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Fraction Calculator — Step-by-Step Solver

Add, subtract, multiply, divide, compare, and solve fractions — chain up to 9 at once with full step-by-step work, mixed numbers, simplify, and decimal conversions.

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Calculator Modes

This calculator offers 7 modes for all your fraction needs:

  • Operations: Add, subtract, multiply, and divide 2-9 fractions
  • Mixed Numbers: Convert between mixed numbers and improper fractions
  • Simplify: Reduce fractions to lowest terms
  • Decimal→Fraction: Convert decimals to fractions
  • Fraction→Decimal: Convert fractions to decimals with repeating pattern detection
  • Compare: Determine which of two fractions is larger using LCD
  • Solve for X: Find the missing value in a fraction equation using cross multiplication

Fraction Operations

Perform basic arithmetic operations on fractions with step-by-step solutions.

Adding Fractions

Formula: a/b + c/d = (ad + bc) / bd

Example: 1/2 + 1/3 = (1×3 + 1×2) / (2×3) = 5/6

Subtracting Fractions

Formula: a/b - c/d = (ad - bc) / bd

Example: 3/4 - 1/2 = (3×2 - 1×4) / (4×2) = 2/8 = 1/4

Multiplying Fractions

Formula: a/b × c/d = ac / bd

Example: 2/3 × 3/4 = 6/12 = 1/2

Dividing Fractions

Formula: a/b ÷ c/d = a/b × d/c = ad / bc

Example: 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2

Understanding Fractions

Parts of a Fraction

  • Numerator: The top number (how many parts you have)
  • Denominator: The bottom number (total equal parts)

Types of Fractions

  • Proper: Numerator < denominator (e.g., 3/4)
  • Improper: Numerator ≥ denominator (e.g., 5/3)
  • Mixed: Whole number + fraction (e.g., 1⅔)

Simplifying Fractions

To simplify, find the greatest common divisor (GCD) and divide both parts by it.

Example: 12/18 → GCD is 6 → 12÷6 / 18÷6 = 2/3

Mixed Number Conversions

Mixed to Improper

Formula: Whole × Denominator + Numerator / Denominator

Example: 2 3/4 = (2×4 + 3)/4 = 11/4

Improper to Mixed

Divide numerator by denominator. Quotient becomes whole number, remainder becomes new numerator.

Example: 7/3 = 2 1/3 (7÷3 = 2 remainder 1)

Decimal Conversions

Decimal to Fraction

Uses continued fractions algorithm for accurate conversion. Handles both terminating and repeating decimals.

Examples:

  • 0.75 → 3/4
  • 0.333... → 1/3
  • 0.125 → 1/8

Fraction to Decimal

Divides numerator by denominator and detects repeating patterns.

Examples:

  • 1/4 = 0.25 (terminating)
  • 1/3 = 0.333... (repeating)
  • 1/7 = 0.142857... (repeating pattern: 142857)

Common Fraction-Decimal-Percent Equivalents

FractionDecimalPercent
180.12512.5%
160.1667...16.7%
150.220%
140.2525%
130.333...33.3%
380.37537.5%
250.440%
120.550%
350.660%
230.666...66.7%
340.7575%
560.8333...83.3%
780.87587.5%
9100.990%

Comparing Fractions

To compare fractions, find the least common denominator (LCD) and scale both numerators:

  • 3/8 vs 5/12 → LCD = 24 → 9/24 vs 10/24 → 5/12 is larger
  • 2/5 vs 3/7 → LCD = 35 → 14/35 vs 15/35 → 3/7 is larger

Solving for X

When two fractions are equal and one value is unknown, use cross multiplication:

  • X/12 = 4/16 → X = (12 × 4) / 16 = 3
  • 7/X = 21/9 → X = (7 × 9) / 21 = 3

Multi-Fraction Operations

Chain operations on 3 or more fractions by clicking "+ Add fraction" in Operations mode. The calculator applies each operation left to right, simplifying intermediate results.

