Fraction Calculator

What This Calculator Does

Add fractions. Enter two fractions with any denominators. The calculator finds the least common denominator, converts both fractions, adds the numerators, and simplifies the result. Works with proper fractions, improper fractions, and mixed numbers.

Subtract fractions. Same process as addition — find the LCD, convert, subtract numerators, simplify. Handles negative results automatically.

Multiply fractions. Multiplies numerator by numerator and denominator by denominator, then reduces to lowest terms. No common denominator needed.

Divide fractions. Flips the second fraction (takes the reciprocal) and multiplies. The result is simplified automatically.

Simplify fractions. Enter a single fraction and the calculator reduces it to lowest terms by dividing both numerator and denominator by their greatest common factor (GCF).

Convert fractions. Converts between improper fractions and mixed numbers, and between fractions and decimals.

How to Use the Calculator

Step 1 — Enter the first fraction. Type the numerator and denominator separately, or type the fraction as "3/4". For mixed numbers, enter the whole number, then the fraction (e.g., "2 3/4" or enter 2 in the whole number field and 3/4 in the fraction field).

Step 2 — Select the operation. Choose addition (+), subtraction (−), multiplication (×), or division (÷).

Step 3 — Enter the second fraction. Same format as the first.

Step 4 — Click calculate. The result appears in simplified form as both a fraction and a decimal, with the full step-by-step solution below.

The Four Operations — Formulas and Examples

Adding Fractions

Same denominator: Add the numerators and keep the denominator. a/d + b/d = (a + b) / d. Example: 2/7 + 3/7 = 5/7.

Different denominators: Find the least common denominator (LCD), convert both fractions, then add. a/b + c/d = (ad + bc) / bd.

Example: 1/3 + 1/4

Step 1 — Find the LCD of 3 and 4. The LCD is 12. Step 2 — Convert: 1/3 = 4/12, and 1/4 = 3/12. Step 3 — Add numerators: 4/12 + 3/12 = 7/12. Step 4 — Simplify: 7/12 is already in lowest terms (GCF of 7 and 12 is 1). Result: 7/12

Subtracting Fractions

The process is identical to addition, except you subtract the numerators. a/b − c/d = (ad − bc) / bd.

Example: 5/6 − 1/4

Step 1 — LCD of 6 and 4 is 12. Step 2 — Convert: 5/6 = 10/12, and 1/4 = 3/12. Step 3 — Subtract: 10/12 − 3/12 = 7/12. Result: 7/12

Multiplying Fractions

No common denominator needed. Multiply straight across. a/b × c/d = (a × c) / (b × d).

Example: 2/3 × 4/5

Step 1 — Multiply numerators: 2 × 4 = 8. Step 2 — Multiply denominators: 3 × 5 = 15. Step 3 — Result: 8/15 (already simplified). Result: 8/15

Tip: You can cross-cancel before multiplying to keep numbers small. In 4/9 × 3/8, notice that 4 and 8 share a factor of 4, and 3 and 9 share a factor of 3. Cancel first: (1/3) × (1/2) = 1/6. Same answer, smaller numbers.

Dividing Fractions

Flip the second fraction (take its reciprocal) and multiply. a/b ÷ c/d = a/b × d/c = (a × d) / (b × c).

Example: 3/4 ÷ 2/5

Step 1 — Flip the second fraction: 2/5 becomes 5/2. Step 2 — Multiply: 3/4 × 5/2 = 15/8. Step 3 — Convert to mixed number: 15/8 = 1 7/8. Result: 15/8 = 1 7/8

Working with Mixed Numbers

A mixed number combines a whole number and a proper fraction — like 3 1/2 (three and one half). To perform any operation with mixed numbers, first convert them to improper fractions.

Conversion formula: whole × denominator + numerator, over the original denominator. Example: 3 1/2 → (3 × 2 + 1) / 2 = 7/2. Example: 2 3/4 → (2 × 4 + 3) / 4 = 11/4.

Once both mixed numbers are improper fractions, use the standard formulas above. After calculating, convert the result back to a mixed number if desired.

Full example: 2 1/3 + 1 3/4

Step 1 — Convert to improper fractions: 2 1/3 = 7/3, and 1 3/4 = 7/4. Step 2 — Find LCD of 3 and 4: LCD = 12. Step 3 — Convert: 7/3 = 28/12, and 7/4 = 21/12. Step 4 — Add: 28/12 + 21/12 = 49/12. Step 5 — Convert to mixed number: 49 ÷ 12 = 4 remainder 1, so 49/12 = 4 1/12. Result: 4 1/12

How to Simplify a Fraction

A fraction is in simplest form (lowest terms) when the numerator and denominator share no common factor other than 1.

Method: Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it.

Example: Simplify 18/24

Step 1 — Find factors of 18: 1, 2, 3, 6, 9, 18. Step 2 — Find factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Step 3 — GCF = 6. Step 4 — Divide: 18 ÷ 6 = 3, and 24 ÷ 6 = 4. Result: 18/24 = 3/4

Quick test: If both the numerator and denominator are even, divide by 2. If both end in 0 or 5, divide by 5. If the digits of each sum to a multiple of 3, divide by 3. Repeat until no common factor remains.

