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Sources & Methodology
All fraction arithmetic uses standard integer mathematics. GCD is computed via the Euclidean algorithm. LCD is derived as LCD = (d1 × d2) / GCD(d1, d2). Verified against NCTM mathematics curriculum standards.
NCTM — Principles and Standards for School Mathematics
National Council of Teachers of Mathematics curriculum reference for fraction arithmetic operations and simplification procedures used in the step-by-step solution generator.
Khan Academy — Fraction Arithmetic
Reference for standard teaching methods of fraction addition, subtraction, multiplication, and division used to verify the step-by-step explanations shown in results.
GCD (Euclidean Algorithm): gcd(a,b) = gcd(b, a mod b) until b=0; return a.
LCD: lcd(d1,d2) = (d1 × d2) / gcd(d1,d2).
Addition/Subtraction: Convert both fractions to LCD denominator, then add/subtract numerators.
Multiplication: Multiply numerators; multiply denominators; simplify by dividing both by GCD.
Division: Keep first fraction; multiply by reciprocal of second fraction.
Simplification: Divide numerator and denominator by their GCD. If numerator ≥ denominator, express as mixed number.
LCD: lcd(d1,d2) = (d1 × d2) / gcd(d1,d2).
Addition/Subtraction: Convert both fractions to LCD denominator, then add/subtract numerators.
Multiplication: Multiply numerators; multiply denominators; simplify by dividing both by GCD.
Division: Keep first fraction; multiply by reciprocal of second fraction.
Simplification: Divide numerator and denominator by their GCD. If numerator ≥ denominator, express as mixed number.
⏱ Last reviewed: April 2026
How to Add, Subtract, Multiply & Divide Fractions
A fraction represents a part of a whole. The top number is the numerator (how many parts you have) and the bottom number is the denominator (how many equal parts the whole is divided into). Fraction arithmetic follows four distinct sets of rules depending on the operation.
Adding and Subtracting Fractions
LCD = (d1 × d2) ÷ GCD(d1, d2)
a/b + c/d = (a×(LCD÷b) + c×(LCD÷d)) / LCD
a/b − c/d = (a×(LCD÷b) − c×(LCD÷d)) / LCD
Example: 1/3 + 1/4
LCD = (3×4) ÷ GCD(3,4) = 12 ÷ 1 = 12
= (1×4 + 1×3) / 12 = (4 + 3) / 12 = 7/12
LCD = (3×4) ÷ GCD(3,4) = 12 ÷ 1 = 12
= (1×4 + 1×3) / 12 = (4 + 3) / 12 = 7/12
Multiplying Fractions
a/b × c/d = (a×c) / (b×d) → simplify by GCD
Example: 2/3 × 3/4 = (2×3) / (3×4) = 6/12
GCD(6,12) = 6 → 6/6 = 1 and 12/6 = 2 → 1/2
Tip: Cross-cancel before multiplying to keep numbers smaller.
GCD(6,12) = 6 → 6/6 = 1 and 12/6 = 2 → 1/2
Tip: Cross-cancel before multiplying to keep numbers smaller.
Dividing Fractions
a/b ÷ c/d = a/b × d/c = (a×d) / (b×c) → simplify
Memory trick: Keep-Change-Flip (KCF)
Keep the first fraction, Change ÷ to ×, Flip the second fraction.
Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Keep the first fraction, Change ÷ to ×, Flip the second fraction.
Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Simplifying Fractions & Converting to Mixed Numbers
| Unsimplified | GCD | Simplified | Mixed Number | Decimal |
|---|---|---|---|---|
| 6/12 | 6 | 1/2 | — | 0.5 |
| 9/6 | 3 | 3/2 | 1 1/2 | 1.5 |
| 14/4 | 2 | 7/2 | 3 1/2 | 3.5 |
| 18/12 | 6 | 3/2 | 1 1/2 | 1.5 |
| 25/10 | 5 | 5/2 | 2 1/2 | 2.5 |
| 36/8 | 4 | 9/2 | 4 1/2 | 4.5 |
Common Fraction — Decimal Conversions
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 3/4 | 0.75 | 75% |
| 2/3 | 0.666… | 66.7% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
💡 Cross-Cancelling Tip: Before multiplying fractions, check if any numerator and any denominator share a common factor. Cancel them first to keep numbers smaller. Example: (2/3) × (3/4) — the 3 in the numerator of the second fraction and the 3 in the denominator of the first cancel to 1: (2/1) × (1/4) = 2/4 = 1/2. Same result, smaller numbers.
Frequently Asked Questions
Find the lowest common denominator (LCD) of the two fractions. Convert each fraction so both have the LCD. Add the numerators and keep the denominator. Simplify. Example: 1/3 + 1/4. LCD = 12. 4/12 + 3/12 = 7/12. GCD(7,12) = 1, so 7/12 is already simplified.
Same process as addition: find the LCD, convert both fractions, subtract the numerators. Example: 3/4 − 1/3. LCD = 12. 9/12 − 4/12 = 5/12. GCD(5,12) = 1, so 5/12 is fully simplified. If the result is negative, the answer is a negative fraction.
Multiply the numerators together and multiply the denominators together, then simplify. Example: 2/3 × 3/4 = 6/12. GCD(6,12) = 6, so 6/12 = 1/2. You can also cross-cancel common factors before multiplying: the 3 in 2/3 and the 3 in 3/4 cancel, leaving (2/1) × (1/4) = 2/4 = 1/2.
Keep the first fraction unchanged, change the ÷ sign to ×, then flip (take the reciprocal of) the second fraction. Then multiply normally. Memory trick: Keep-Change-Flip (KCF). Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
Find the greatest common divisor (GCD) of the numerator and denominator using the Euclidean algorithm. Divide both the numerator and denominator by the GCD. The result is the simplified fraction. Example: 12/18. GCD(12,18) = 6. 12/6 = 2, 18/6 = 3. Simplified: 2/3.
The lowest common denominator (LCD) is the smallest number that both denominators divide into evenly. It equals the least common multiple (LCM) of the denominators. Formula: LCD = (d1 × d2) ÷ GCD(d1, d2). Example: LCD of 4 and 6 = (4×6) ÷ GCD(4,6) = 24 ÷ 2 = 12.
Divide the numerator by the denominator. The quotient is the whole number part; the remainder becomes the numerator of the fractional part, over the original denominator. Example: 7/4 = 1 remainder 3, so 7/4 = 1 and 3/4 (written as 1 3/4).
Convert each mixed number to an improper fraction first. For 2 3/4: (2×4)+3 = 11, so 2 3/4 = 11/4. Then add the improper fractions using the standard LCD method. Convert the result back to a mixed number. Example: 1 1/2 + 2 1/3 = 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3 5/6.
Use the Euclidean algorithm: gcd(a,b) = gcd(b, a mod b), repeating until the remainder is 0. The last non-zero value is the GCD. Example: GCD(48,18). 48 mod 18 = 12. 18 mod 12 = 6. 12 mod 6 = 0. GCD = 6. This is far faster than listing factors for large numbers.
Divide the numerator by the denominator. Example: 3/4 = 3 ÷ 4 = 0.75. Some fractions produce repeating decimals: 1/3 = 0.333… (repeating 3), 1/7 = 0.142857… (repeating 6-digit block). This calculator shows the decimal equivalent (rounded to 6 decimal places) alongside every result.
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