Fractions in Real Life

  • Cooking: Recipes use fractions constantly — 1/2 cup flour, 3/4 tsp salt, 1/3 cup milk. Scaling a recipe requires multiplying fractions
  • Construction: Tape measures are marked in 1/16-inch increments. Lumber dimensions use fractions (a 2×4 is actually 1 1/2 × 3 1/2 inches)
  • Music: Note durations are fractions — whole, half, quarter, eighth, sixteenth notes. Time signatures like 3/4 and 6/8 are fractions
  • Finance: Stock prices historically used fractions (1/8 increments) before switching to decimals in 2001

Common Fraction Mistakes

  • Adding denominators: 1/4 + 1/4 = 2/4, NOT 2/8. Combining same-size pieces does not change piece size
  • Forgetting to simplify: 6/8 is correct, but 3/4 is the lowest-terms answer expected on most tests
  • Cross-multiplying when adding: Cross multiplication is for comparing or solving proportions, not for adding or subtracting
  • Inverting the wrong fraction in division: Always flip the divisor (the second fraction), not the dividend
  • Mixing mixed and improper carelessly: Convert 2 1/3 to 7/3 before multiplying or dividing — never multiply the whole number through and the fraction separately
  • Sign errors with negatives: -3/4 means -(3/4). When subtracting a negative fraction, the two minuses become a plus: 1/2 − (−1/4) = 1/2 + 1/4 = 3/4

Quick Mental Math Tricks

  • Halving: Multiplying by 1/2 is the same as dividing by 2. 3/8 × 1/2 = 3/16
  • Same denominator: a/n + b/n = (a+b)/n. Skip the LCD step entirely
  • Cancelling early: 4/9 × 3/8 → cancel 4 and 8 to 1 and 2, cancel 3 and 9 to 1 and 3 → 1/6. Easier than multiplying then simplifying 12/72
  • Decimal sense check: 5/8 = 0.625. If your calculation produces 0.4 you have made an error

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Frequently Asked Questions

How do I add fractions?

Find a common denominator, add the numerators, then simplify. Formula: a/b + c/d = (ad + bc) / bd

How do I subtract fractions?

Find a common denominator, subtract the numerators, then simplify. Formula: a/b - c/d = (ad - bc) / bd

How do I multiply fractions?

Multiply numerators together and denominators together, then simplify. Formula: a/b × c/d = ac / bd

How do I divide fractions?

Multiply by the reciprocal (flip the second fraction). Formula: a/b ÷ c/d = a/b × d/c = ad / bc

What is simplifying a fraction?

Dividing both numerator and denominator by their greatest common divisor (GCD). Example: 6/8 simplifies to 3/4.

How do I convert a mixed number to an improper fraction?

Multiply the whole number by the denominator, add the numerator, then place over the original denominator. Example: 2 3/4 = (2×4 + 3)/4 = 11/4

How do I convert a decimal to a fraction?

The calculator uses continued fractions algorithm to find the best fractional representation. Example: 0.75 = 3/4, 0.333... = 1/3

What is a repeating decimal?

A decimal with digits that repeat infinitely. Example: 1/3 = 0.333..., 1/7 = 0.142857... (the pattern 142857 repeats)

How do I compare two fractions?

Find a common denominator (LCD), then compare the scaled numerators. The fraction with the larger numerator is bigger. Example: 3/8 vs 5/12 → LCD=24 → 9/24 vs 10/24 → 5/12 is larger.

How do I solve for X in a fraction equation?

Use cross multiplication. For X/b = c/d, multiply both sides: X = (b × c) / d. For a/X = c/d, then X = (a × d) / c.

Can I add more than two fractions?

Yes — use the "+ Add fraction" button in Operations mode to chain up to 9 fractions together with the same operation.

What is the difference between LCM and LCD?

They are the same number — just named for context. LCM (least common multiple) applies to any integers; LCD (least common denominator) is the LCM used specifically for the denominators of fractions you are adding or subtracting.

Why do I "flip" the second fraction when dividing?

Dividing by a fraction is the same as multiplying by its reciprocal. Dividing by 1/2 means asking "how many halves fit?" — equivalent to multiplying by 2 (which is 2/1, the reciprocal of 1/2).

What is the most common mistake when adding fractions?

Adding the denominators. 1/3 + 1/4 is NOT 2/7. You must convert to a common denominator first: 4/12 + 3/12 = 7/12. The denominator names the size of each piece — it stays the same when you combine pieces of the same size.

How do I write a fraction in lowest terms?

Divide the numerator and denominator by their greatest common divisor (GCD). Example: 24/36. GCD = 12. 24÷12 = 2, 36÷12 = 3 → 2/3. The Simplify mode does this automatically.