Fraction Rules Cheat Sheet
Last updated: February 15, 2026 · All formulas verified against CCSS.Math.Content.NF standards
Every fraction formula you'll ever need, all on one page. Bookmark it, print it, or grab the free PDF below.
Fraction rules are the formulas that tell you how to add, subtract, multiply, divide, simplify, and convert fractions. The short version: addition and subtraction need a common denominator. Multiplication goes straight across. Division means flipping the second fraction and multiplying. And you should always simplify your answer by dividing the top and bottom by their greatest common factor. That's the whole game, and every formula on this page is a variation of those four ideas.
A fraction is a rational number written as one integer over another, like ¾ or 5/8. The reason they confuse people is that each operation has its own rule. You can't just memorize one process and use it everywhere. Adding fractions works nothing like multiplying them. This page covers all of it so you don't have to flip between five different websites. The formulas here match what's taught in Common Core grades 3 through 5, but honestly, plenty of adults still look this stuff up too.
📋 Quick Summary
- Add or subtract: get a common denominator first, then combine the numerators. a/c + b/c = (a+b)/c
- Multiply: go straight across. a/b × c/d = (a×c)/(b×d). No common denominator needed.
- Divide: flip the second fraction, then multiply. a/b ÷ c/d = a/b × d/c
- Mixed numbers: convert to an improper fraction first (whole × denominator + numerator, over the denominator), then use the rules above.
- Simplify: divide the numerator and denominator by their greatest common factor (GCF).
- Convert: fraction → decimal: divide top by bottom. Decimal → fraction: write the digits over 10, 100, etc. and simplify. Fraction → percent: divide and multiply by 100. Percent → fraction: put it over 100 and simplify.
Key Terms You'll See on This Page
These words show up constantly in fraction problems. If any of them feel fuzzy, skim this list before you start.
Denominator: the bottom number. It tells you how many equal parts make up the whole.
Reciprocal: a fraction flipped upside down. The reciprocal of a/b is b/a.
Greatest Common Factor (GCF): the biggest number that divides evenly into both the numerator and denominator.
Least Common Denominator (LCD): the smallest number that both denominators divide into evenly.
Equivalent fractions: fractions that look different but mean the same thing, like 2/4 and 1/2.
Improper fraction: one where the numerator is as big as or bigger than the denominator, like 7/4.
Mixed number: a whole number sitting next to a fraction, like 1¾.
How Do You Add and Subtract Fractions?
Both addition and subtraction follow the same core rule: the denominators have to match before you touch the numerators. Think of it this way. You can add 3 apples and 2 apples because they're the same unit. But you can't directly add 3 apples and 2 oranges. Denominators work the same way. If they already match, great. If not, you'll need to rewrite the fractions as equivalent fractions that share a common denominator. Khan Academy's fraction arithmetic course goes deep on this if you want more practice.
How to Add Fractions With the Same Denominator
This is the easy case. Add the numerators, keep the denominator, done.
a/c − b/c = (a − b)/c
Example: 5/9 − 2/9 = (5 − 2)/9 = 3/9 = 1/3
How to Add Fractions With Different Denominators
When the denominators don't match, you need to find the least common denominator (LCD) first. Rewrite each fraction using the LCD, then add the numerators like normal.
or use the LCD for smaller numbers
LCD of 3 and 4 = 12
1/3 = 4/12, and 1/4 = 3/12
4/12 + 3/12 = 7/12
How Do You Multiply and Divide Fractions?
Good news. Multiplication and division are actually simpler than addition because you don't need a common denominator for either one. Multiplying goes straight across, and dividing is really just multiplying in disguise (you flip the second fraction first). Once you see the pattern, these two are the fastest fraction operations to do by hand.
What Is the Rule for Multiplying Fractions?
Multiply the numerators together. Multiply the denominators together. You're done. No LCD, no rewriting, nothing extra.
What Is the Rule for Dividing Fractions?
Flip the second fraction (that's the reciprocal) and multiply. You might hear teachers call this "keep, change, flip" or "invert and multiply." It isn't a trick. Dividing by any number gives the same result as multiplying by its reciprocal, so flipping the fraction and multiplying is the real math behind the shortcut.
Flip the 2/5 → 5/2
3/4 × 5/2 = 15/8 = 1⅞
How Do You Use Fraction Rules With Mixed Numbers?
Convert the mixed number to an improper fraction first, then use the same rules as above. A mixed number like 2⅓ is really just 7/3 wearing a disguise. Once you rewrite it, all four operations work exactly the same way.
Example: 2⅓ = (2×3 + 1)/3 = 7/3
Convert: 7/3 × 3/2
Multiply: 21/6 = 7/2 = 3½
How Do You Simplify a Fraction?
Simplifying a fraction (some textbooks say "reducing") means dividing the numerator and denominator by their greatest common factor (GCF) until you can't go any further. You know you've hit lowest terms when no number except 1 divides evenly into both the top and bottom. If you think about it, this is the opposite of finding a common denominator: instead of scaling fractions up to equivalent values, you're scaling them down. For big numbers, the Euclidean algorithm finds the GCF faster than listing factors by hand.
How to Find the GCF and Reduce a Fraction
The GCF is the biggest number that goes into both the numerator and denominator evenly. List the factors of each, find the largest one they share, then divide both parts of the fraction by it.
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 → 3/4
Quick Divisibility Shortcuts for Simplifying
You don't always need to compute the GCF from scratch. Try these checks first:
The digits add up to a multiple of 3? Divide by 3. (For example, 18: 1 + 8 = 9, which is divisible by 3.)
Both end in 0 or 5? Divide by 5.
The numerator goes into the denominator with no remainder? Then you've actually got a whole number.
