How to Add, Subtract, Multiply & Divide Fractions (Complete Guide 2026)
Quick Answer
- *Adding/subtracting: find the LCD, convert both fractions, then add or subtract numerators.
- *Multiplying: multiply numerators together and denominators together — straight across.
- *Dividing: keep, change, flip — multiply by the reciprocal of the second fraction.
- *Simplifying: divide numerator and denominator by their greatest common factor (GCF).
What Is a Fraction?
A fraction represents a part of a whole. It has two parts: the numerator (top number) tells you how many parts you have, and the denominator (bottom number) tells you how many equal parts the whole is divided into.
In 3/4, the denominator 4 means the whole is cut into 4 equal pieces. The numerator 3 means you have 3 of those pieces. Simple enough. But fractions trip people up the moment you try to combine them — and that's where the rules matter.
Fractions aren't an obscure math concept. They're everywhere: a recipe calls for 2/3 cup of flour, a contractor quotes lumber in 3/4-inch boards, music is written in 3/4 or 4/4 time, and financial ratios like price-to-earnings are fractions at their core. According to the National Assessment of Educational Progress (NAEP, 2024), only 24% of 8th graders scored "proficient or above" in fractions — making it one of the most challenging areas in K–8 math. The concept itself is ancient: the Rhind Mathematical Papyrus (circa 1650 BCE), now held at the British Museum, contains fraction tables used by Egyptian scribes for land measurement and grain distribution.
Fractions are introduced in the US curriculum in 3rd grade under the Common Core State Standards and are expected to be mastered by 5th grade. A 2023 study in the Journal of Educational Psychologyfound that students who struggle with fractions in 5th grade are significantly more likely to struggle with algebra in middle school — making fraction fluency one of the strongest academic predictors for later math success.
Adding Fractions
Same Denominator
When the denominators match, addition is straightforward. Add the numerators and keep the denominator unchanged.
Example: 2/7 + 3/7 = 5/7
That's it. The denominator stays at 7 because the size of each piece hasn't changed — you just have more of them.
Different Denominators — The LCD Method
You can't add 1/3 and 1/4 directly because the pieces are different sizes. A third of a pizza and a quarter of a pizza are different amounts. To add them, you need to cut everything into equal-sized pieces first. That means finding a common denominator.
The most efficient approach uses the Least Common Multiple (LCM) of the two denominators, which becomes the Least Common Denominator (LCD).
| Step | Action | Result |
|---|---|---|
| 1 | Find LCM of 3 and 4 | LCM = 12 |
| 2 | Convert 1/3 → ?/12 | 1 × 4 = 4, so 4/12 |
| 3 | Convert 1/4 → ?/12 | 1 × 3 = 3, so 3/12 |
| 4 | Add numerators | 4/12 + 3/12 = 7/12 |
Worked example: 1/3 + 1/4 → LCM(3, 4) = 12 → 4/12 + 3/12 = 7/12
To find the LCM, list multiples of each denominator until you find the first one they share. Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. First common value: 12.
Subtracting Fractions
Subtraction follows the same logic as addition. If the denominators match, subtract the numerators and keep the denominator. If they differ, find the LCD first.
Worked example:3/4 − 1/6
| Step | Action | Result |
|---|---|---|
| 1 | Find LCM of 4 and 6 | LCM = 12 |
| 2 | Convert 3/4 → ?/12 | 3 × 3 = 9, so 9/12 |
| 3 | Convert 1/6 → ?/12 | 1 × 2 = 2, so 2/12 |
| 4 | Subtract numerators | 9/12 − 2/12 = 7/12 |
3/4 − 1/6 = 7/12
Watch out for subtraction with mixed numbers when the fraction part of the first number is smaller than the second. You'll need to borrow from the whole number — covered in the mixed numbers section below.
Multiplying Fractions
Multiplication is the easiest operation. No common denominator needed. Multiply straight across: numerator × numerator, denominator × denominator.
Worked example:2/3 × 3/4
Numerators: 2 × 3 = 6
Denominators: 3 × 4 = 12
Result: 6/12 = 1/2 (simplified by dividing both by 6)
A useful shortcut: look for common factors between any numerator and any denominator beforemultiplying. In 2/3 × 3/4, the 3 in the numerator of the second fraction cancels with the 3 in the denominator of the first. And the 2 in the first numerator and the 4 in the second denominator share a factor of 2. Cross-cancel first: (2/3) × (3/4) → (1/1) × (1/2) = 1/2. Same answer, less arithmetic.
Dividing Fractions
Division has one rule worth memorizing: keep, change, flip.
- Keep the first fraction as-is.
- Change the division sign to multiplication.
- Flip the second fraction (write its reciprocal).
Then multiply straight across.
Worked example:2/3 ÷ 3/4
Keep 2/3. Change ÷ to ×. Flip 3/4 to 4/3.
2/3 × 4/3 = 8/9
Result: 8/9
Why does this work? Dividing by a fraction is the same as asking "how many times does this fraction fit?" Multiplying by the reciprocal gives you the same answer through a cleaner calculation. The reciprocal of 3/4 is 4/3 — just swap numerator and denominator.
