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Sources & Methodology
Add/Subtract: a/b ± c/d = (a×d ± c×b) / (b×d), then simplify by GCF Multiply: a/b × c/d = (a×c) / (b×d), then simplify Divide: a/b ÷ c/d = a/b × d/c = (a×d) / (b×c), then simplify Simplify: GCF = Euclidean algorithm; result = (num/GCF) / (den/GCF) All arithmetic performed in integers. No floating-point rounding until final decimal conversion.
Last reviewed: April 2026
Complete Guide to Fraction Arithmetic — Every Operation with Steps
Fractions represent parts of a whole. They appear everywhere in daily life: cooking measurements (1/2 cup, 3/4 teaspoon), construction dimensions (3/8 inch, 7/16 inch), time calculations (3/4 of an hour), probability (1 in 6 chance), and finance (2/3 of a budget). Understanding fraction arithmetic is a foundational math skill that underpins algebra, calculus, physics, and engineering.
Step 1: Find LCD of 3 and 4. Multiples of 3: 3, 6, 12. Multiples of 4: 4, 8, 12. LCD = 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: Check simplification: GCF(7,12) = 1. Already simplified. Answer = 7/12
How to Add Fractions — Step by Step
To add fractions, both must share the same denominator (the bottom number). The process: (1) find the Least Common Denominator (LCD) of both denominators, (2) convert each fraction so both have the LCD, (3) add the numerators (top numbers) while keeping the denominator, (4) simplify the result by dividing both parts by their GCF. For fractions with the same denominator (e.g. 3/8 + 1/8), just add numerators: 4/8 = 1/2.
How to Subtract Fractions
Subtraction follows the identical process as addition. Find the LCD, convert both fractions to the LCD, then subtract the second numerator from the first. Example: 3/4 − 1/3. LCD = 12. Convert: 9/12 − 4/12 = 5/12. GCF(5,12) = 1, so the result is already simplified. When the result is negative (e.g. 1/4 − 3/4 = −2/4 = −1/2), the negative sign applies to the numerator.
How to Multiply Fractions
Multiplication is the simplest fraction operation: multiply numerators together and denominators together. Formula: (a/b) × (c/d) = (a×c)/(b×d). Example: 2/3 × 3/4 = 6/12 = 1/2. No LCD is needed. To simplify, divide by GCF before or after multiplying. Cross-cancelling simplifies before multiplying: 2/3 × 3/4 → cancel the 3s → 2/1 × 1/4 = 2/4 = 1/2. This keeps numbers small and avoids large intermediate values.
How to Divide Fractions — Keep, Change, Flip
Division uses the KCF method: Keep the first fraction, Change division to multiplication, Flip (find the reciprocal of) the second fraction. Formula: (a/b) ÷ (c/d) = (a/b) × (d/c). Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. The reciprocal of any fraction a/b is b/a — simply swap numerator and denominator. Never divide by zero: if the second fraction is 0/any or any/0, the operation is undefined.
How to Simplify Fractions (Reduce to Lowest Terms)
A fraction is in its simplest form (lowest terms) when the numerator and denominator share no common factors other than 1. Process: find the GCF (Greatest Common Factor) of numerator and denominator, then divide both by the GCF. Example: simplify 18/24. Factors of 18: 1,2,3,6,9,18. Factors of 24: 1,2,3,4,6,8,12,24. GCF = 6. So 18/24 = (18/6)/(24/6) = 3/4. The GCF can be found using the Euclidean algorithm: GCF(18,24) = GCF(18, 24−18) = GCF(18,6) = GCF(6,0) = 6.
Working with Mixed Numbers
Mixed numbers like 2½ combine a whole number and a fraction. To perform arithmetic, convert to improper fractions first: multiply the whole number by the denominator and add the numerator. Example: 2 3/4 = (2×4 + 3)/4 = 11/4. After calculation, convert back: 11/4 ÷ 4 = 2 remainder 3, so 11/4 = 2 3/4. This calculator handles mixed numbers automatically when the “Use mixed numbers” checkbox is checked.
Fraction Reference Table
| Operation | Formula | Example | Result |
|---|---|---|---|
| Addition | (ad + bc) / bd | 1/3 + 1/4 | 7/12 |
| Subtraction | (ad − bc) / bd | 3/4 − 1/3 | 5/12 |
| Multiplication | (a×c) / (b×d) | 2/3 × 3/4 | 1/2 |
| Division | (a×d) / (b×c) | 2/3 ÷ 4/5 | 5/6 |
| Simplify | num/GCF ÷ den/GCF | 18/24 | 3/4 |
Common Fraction Mistakes to Avoid
- Adding denominators: 1/2 + 1/3 ≠ 2/5. You must find the LCD, not add denominators.
- Forgetting to simplify: 6/8 = 3/4. Always check if GCF > 1.
- Wrong reciprocal in division: Flip the second fraction only, not the first.
- Mixed number error: Converting 2 3/4 incorrectly as 23/4 instead of 11/4.
- Negative fraction confusion: −3/4 = 3/−4 = −(3/4). All three are the same value.
Real-World Uses of Fraction Arithmetic
- Cooking: Scaling recipes (double 3/4 cup = 1 1/2 cups), combining ingredient amounts
- Construction: Adding board lengths in fractions of inches (3/8″ + 7/16″ = 13/16″)
- Finance: Fractional shares, interest rate calculations, budget allocations
- Time: Adding durations (1/4 hour + 1/3 hour = 7/12 hour = 35 minutes)
- Probability: Combining probabilities of independent events
- Sewing & fabric: Adding yardage amounts in fractional cuts