Free Math Tool · Step-by-Step Working
Fraction Calculator
Add, subtract, multiply and divide fractions — with full step-by-step working. Supports mixed numbers, auto-simplifies results, and shows the decimal and percentage equivalent instantly.
How to Use the Fraction Calculator
The calculator offers four modes: arithmetic operations, simplification, comparison and conversion. Results appear instantly as you type — no calculate button is needed. Furthermore, the step-by-step panel shows every stage of the working, making it useful for checking homework and understanding the method.
- Arithmetic — add, subtract, multiply or divideSelect the + − × ÷ tab. Enter the numerator and denominator for each fraction in the stacked input boxes. Click the operation symbol between the fractions to change between addition, subtraction, multiplication and division. Furthermore, enable the Mixed Numbers toggle in the toolbar to add whole number fields — enter 2 3/4 as whole = 2, numerator = 3, denominator = 4.
- Simplify — reduce to lowest termsSelect the Simplify tab and enter the numerator and denominator of the fraction you want to reduce. The calculator finds the greatest common divisor (GCD) and divides both numbers by it to produce the simplified form. Furthermore, the step-by-step panel shows how the GCD was found using the Euclidean algorithm.
- Compare — find which fraction is largerSelect the Compare tab and enter two fractions. A live symbol (>, < or =) updates between the fractions as you type. Furthermore, the step-by-step working shows the cross-multiplication used to reach the conclusion, so you can verify the result manually.
- Convert — switch between fractions, decimals and percentagesThe Convert tab works in two directions. Enter a fraction to see its decimal and percentage equivalents instantly. Furthermore, enter a decimal in the lower field to convert it to a fraction in its simplest form — for example, 0.625 becomes 5/8.
- Read the visual fraction barBelow the result, a visual bar shows the fraction as a proportion of a whole. The green portion represents the numerator and the empty portion represents the remaining denominator. Furthermore, this helps visualise what a fraction actually means as a part of a whole — particularly useful for students learning fractions for the first time.
What Is a Fraction?
A fraction represents a part of a whole. It is written as two numbers separated by a horizontal bar — the numerator on top and the denominator on the bottom. The numerator tells you how many parts you have. The denominator tells you how many equal parts the whole is divided into. Furthermore, the fraction 3/4 means three out of four equal parts.
Fractions can be classified into several types. A proper fraction has a numerator smaller than its denominator — its value is less than 1. For example, 3/8 is a proper fraction. An improper fraction has a numerator greater than or equal to its denominator — its value is 1 or more. For example, 7/4 is an improper fraction. Furthermore, a mixed number combines a whole number with a proper fraction — 1 3/4 is a mixed number equivalent to 7/4.
Fractions appear throughout daily life. Cooking recipes use fractions for ingredient quantities. Construction measurements use fractions of inches. Financial calculations use fractions for interest rates and proportions. Furthermore, understanding fractions is fundamental to all higher mathematics — they underpin algebra, geometry, calculus and statistics.
Proper fraction
Numerator < Denominator. Value is between 0 and 1. Examples: 1/2, 3/4, 7/8, 5/6. Furthermore, all proper fractions are less than 1 on the number line.
Improper fraction
Numerator ≥ Denominator. Value is 1 or greater. Examples: 5/4, 7/3, 11/8. Furthermore, any improper fraction can be written as a mixed number by dividing numerator by denominator.
Mixed number
A whole number plus a proper fraction. Examples: 1 1/2, 2 3/4, 5 7/8. Furthermore, convert to an improper fraction by multiplying the whole number by the denominator and adding the numerator.
How to Add and Subtract Fractions
Adding and subtracting fractions requires a common denominator. If the denominators are the same (like fractions), simply add or subtract the numerators and keep the denominator. Furthermore, if the denominators differ (unlike fractions), you must first find the least common denominator (LCD).
The LCD is the least common multiple (LCM) of the two denominators. Convert each fraction to an equivalent fraction with the LCD as the new denominator. Then add or subtract the numerators and simplify the result. Furthermore, this process ensures you are combining equal-sized parts — which is the mathematical requirement for addition and subtraction.
| Problem | LCD | Converted | Result | Simplified |
|---|---|---|---|---|
| 1/3 + 1/4 | 12 | 4/12 + 3/12 | 7/12 | 7/12 |
| 3/4 − 1/6 | 12 | 9/12 − 2/12 | 7/12 | 7/12 |
| 1/2 + 2/3 | 6 | 3/6 + 4/6 | 7/6 | 1 1/6 |
| 5/8 − 1/4 | 8 | 5/8 − 2/8 | 3/8 | 3/8 |
| 2 1/2 + 1 1/3 | 6 | 5/2 + 4/3 = 15/6 + 8/6 | 23/6 | 3 5/6 |
How to Multiply Fractions
Multiplying fractions is the simplest of the four operations. Multiply the numerators together to get the new numerator. Multiply the denominators together to get the new denominator. Then simplify the result. Furthermore, no common denominator is needed — multiplication works directly on numerator and denominator.