Finding the Least Common Denominator (LCD)

The LCD is the smallest number that both denominators divide into evenly. You need it whenever you add or subtract fractions with different denominators.

Method 1 — List multiples: For denominators 6 and 8: Multiples of 6: 6, 12, 18, 24, 30... Multiples of 8: 8, 16, 24, 32... LCD = 24.

Method 2 — Prime factorization: 6 = 2 × 3, 8 = 2³. LCD = 2³ × 3 = 24.

Method 3 — Formula: LCD(a, b) = (a × b) / GCF(a, b). LCD(6, 8) = 48 / 2 = 24. The formula method is fastest when you already know the GCF.

Converting Between Fractions and Decimals

Fraction to Decimal

Divide the numerator by the denominator.

Why do some fractions produce repeating decimals? A fraction produces a terminating decimal only if the denominator (in simplest form) has no prime factors other than 2 and 5. If the denominator contains any other prime factor (3, 7, 11, etc.), the decimal repeats.

Decimal to Fraction

Write the decimal over a power of 10, then simplify. Example: 0.75 = 75/100 = 3/4 (divide by GCF of 25). Example: 0.125 = 125/1000 = 1/8 (divide by GCF of 125).

For repeating decimals, the conversion uses algebra: Let x = 0.333... Then 10x = 3.333... So 10x − x = 3, meaning 9x = 3, so x = 3/9 = 1/3.

Common Fraction Reference Table

This table covers the fractions that appear most often in everyday calculations — cooking, construction, finance, and school work.

Real-World Applications

Cooking and baking. Recipes are full of fractions: 3/4 cup flour, 1/3 cup sugar, 2 1/2 teaspoons salt. Doubling or halving a recipe means multiplying fractions by 2 or 1/2. Combining partial measurements means adding fractions.

Construction and woodworking. Lumber dimensions, drill bit sizes, and tape measures use fractions of an inch — 5/16", 7/8", 1 3/4". Calculating material cuts requires adding and subtracting fractions constantly. A carpenter cutting a board to fit a 36 3/4" opening from a 48" board needs to subtract: 48 − 36 3/4 = 11 1/4" of waste.

Finance. Interest rates and stock prices historically used fractions (stocks traded in eighths until 2001). Bond yields, tax brackets, and ownership percentages are still expressed as fractions in many contexts. "A one-third stake in the company" means multiplying the company value by 1/3.

Science and engineering. Unit conversions, chemical ratios, and scaling calculations all use fraction arithmetic. A chemist mixing solutions at 2/5 concentration with 3/7 concentration needs to add weighted fractions.

Everyday problem-solving. Splitting a bill three ways, dividing pizza evenly, figuring out how much paint covers 2/3 of a wall — fractions are embedded in daily decisions that most people solve by converting to decimals. Understanding fraction arithmetic lets you work with exact values instead of rounded approximations.

Common Mistakes to Avoid

Adding numerators and denominators separately. The most common fraction error: 1/3 + 1/4 ≠ 2/7. You cannot add fractions by adding tops and bottoms. You must find a common denominator first.

Forgetting to simplify. The answer 6/8 is not wrong, but it is not in lowest terms. Always check whether the numerator and denominator share a common factor and divide both by the GCF. The simplified form, 3/4, is the expected answer.

Not converting mixed numbers first. The safest method is always to convert to improper fractions first. While sometimes you can add whole numbers and fractions separately, this shortcut fails when the fractional parts produce an improper fraction.

Flipping the wrong fraction when dividing. In a ÷ b, you flip b (the divisor), not a. Flipping the wrong one produces the reciprocal of the correct answer.

Canceling across addition or subtraction. Cross-canceling only works when multiplying or dividing. In (2 + 4)/4, you cannot cancel the 4s — 2 + 4 = 6, so the answer is 6/4 = 3/2, not 2/1.

Improper Fractions vs. Mixed Numbers

An improper fraction has a numerator larger than (or equal to) its denominator: 7/4, 11/3, 9/9. It represents a value greater than or equal to 1.

A mixed number expresses the same value as a whole number plus a proper fraction: 1 3/4, 3 2/3, 1.

To convert improper to mixed: Divide numerator by denominator. The quotient is the whole number; the remainder is the new numerator. 7/4: 7 ÷ 4 = 1 remainder 3 → 1 3/4.

To convert mixed to improper: Multiply whole number by denominator, add numerator. 2 3/5: (2 × 5) + 3 = 13 → 13/5.

Both forms are mathematically equivalent. Schools typically expect mixed numbers in final answers. In algebra and advanced math, improper fractions are preferred because they are easier to work with in equations.

Equivalent Fractions

Equivalent fractions represent the same value with different numerators and denominators: 1/2 = 2/4 = 3/6 = 4/8 = 50/100. You create equivalent fractions by multiplying (or dividing) both the numerator and denominator by the same non-zero number.

This is the principle behind finding common denominators. When you convert 1/3 to 4/12, you have multiplied both top and bottom by 4. The value has not changed — only the representation.

Test for equivalence: Cross-multiply. If a/b and c/d are equivalent, then a × d = b × c. Example: Is 3/4 equivalent to 9/12? 3 × 12 = 36, and 4 × 9 = 36. Equal — they are equivalent.