Fraction 1
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numerator / denominator
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Fraction 2
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Enter denominator (not 0).
numerator / denominator
Sum of Fractions
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Step-by-Step Working (LCD Method)
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Sources & Methodology
Fraction addition algorithm verified against Khan Academy mathematics curriculum and standard textbook methods (LCD approach).
Khan Academy — Adding Fractions with Different Denominators
Curriculum-standard LCD method for adding fractions used as reference for step-by-step output and formula verification
National Council of Teachers of Mathematics (NCTM)
Standards for fraction arithmetic confirming LCD method as the appropriate algorithm for grade-level instruction
Methodology: Step 1: compute GCD of denominators using Euclidean algorithm. Step 2: LCD = (d1 × d2) / GCD(d1,d2). Step 3: convert both fractions to LCD denominator. Step 4: add numerators. Step 5: simplify result fraction by dividing both terms by GCD(resultNumerator, LCD). Step 6: convert to mixed number if improper.
⏱ Last reviewed: April 2026
How to Add Fractions Step by Step
Adding fractions requires the denominators to match before you can add the numerators. If denominators are different, you must find the Least Common Denominator (LCD) first and convert each fraction. Once denominators match, addition is straightforward — add the numerators and simplify.
Adding Fractions with Different Denominators (LCD Method)
a/b + c/d = (a×(LCD/b) + c×(LCD/d)) / LCD
Step 1: Find the LCD (Least Common Denominator) of b and d
Step 2: Multiply each numerator by (LCD ÷ its denominator)
Step 3: Add the new numerators over the LCD
Step 4: Simplify by dividing by the GCD
Example: 1/3 + 1/4
LCD = 12 | 1/3 = 4/12 | 1/4 = 3/12
4/12 + 3/12 = 7/12
Step 2: Multiply each numerator by (LCD ÷ its denominator)
Step 3: Add the new numerators over the LCD
Step 4: Simplify by dividing by the GCD
Example: 1/3 + 1/4
LCD = 12 | 1/3 = 4/12 | 1/4 = 3/12
4/12 + 3/12 = 7/12
Adding Fractions with the Same Denominator
a/b + c/b = (a + c) / b
Simply add the numerators and keep the denominator unchanged. Then simplify.
Example: 3/8 + 5/8 = (3+5)/8 = 8/8 = 1
Example: 3/8 + 5/8 = (3+5)/8 = 8/8 = 1
How to Find the LCD (Least Common Denominator)
The LCD is the smallest number that both denominators divide into evenly. There are two methods:
- Listing multiples: List multiples of each denominator until you find the first shared value. Denominators 4 and 6: multiples of 4 = 4, 8, 12…; multiples of 6 = 6, 12… LCD = 12.
- Formula: LCD = (d1 × d2) ÷ GCD(d1, d2). For 4 and 6: GCD = 2, LCD = (4×6)/2 = 12.
Common Fraction Addition Reference Table
| Addition | LCD | Equivalent | Simplified Result |
|---|---|---|---|
| 1/2 + 1/2 | 2 | 1/2 + 1/2 | 1 |
| 1/2 + 1/3 | 6 | 3/6 + 2/6 | 5/6 |
| 1/4 + 1/3 | 12 | 3/12 + 4/12 | 7/12 |
| 2/3 + 3/4 | 12 | 8/12 + 9/12 | 17/12 = 1 5/12 |
| 1/3 + 1/6 | 6 | 2/6 + 1/6 | 3/6 = 1/2 |
| 3/5 + 2/5 | 5 | 3/5 + 2/5 | 5/5 = 1 |
| 3/8 + 1/4 | 8 | 3/8 + 2/8 | 5/8 |
How to Simplify a Fraction (Reduce to Lowest Terms)
After adding, always check if the result can be simplified. Find the Greatest Common Divisor (GCD) of the numerator and denominator, then divide both by it. Example: result is 6/8 — GCD(6,8) = 2, so 6/8 ÷ 2/2 = 3/4. If the GCD is 1, the fraction is already in simplest form.
💡 Shortcut: If one denominator is a multiple of the other, use the larger one as the LCD. Example: 1/4 + 1/8 — since 8 is a multiple of 4, LCD = 8. Convert 1/4 = 2/8. Add: 2/8 + 1/8 = 3/8. No Euclidean algorithm needed.
Frequently Asked Questions
Find the LCD of both denominators. Convert each fraction to an equivalent fraction with the LCD as denominator. Add the numerators and keep the LCD as denominator. Simplify. Example: 1/3 + 1/4: LCD = 12, so 4/12 + 3/12 = 7/12.
Add the numerators and keep the denominator the same. Then simplify if possible. Example: 2/7 + 3/7 = (2+3)/7 = 5/7. Another: 3/8 + 5/8 = 8/8 = 1.
1/2 + 1/3 = 5/6. LCD of 2 and 3 is 6. Convert: 1/2 = 3/6, 1/3 = 2/6. Add: 3/6 + 2/6 = 5/6. Already in simplest form (GCD of 5 and 6 is 1).
1/4 + 1/3 = 7/12. LCD of 4 and 3 is 12. Convert: 1/4 = 3/12, 1/3 = 4/12. Add: 3/12 + 4/12 = 7/12. GCD(7,12) = 1, so 7/12 is already fully simplified.
The LCD is the smallest number both denominators divide into evenly. Method 1: list multiples of each denominator and find the first common one. Method 2: use the formula LCD = (d1 × d2) / GCD(d1, d2). Example: LCD of 6 and 8 = (6×8)/GCD(6,8) = 48/2 = 24.
2/3 + 3/4 = 17/12 = 1 and 5/12. LCD of 3 and 4 is 12. Convert: 2/3 = 8/12, 3/4 = 9/12. Add: 8/12 + 9/12 = 17/12. Improper fraction 17/12 = 1 remainder 5, so 1 and 5/12.
Find the GCD (Greatest Common Divisor) of the numerator and denominator, then divide both by it. Example: 6/8 — GCD(6,8) = 2, so 6/8 = 3/4. If GCD = 1, the fraction cannot be simplified further and is already in lowest terms.
3/5 + 2/5 = 5/5 = 1. Since denominators are already the same, add numerators: 3 + 2 = 5. The result 5/5 simplifies to 1 (a whole number), because any number divided by itself equals 1.
Yes. The LCD method works with negative numerators too. Example: −1/3 + 1/2: LCD = 6, −1/3 = −2/6, 1/2 = 3/6. Add: −2/6 + 3/6 = 1/6. Standard signed arithmetic applies to the numerators while the LCD process handles the denominators.
1/2 + 1/4 = 3/4. Since 4 is a multiple of 2, the LCD is simply 4. Convert: 1/2 = 2/4. Add: 2/4 + 1/4 = 3/4. GCD(3,4) = 1, so 3/4 is already fully simplified.
Divide the numerator by the denominator. The quotient is the whole number, and the remainder over the original denominator is the fraction. Example: 17/12 — 17 ÷ 12 = 1 remainder 5, so 17/12 = 1 and 5/12.
1/3 + 1/6 = 3/6 = 1/2. LCD of 3 and 6 is 6. Convert: 1/3 = 2/6. Add: 2/6 + 1/6 = 3/6. Simplify: GCD(3,6) = 3, so 3/6 ÷ 3/3 = 1/2.
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