For example: 2/3 × 3/4 = (2×3)/(3×4) = 6/12. Simplified by dividing both by GCD(6,12) = 6, the result is 1/2. Furthermore, you can cross-cancel before multiplying to keep numbers smaller. In 2/3 × 3/4, the 3 in the numerator of the second fraction cancels with the 3 in the denominator of the first: (2/1) × (1/4) = 2/4 = 1/2.
Standard method
Multiply numerators: 2 × 3 = 6. Multiply denominators: 3 × 4 = 12. Result: 6/12. Simplify by GCD(6,12) = 6: answer is 1/2. Furthermore, always check if the result can be simplified before presenting it.
Cross-cancellation method
Before multiplying, cancel any common factor in a numerator with any denominator. In 2/3 × 3/4: the 3s cancel, leaving 2/1 × 1/4 = 2/4 = 1/2. Furthermore, this avoids dealing with large numbers in intermediate steps.
For mixed numbers, convert to improper fractions first. For 1 1/2 × 2 1/3: convert to 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2. Furthermore, multiplying mixed numbers without converting first leads to errors — always convert first.
How to Divide Fractions
Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction is produced by swapping the numerator and denominator — the reciprocal of 3/4 is 4/3. Furthermore, the phrase "keep, change, flip" is a useful memory aid: keep the first fraction, change the operation from division to multiplication, and flip the second fraction.
For example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Furthermore, the step-by-step panel in this calculator shows the reciprocal step explicitly, so you can follow exactly how the division is converted to multiplication.
| Division problem | Reciprocal applied | Multiplied | Simplified |
|---|---|---|---|
| 1/2 ÷ 1/4 | 1/2 × 4/1 | 4/2 | 2 |
| 3/4 ÷ 2/3 | 3/4 × 3/2 | 9/8 | 1 1/8 |
| 5/6 ÷ 5/9 | 5/6 × 9/5 | 45/30 | 3/2 = 1 1/2 |
| 2 1/2 ÷ 1 1/4 | 5/2 × 4/5 | 20/10 | 2 |
Division by zero is undefined and has no result. This calculator shows an error if you attempt to divide by a fraction with a zero numerator. Furthermore, a denominator of zero is also invalid in any fraction — the denominator field cannot be set to zero in any mode.
How to Simplify Fractions
A fraction is in its simplest form (lowest terms) when the numerator and denominator share no common factors other than 1. Furthermore, simplifying makes fractions easier to work with and compare. The standard method uses the greatest common divisor (GCD).
The GCD of two numbers is the largest number that divides both without a remainder. The most efficient way to find it is the Euclidean algorithm: divide the larger number by the smaller and take the remainder. Repeat with the smaller number and the remainder until the remainder is zero. The last non-zero remainder is the GCD. Furthermore, this calculator shows each step of the Euclidean algorithm in the simplify mode step-by-step panel.
A fraction with a negative numerator or denominator is still valid. By convention, the sign is placed on the numerator. Furthermore, a fraction like −6/−8 has both a negative numerator and negative denominator — it simplifies to 3/4 (positive) because the two negatives cancel.
Mixed Numbers and Improper Fractions
A mixed number consists of a whole number and a proper fraction — for example, 3 2/5. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the total over the original denominator. For 3 2/5: (3×5 + 2)/5 = 17/5. Furthermore, this conversion is the essential first step before performing any arithmetic operation involving mixed numbers.
To convert an improper fraction back to a mixed number, divide the numerator by the denominator. The quotient is the whole number part and the remainder is the new numerator over the original denominator. For 17/5: 17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5. Furthermore, results are always presented in mixed number form when the numerator exceeds the denominator, so you can immediately see the whole and fractional parts.
| Mixed number | Conversion steps | Improper fraction |
|---|---|---|
| 1 1/2 | (1×2 + 1) / 2 | 3/2 |
| 2 3/4 | (2×4 + 3) / 4 | 11/4 |
| 5 2/3 | (5×3 + 2) / 3 | 17/3 |
| 10 7/8 | (10×8 + 7) / 8 | 87/8 |
Fractions as Decimals and Percentages
Every fraction can be expressed as a decimal by dividing the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75. Furthermore, some fractions produce terminating decimals (0.75, 0.5, 0.125) while others produce repeating decimals (1/3 = 0.333…, 1/7 = 0.142857142857…). Terminating decimals occur when the denominator's only prime factors are 2 and 5.
Converting a decimal to a fraction requires identifying the place value of the last decimal digit. For 0.75: the last digit is in the hundredths place, so write 75/100 and simplify by GCD(75,100) = 25 to get 3/4. Furthermore, for repeating decimals, an algebraic method is needed — the Convert mode in this calculator handles the common case of terminating decimals.
To convert a fraction to a percentage, convert to a decimal first then multiply by 100. For 3/4: decimal = 0.75; percentage = 75%. Furthermore, percentages are simply fractions with a denominator of 100 — 75% means 75/100, which simplifies to 3/4.
Frequently Asked Questions
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