Fraction Calculator
Add, subtract, multiply, or divide fractions and mixed numbers. Get step-by-step solutions, simplified results, and decimal equivalents instantly.
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Step-by-Step Solution
How to Calculate Fractions: Complete Guide
A fraction represents a part of a whole. The top number (numerator) tells you how many parts you have, and the bottom number (denominator) tells you how many equal parts the whole is divided into. For example, 3/4 means 3 out of 4 equal parts. This calculator handles all four operations — addition, subtraction, multiplication, and division — with automatic simplification using the Greatest Common Divisor (GCD).
Adding and Subtracting Fractions
To add or subtract fractions, you need a common denominator — both fractions must be divided into the same number of parts before you can combine them. The most efficient common denominator is the Least Common Multiple (LCM) of the two denominators.
Worked Example: 2/3 + 3/4
Step 1: Find LCM of 3 and 4 → LCM = 12
Step 2: Convert: 2/3 = 8/12 and 3/4 = 9/12
Step 3: Add numerators: 8 + 9 = 17
Step 4: Result: 17/12 = 1 5/12 ≈ 1.417
Multiplying Fractions
Multiplication is the simplest fraction operation. Multiply the numerators together and the denominators together. No common denominator is needed. Cross-cancellation before multiplying can simplify the work.
Worked Example: 3/5 × 2/7
Step 1: Multiply numerators: 3 × 2 = 6
Step 2: Multiply denominators: 5 × 7 = 35
Step 3: Result: 6/35 (already simplified) ≈ 0.171
Dividing Fractions
To divide by a fraction, multiply by its reciprocal (flip the second fraction). This is often taught as "keep, change, flip" — keep the first fraction, change division to multiplication, and flip the second fraction.
Worked Example: 3/4 ÷ 2/5
Step 1: Keep 3/4, flip 2/5 → 5/2
Step 2: Multiply: 3/4 × 5/2
Step 3: Numerators: 3 × 5 = 15
Step 4: Denominators: 4 × 2 = 8
Step 5: Result: 15/8 = 1 7/8 = 1.875
Simplifying Fractions
A fraction is in simplest form when the numerator and denominator share no common factors other than 1. To simplify, divide both by their Greatest Common Divisor (GCD). For example, 24/36: GCD(24, 36) = 12, so 24/36 = 2/3.
Common Fractions, Decimals, and Percentages
| Fraction | Decimal | Percentage | Fraction | Decimal | Percentage |
|---|---|---|---|---|---|
| 1/2 | 0.5 | 50% | 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333... | 33.3% | 3/8 | 0.375 | 37.5% |
| 2/3 | 0.667... | 66.7% | 5/8 | 0.625 | 62.5% |
| 1/4 | 0.25 | 25% | 7/8 | 0.875 | 87.5% |
| 3/4 | 0.75 | 75% | 1/10 | 0.1 | 10% |
| 1/5 | 0.2 | 20% | 1/16 | 0.0625 | 6.25% |
| 2/5 | 0.4 | 40% | 1/100 | 0.01 | 1% |
| 1/6 | 0.167... | 16.7% | 1/1000 | 0.001 | 0.1% |
Why Fractions Matter: Research Findings
A landmark 2012 study by Siegler et al. published in Psychological Science analyzed data from over 3,400 students and found that fifth-graders' knowledge of fractions uniquely predicted their algebra achievement and overall math attainment in high school, even after controlling for IQ, reading ability, working memory, family income, and whole-number arithmetic skill. The researchers concluded that fractions are "the gatekeeper to advanced mathematics."
The National Mathematics Advisory Panel (2008) identified fractions as the most important foundational skill for algebra readiness. Despite this, the 2013 National Assessment of Educational Progress (NAEP) found that only 24% of eighth-graders could correctly order three fractions from least to greatest — revealing a widespread gap in fraction fluency.
For practical applications, fractions appear constantly in cooking (halving or doubling recipes), construction (measurements in inches and feet), music (time signatures and note values), statistics (probability expressed as fractions), and finance (interest rates and stock prices). Our Percentage Calculator converts between fractions, decimals, and percentages — the three representations of the same value.
Fraction Operations Reference
| Operation | Formula | Common Denominator Needed? | Example |
|---|---|---|---|
| Addition | a/b + c/d = (ad + bc) / bd | Yes | 1/3 + 1/4 = 7/12 |
| Subtraction | a/b − c/d = (ad − bc) / bd | Yes | 3/4 − 1/3 = 5/12 |
| Multiplication | a/b × c/d = ac / bd | No | 2/3 × 3/5 = 6/15 = 2/5 |
| Division | a/b ÷ c/d = a/b × d/c | No | 3/4 ÷ 2/5 = 15/8 |
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