How Do You Convert Between Fractions, Decimals, and Percents?
Fractions, decimals, and percentages are three ways to write the same rational number. You'll bounce between them constantly in real life: reading a tape measure (fractions), punching something into a calculator (decimals), figuring out a sale price (percents). The conversions are all pretty short once you know the steps. Math is Fun's conversion guide has some good visual examples if you want a second explanation.
How to Convert a Fraction to a Decimal
Divide the numerator by the denominator.
How to Convert a Decimal to a Fraction
Count the decimal places. Write the digits over the matching power of 10, then simplify.
How to Convert a Fraction to a Percent
Divide the numerator by the denominator, then multiply by 100.
How to Convert a Percent to a Fraction
Put the percent over 100, then simplify.
Common Conversions Reference
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3…% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 2/3 | 0.666… | 66.6…% |
| 3/4 | 0.75 | 75% |
| 3/5 | 0.6 | 60% |
| 7/8 | 0.875 | 87.5% |
Which Fraction Rule Should You Use?
Knowing the formulas is one thing. Knowing which one to use on a specific problem is another. This table makes it easy to check at a glance.
| Operation | Need Common Denominator? | Formula |
|---|---|---|
| Addition | Yes | a/c + b/c = (a+b)/c |
| Subtraction | Yes | a/c − b/c = (a−b)/c |
| Multiplication | No | a/b × c/d = (a×c)/(b×d) |
| Division | No | a/b ÷ c/d = a/b × d/c |
| Simplification | N/A | (a÷GCF)/(b÷GCF) |
See the split? Addition and subtraction are about combining pieces of the same size, so the denominators have to match. Multiplication and division operate across the fractions directly, no matching needed. And simplification is its own thing, something you do after any operation to get your answer into lowest terms.
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Keep Learning
If you want the full walkthrough on any of these rules, these guides break each one down step by step with extra practice problems.
Frequently Asked Questions About Fraction Rules
Four rules cover almost everything. To add or subtract fractions, the denominators have to match first. Multiplication is straight across, numerator times numerator and denominator times denominator. For division, flip the second fraction and then multiply. After any of those, simplify your result by dividing the top and bottom by their greatest common factor.
You need the least common denominator (LCD) of the two fractions. Rewrite each fraction so they both use the LCD, then add the numerators and keep the denominator. Quick example: 1/3 + 1/4. The LCD of 3 and 4 is 12, which gives you 4/12 + 3/12 = 7/12.
Because dividing by a number gives the same result as multiplying by its reciprocal. The reciprocal of a/b is b/a. So "keep, change, flip" isn't just a classroom trick; it's the actual math. When you flip and multiply, you're really asking "how many groups of this fraction fit inside that one?"
Figure out the greatest common factor (GCF) of the numerator and denominator, then divide both by it. Say you've got 18/24. The GCF of 18 and 24 is 6, so 18 ÷ 6 = 3 and 24 ÷ 6 = 4. That gives you 3/4. Not sure where to start? Check if both numbers are divisible by 2 first, then try 3, then 5.
Just divide the top number by the bottom number. 3/8 = 3 ÷ 8 = 0.375. That's literally it. The one catch is that some fractions produce repeating decimals. 1/3 becomes 0.333… and it never stops. You can round those, or use a bar over the repeating digit to show it goes on forever.
Look at how many digits come after the decimal point. Write those digits over the matching power of 10. So 0.75 has two decimal places, which makes it 75/100. Simplify and you get 3/4. Repeating decimals are trickier. For something like 0.333…, there's an algebra trick: set x = 0.333…, multiply both sides by 10 to get 10x = 3.333…, subtract the original equation, and you land on x = 1/3.
Divide the numerator by the denominator, then multiply by 100. So 3/5 = 0.6, and 0.6 × 100 = 60%. There's also a shortcut: if you can scale the denominator to 100, the numerator becomes the percent directly. 3/5 = 60/100, so it's 60%. That trick saves a step whenever 100 divides evenly by your denominator.
A proper fraction is less than 1. The numerator is smaller than the denominator, like 3/4. An improper fraction is 1 or more because the numerator is at least as big as the denominator, like 7/4. The name "improper" makes it sound wrong, but it's perfectly valid. You can convert improper fractions to mixed numbers if you want: 7/4 = 1¾.
Same rules, one extra step at the beginning. Convert the mixed numbers to improper fractions first, then everything else works exactly the same. Take 2⅓ × 1½: rewrite as 7/3 × 3/2, multiply straight across to get 21/6, simplify to 7/2, and that's 3½.
It's the smallest number that both denominators divide into evenly. Easiest way to find it: list the multiples of each denominator and pick the first one they share. With 4 and 6, the multiples of 4 go 4, 8, 12, 16… and the multiples of 6 go 6, 12, 18… so the LCD is 12. If the numbers are bigger and listing multiples takes too long, use the formula LCD = (a × b) ÷ GCF(a, b).
Not really, no. But there is a shortcut that gets around it. Cross-multiply: take each numerator times the other fraction's denominator, add those results, and put the total over the product of both denominators. Works every time. The only downside is it can leave you with bigger numbers than the LCD method would, so you'll often need to simplify at the end.
Try this: same denominator? Just combine the tops. Different denominators? Find the LCD first. Multiplying? Go straight across. Dividing? Flip and multiply. The one thing worth drilling into your head is that only addition and subtraction require matching denominators. Multiplication and division don't. Once that clicks, the rest follows pretty quickly.
Technically, no. 6/8 and 3/4 are the same value, so leaving a fraction unsimplified isn't "wrong." But most teachers and tests expect the simplified version, and you'll lose points if you don't reduce. Good habit: after every problem, check whether the numerator and denominator share a common factor. Takes two seconds and saves you marks.