Fractions show up everywhere: in recipes, measurements, money, test scores, and just about every math class from fourth grade onward. Yet they trip up more students than almost any other topic, usually for one simple reason. The rules for the four operations are not the same. You handle a common denominator for one operation, flip a fraction for another, and multiply straight across for a third. Once you know which rule belongs to which operation, the confusion disappears.
In this guide you will learn exactly how to add, subtract, multiply and divide fractions, with every step shown and double-checked. We will start with the vocabulary you need, then walk through each operation with fully worked examples, work through a real-world word problem, and share the most common mistakes so you can avoid them. If you want to check an answer along the way, our free fraction calculator shows the worked steps instantly.
This is written for students from middle school through early college, and for parents helping with homework. No prior memorization is assumed. By the end, you will have a reliable method for any fraction problem you meet.
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Fraction Basics: The Words You Need First
A fraction represents a part of a whole. It is written as one number over another, separated by a bar:
$$\frac{\text{numerator}}{\text{denominator}}$$The numerator (top number) tells you how many parts you have. The denominator (bottom number) tells you how many equal parts the whole is divided into. In \(\frac{3}{4}\), the whole is split into 4 equal parts and you have 3 of them.
Three kinds of fractions appear constantly:
| Term | What it means | Example |
|---|---|---|
| Proper fraction | Numerator smaller than denominator (value less than 1) | \(\frac{2}{5}\) |
| Improper fraction | Numerator equal to or larger than denominator (value 1 or more) | \(\frac{7}{4}\) |
| Mixed number | A whole number combined with a proper fraction | \(1\frac{3}{4}\) |
Equivalent fractions are different-looking fractions with the same value, such as \(\frac{1}{2}=\frac{2}{4}=\frac{4}{8}\). You create them by multiplying or dividing the numerator and denominator by the same number. This single idea powers addition and subtraction, so keep it close.
Key idea
Multiplying or dividing the top and bottom of a fraction by the same nonzero number never changes its value. That is why \(\frac{2}{3}\) and \(\frac{8}{12}\) are the same amount, just written with different parts.
Converting Between Mixed Numbers and Improper Fractions
Most operations are easier with improper fractions, so it helps to switch back and forth quickly.
Mixed number to improper: multiply the whole number by the denominator, add the numerator, and keep the same denominator. For \(3\frac{1}{2}\):
$$3\frac{1}{2}=\frac{(3\times 2)+1}{2}=\frac{7}{2}$$Improper to mixed: divide the numerator by the denominator. The quotient is the whole number and the remainder becomes the new numerator. For \(\frac{11}{6}\), since \(11\div 6=1\) remainder \(5\):
$$\frac{11}{6}=1\frac{5}{6}$$How to Add and Subtract Fractions
Addition and subtraction share one golden rule: the denominators must match before you combine. You can only add or subtract parts that are the same size. Adding \(\frac{1}{2}\) and \(\frac{1}{3}\) directly makes no sense because halves and thirds are different-sized pieces.
Same Denominator: The Easy Case
When the denominators already match, simply add or subtract the numerators and keep the denominator the same.
$$\frac{3}{8}+\frac{2}{8}=\frac{3+2}{8}=\frac{5}{8}$$Do not add the denominators. The bottom number names the size of the pieces, and that size does not change just because you have more of them.
Different Denominators: Find the LCD
When denominators differ, you need a common denominator, ideally the least common denominator (LCD), which is the least common multiple of the two denominators. Then rewrite each fraction as an equivalent fraction over that denominator.
The five-step method
- Find the LCD of the denominators.
- Rewrite each fraction with the LCD.
- Add or subtract the numerators.
- Keep the common denominator.
- Simplify the result.
Worked Example 1: Adding unlike fractions. Solve \(\frac{2}{3}+\frac{1}{4}\).
Step 1 — LCD. The multiples of 3 are 3, 6, 9, 12; multiples of 4 are 4, 8, 12. The smallest shared multiple is 12, so the LCD is 12.
Step 2 — Rewrite each fraction over 12.
$$\frac{2}{3}=\frac{2\times 4}{3\times 4}=\frac{8}{12}\qquad\qquad \frac{1}{4}=\frac{1\times 3}{4\times 3}=\frac{3}{12}$$Step 3 — Add the numerators.
$$\frac{8}{12}+\frac{3}{12}=\frac{8+3}{12}=\frac{11}{12}$$Step 4 — Simplify. The numerator 11 and denominator 12 share no common factor, so the fraction is already in lowest terms.
Worked Example 2: Subtracting mixed numbers. Solve \(3\frac{1}{2}-1\frac{2}{3}\).