Simplifying Fractions
A fraction is in simplest form (also called lowest terms) when the numerator and denominator share no common factor other than 1. To simplify, find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both by it.
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
18 ÷ 6 = 3 24 ÷ 6 = 4
Simplified: 3/4
If you're not sure of the GCF right away, divide by any common factor you can see and repeat until no common factors remain. Dividing 18/24 by 2 gives 9/12. Dividing 9/12 by 3 gives 3/4. Same result, two steps instead of one.
Mixed Numbers and Improper Fractions
A mixed number combines a whole number and a fraction, like 2 3/4. An improper fraction has a numerator larger than or equal to its denominator, like 11/4. Both represent the same value.
Converting Mixed Number to Improper Fraction
Multiply the whole number by the denominator, then add the numerator. Place the result over the original denominator.
2 3/4 → (2 × 4) + 3 = 8 + 3 = 11 → 11/4
Converting Improper Fraction to Mixed Number
Divide the numerator by the denominator. The quotient becomes the whole number; the remainder becomes the new numerator over the same denominator.
11/4 → 11 ÷ 4 = 2 remainder 3 → 2 3/4
Operating with Mixed Numbers
The cleanest approach: convert mixed numbers to improper fractions first, perform the operation, then convert back if needed.
Example: 1 1/2 + 2 2/3
Convert: 1 1/2 = 3/2 2 2/3 = 8/3
Find LCD of 2 and 3: LCD = 6
3/2 = 9/6 8/3 = 16/6
9/6 + 16/6 = 25/6
Convert back: 25 ÷ 6 = 4 remainder 1 → 4 1/6
Summary of All Four Operations
| Operation | Rule | Example | Answer |
|---|---|---|---|
| Addition | Find LCD, convert, add numerators | 1/3 + 1/4 | 7/12 |
| Subtraction | Find LCD, convert, subtract numerators | 3/4 − 1/6 | 7/12 |
| Multiplication | Multiply straight across | 2/3 × 3/4 | 1/2 |
| Division | Keep, change, flip — then multiply | 2/3 ÷ 3/4 | 8/9 |
Where Fractions Show Up in Real Life
Fractions aren't just a classroom exercise. You use them constantly without thinking about it:
- Cooking and baking: Scaling a recipe from 4 servings to 6 means multiplying every ingredient by 3/2 (1.5). A recipe for 2/3 cup of butter doubled needs 4/3 cups — or 1 1/3 cups.
- Construction and home improvement: Lumber dimensions (a "2x4" is actually 1 1/2" × 3 1/2"), tile layouts, and paint coverage all involve fraction arithmetic.
- Music: Time signatures are fractions. 3/4 time means 3 quarter notes per measure. 6/8 means 6 eighth notes. Subdividing rhythms is fraction division.
- Finance: Interest rates, stock price-to-earnings ratios, and debt-to-income ratios are all fractions. A mortgage rate of 6.875% is 6 7/8 percent — a mixed number.
- Probability: The chance of rolling a 3 on a die is 1/6. Drawing two aces from a deck? That's a multiplication of fractions: 4/52 × 3/51.
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Frequently Asked Questions
How do you add fractions with different denominators?
Find the least common denominator (LCD) of the two fractions. Convert each fraction so it has the LCD as its denominator, then add the numerators and keep the denominator. For example, 1/3 + 1/4: the LCD of 3 and 4 is 12. Convert to 4/12 + 3/12 = 7/12.
What is the rule for dividing fractions?
Use the "keep, change, flip" rule: keep the first fraction as-is, change the division sign to multiplication, and flip (take the reciprocal of) the second fraction. Then multiply straight across. For example, 2/3 ÷ 3/4 becomes 2/3 × 4/3 = 8/9.
How do you simplify a fraction?
Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. To simplify 6/12: the GCF of 6 and 12 is 6. Dividing both by 6 gives 1/2. A fraction is fully simplified when the numerator and denominator share no common factor other than 1.
How do you convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, then add the numerator. Place the result over the original denominator. For example, 2 3/4: multiply 2 × 4 = 8, then add the numerator: 8 + 3 = 11. The improper fraction is 11/4.
What does LCD mean in fractions?
LCD stands for Least Common Denominator. It is the smallest number that is a multiple of both denominators. Finding the LCD lets you rewrite fractions with the same denominator so you can add or subtract them. The LCD is the same as the LCM (Least Common Multiple) of the two denominators.
Can you add fractions with different denominators without finding the LCD?
Yes. You can multiply the denominators together to get a common denominator, then cross-multiply to find the new numerators. For 1/3 + 1/4, use denominator 3 × 4 = 12. The numerators become 1 × 4 = 4 and 1 × 3 = 3, giving 4/12 + 3/12 = 7/12. This always works, but it may give a larger denominator that requires more simplification afterward. The LCD method produces the smallest possible common denominator from the start.