smaller than denominator: ¾), improper (numerator larger: 7/4), or mixed numbers (a whole number plus a proper fraction: 1¾). Improper fractions and mixed numbers are equivalent representations: 7/4 = 1¾.
Simplifying fractions — reducing them to lowest terms — divides both numerator and denominator by their highest common factor. 12/16 simplifies to 3/4 (dividing both by 4). 15/25 simplifies to 3/5 (dividing both by 5). Simplified fractions are easier to compare and work with.
Fractions to Decimals
Converting a fraction to a decimal is straightforward: divide the numerator by the denominator. 3 ÷ 4 = 0.75. 2 ÷ 5 = 0.4. 1 ÷ 3 = 0.333... (a repeating decimal).
The key fractions to know by heart (they appear everywhere):
- 1/2 = 0.5
- 1/4 = 0.25; 3/4 = 0.75
- 1/5 = 0.2; 2/5 = 0.4; 3/5 = 0.6; 4/5 = 0.8
- 1/8 = 0.125; 3/8 = 0.375; 5/8 = 0.625; 7/8 = 0.875
- 1/3 = 0.333...; 2/3 = 0.667...
- 1/10 = 0.1; 1/100 = 0.01; 1/1000 = 0.001
Recognising these on sight removes the need for calculation in many everyday situations. A 25% discount is ¼ off. A 37.5% interest rate is 3/8 expressed as a percentage. Fluency between forms makes financial, statistical, and measurement calculations faster and less error-prone.
Decimals to Fractions
Converting a terminating decimal back to a fraction: write the decimal over the appropriate power of 10, then simplify. 0.75 = 75/100 = 3/4. 0.125 = 125/1000 = 1/8. 0.6 = 6/10 = 3/5.
Recurring decimals (like 0.333...) convert differently: 0.333... = 1/3 exactly. 0.666... = 2/3. 0.142857142857... = 1/7. These identities are worth knowing for probability and division calculations.
Fractions and Decimals to Percentages
Converting to a percentage: multiply by 100. A fraction 3/5 becomes (3 ÷ 5) × 100 = 60%. A decimal 0.85 becomes 0.85 × 100 = 85%.
Use our percentage calculator to verify any conversion or to calculate percentage values, changes, and comparisons quickly. The calculator handles all the standard percentage operations: finding what percentage one number is of another, calculating a percentage increase or decrease, and finding a value given a percentage.
Percentages to Fractions and Decimals
Going the other direction: divide a percentage by 100. 35% becomes 35/100 = 7/20 as a simplified fraction, or 0.35 as a decimal. 12.5% becomes 12.5/100 = 1/8 as a fraction, or 0.125 as a decimal.
This conversion is used constantly in finance: an interest rate of 4.5% is 0.045 as a decimal multiplier. Applying 4.5% interest to £10,000 means multiplying by 0.045 to get £450, or using the multiplier 1.045 to find the total including interest: £10,450.
Percentages in Probability
Probability is expressed in three equivalent forms: as a fraction (3/10), as a decimal (0.3), or as a percentage (30%). The probability of rolling a 3 or 4 on a six-sided die is 2/6 = 1/3 ≈ 33.3%. Weather forecasts use percentages. Medical risk is expressed in all three forms depending on context. Understanding probability requires fluency with all three representations.
Our probability calculator works in all three formats — enter your probability as a fraction, decimal, or percentage and convert between them. Combined events (multiple independent probabilities) multiply as decimals most easily: a 70% chance of rain today and a 60% chance tomorrow gives 0.7 × 0.6 = 0.42 = 42% probability of rain on both days.
Percentage Increase and Decrease
Percentage change = (new value − original value) ÷ original value × 100. A price rising from £80 to £96: (96 − 80) ÷ 80 × 100 = 20% increase. A salary falling from £45,000 to £40,500: (40,500 − 45,000) ÷ 45,000 × 100 = −10% (a 10% decrease).
A common error: applying a percentage increase and then the same percentage decrease does not return to the original value. A 20% increase on £100 gives £120. A subsequent 20% decrease on £120 gives £96 — not the original £100. The base changed. This asymmetry matters in finance, retail discounting, and compound interest calculations.
Compound Percentage Changes
When percentages are applied repeatedly — annual interest rates, inflation over multiple years, successive discounts — the correct approach multiplies the decimal multipliers rather than adding the percentages. Three successive annual increases of 5% applied to £1,000: £1,000 × 1.05 × 1.05 × 1.05 = £1,000 × 1.157625 = £1,157.63. Adding 5+5+5 = 15% would give only £1,150 — an underestimate that grows more significant over more periods.
BBC Bitesize has comprehensive worked examples on fractions, decimals and percentages with interactive practice, useful for building speed and confidence before applying these conversions in professional or financial contexts.