Step 1 — Convert to improper fractions.
$$3\frac{1}{2}=\frac{(3\times 2)+1}{2}=\frac{7}{2}\qquad\qquad 1\frac{2}{3}=\frac{(1\times 3)+2}{3}=\frac{5}{3}$$Step 2 — Find the LCD. The denominators are 2 and 3, so the LCD is 6.
Step 3 — Rewrite both fractions over 6.
$$\frac{7}{2}=\frac{7\times 3}{2\times 3}=\frac{21}{6}\qquad\qquad \frac{5}{3}=\frac{5\times 2}{3\times 2}=\frac{10}{6}$$Step 4 — Subtract the numerators.
$$\frac{21}{6}-\frac{10}{6}=\frac{21-10}{6}=\frac{11}{6}$$Step 5 — Convert back to a mixed number. Since \(11\div 6=1\) remainder \(5\):
$$\frac{11}{6}=1\frac{5}{6}$$Common mistake
Never add or subtract the denominators. \(\frac{1}{4}+\frac{1}{4}\) equals \(\frac{2}{4}=\frac{1}{2}\), not \(\frac{2}{8}\). The denominator stays the same; only the numerators combine.
How to Multiply Fractions
Here is the good news: multiplication is the simplest of the four operations. You do not need a common denominator. Just multiply straight across, numerator times numerator and denominator times denominator, then simplify.
$$\frac{a}{b}\times\frac{c}{d}=\frac{a\times c}{b\times d}$$Worked Example 3: Multiplying two fractions. Solve \(\frac{3}{4}\times\frac{2}{5}\).
Step 1 — Multiply across.
$$\frac{3}{4}\times\frac{2}{5}=\frac{3\times 2}{4\times 5}=\frac{6}{20}$$Step 2 — Simplify. Both 6 and 20 are divisible by 2:
$$\frac{6}{20}=\frac{6\div 2}{20\div 2}=\frac{3}{10}$$A Shortcut: Cross-Cancel Before You Multiply
You can simplify before multiplying by canceling a common factor between any numerator and any denominator. This keeps the numbers small. Using the same example, the 2 in the second numerator and the 4 in the first denominator share a factor of 2:
$$\frac{3}{\cancel{4}^{\,2}}\times\frac{\cancel{2}^{\,1}}{5}=\frac{3}{2}\times\frac{1}{5}=\frac{3}{10}$$Same answer, less reducing at the end. This is especially helpful with large numbers.
Multiplying With Whole Numbers and Mixed Numbers
Write any whole number as a fraction over 1. For instance, \(6=\frac{6}{1}\). Convert any mixed number to an improper fraction first, then multiply. To find \(\frac{2}{3}\) of \(1\frac{1}{2}\):
$$\frac{2}{3}\times 1\frac{1}{2}=\frac{2}{3}\times\frac{3}{2}=\frac{6}{6}=1$$Study tip
In math, the word “of” usually means multiply. “What is \(\frac{1}{2}\) of \(\frac{3}{4}\)?” is just \(\frac{1}{2}\times\frac{3}{4}=\frac{3}{8}\). Spotting that keyword makes word problems much easier.
How to Divide Fractions
Division looks intimidating but uses one memorable rule: Keep, Change, Flip. Keep the first fraction, change the division sign to multiplication, and flip the second fraction to its reciprocal (swap its numerator and denominator). Then you simply multiply.
$$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$$The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\); the reciprocal of 5 (that is, \(\frac{5}{1}\)) is \(\frac{1}{5}\). Dividing by a number is the same as multiplying by its reciprocal, which is why this works.
Worked Example 4: Dividing two fractions. Solve \(\frac{5}{6}\div\frac{2}{3}\).
Step 1 — Keep, Change, Flip.
$$\frac{5}{6}\div\frac{2}{3}=\frac{5}{6}\times\frac{3}{2}$$Step 2 — Multiply across.
$$\frac{5}{6}\times\frac{3}{2}=\frac{5\times 3}{6\times 2}=\frac{15}{12}$$Step 3 — Simplify. Both 15 and 12 divide by 3:
$$\frac{15}{12}=\frac{15\div 3}{12\div 3}=\frac{5}{4}=1\frac{1}{4}$$Dividing Mixed Numbers
Convert each mixed number to an improper fraction before applying Keep, Change, Flip. Solve \(2\frac{1}{4}\div\frac{3}{8}\):
Step 1 — Convert. \(2\frac{1}{4}=\frac{(2\times 4)+1}{4}=\frac{9}{4}\).
Step 2 — Keep, Change, Flip.
$$\frac{9}{4}\div\frac{3}{8}=\frac{9}{4}\times\frac{8}{3}$$Step 3 — Multiply and simplify.
$$\frac{9}{4}\times\frac{8}{3}=\frac{72}{12}=6$$Common mistake
Only flip the second fraction (the divisor). A frequent error is flipping the first fraction or both. Keep the first one exactly as it is.
Putting It All Together: A Word Problem
Real problems often combine operations. Working carefully one step at a time is the key, just as in our guide on How to Solve Math Word Problems: A Step-by-Step Framework.
Worked Example 5. A cookie recipe calls for \(\frac{2}{3}\) cup of sugar for one batch. You want to make \(1\frac{1}{2}\) batches. You already have \(\frac{1}{4}\) cup of sugar measured out. How much more sugar do you need?
Step 1 — Find the total sugar needed (multiply). Multiply the per-batch amount by the number of batches.
$$\frac{2}{3}\times 1\frac{1}{2}=\frac{2}{3}\times\frac{3}{2}=\frac{6}{6}=1 \text{ cup}$$Step 2 — Subtract what you already have. You need 1 cup and have \(\frac{1}{4}\) cup. Write 1 as \(\frac{4}{4}\) so the denominators match.
$$1-\frac{1}{4}=\frac{4}{4}-\frac{1}{4}=\frac{3}{4} \text{ cup}$$Try it yourself
Want to confirm a multi-step answer in seconds? Solve it instantly with our free fraction calculator, which lays out each operation so you can compare it with your own work.
Simplifying Fractions and Final Answers
A complete answer is usually written in lowest terms. To simplify, divide the numerator and denominator by their greatest common factor (GCF), the largest number that divides both evenly.
For \(\frac{12}{18}\), the GCF of 12 and 18 is 6:
$$\frac{12}{18}=\frac{12\div 6}{18\div 6}=\frac{2}{3}$$If you cannot spot the GCF right away, divide by any common factor and repeat. Dividing \(\frac{12}{18}\) by 2 gives \(\frac{6}{9}\), then dividing by 3 gives \(\frac{2}{3}\). You reach the same place either way. Teachers also generally prefer improper fractions converted to mixed numbers in a final answer, so \(\frac{11}{6}\) becomes \(1\frac{5}{6}\).
Quick Reference: The Four Operations
Keep this table handy until the rules become automatic.
| Operation | What to do | Example |
|---|---|---|
| Add | Get a common denominator, then add the numerators | \(\frac{1}{3}+\frac{1}{6}=\frac{2}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}\) |
| Subtract | Get a common denominator, then subtract the numerators | \(\frac{3}{4}-\frac{1}{2}=\frac{3}{4}-\frac{2}{4}=\frac{1}{4}\) |
| Multiply | Multiply straight across, then simplify | \(\frac{2}{3}\times\frac{1}{4}=\frac{2}{12}=\frac{1}{6}\) |
| Divide | Keep, Change, Flip, then multiply | \(\frac{1}{2}\div\frac{1}{4}=\frac{1}{2}\times\frac{4}{1}=2\) |
Notice the pattern: addition and subtraction need a common denominator, while multiplication and division do not. That single distinction prevents most errors.
Common Mistakes to Avoid
- Adding denominators. When adding or subtracting, the denominator stays put. Only the numerators change.
- Forgetting a common denominator. You cannot combine \(\frac{1}{2}\) and \(\frac{1}{3}\) until both are sixths.
- Looking for a common denominator when multiplying or dividing. You do not need one. Multiply across, or flip and multiply.
- Flipping the wrong fraction. In division, only the divisor (the second fraction) is inverted.
- Skipping simplification. A correct but unreduced answer like \(\frac{6}{20}\) often loses points; reduce it to \(\frac{3}{10}\).
- Leaving mixed numbers as mixed when multiplying or dividing. Always convert to improper fractions first.
Practice strategy
Work three problems of each operation per day for a week, and always check your final answer by simplifying. If your steps follow the correct rule for each operation, your result will be reliable. For a refresher on operation order in longer expressions, see our guide to the Order of Operations (PEMDAS) Explained.
Where Fractions Connect to Other Math
Fraction skills are the foundation for many later topics. A fraction is really a division statement, so \(\frac{3}{4}\) equals \(3\div 4=0.75\). That link is exactly how fractions, decimals, and percentages relate to one another. If you want to extend these skills, our walkthrough on How to Calculate Percentages (with Examples) builds directly on the multiplication and division you practiced here.
Algebra leans on the same rules too. Solving equations with fractional coefficients, simplifying rational expressions, and working with ratios all reuse common denominators and reciprocals. Mastering the four operations now pays off for years.
Conclusion
You now have a complete, reliable method for every fraction operation. Add and subtract by finding a common denominator and combining numerators. Multiply straight across, canceling common factors when you can. Divide with Keep, Change, Flip. Then simplify and, if needed, convert back to a mixed number. With those four habits, no fraction problem can surprise you.
The best way to lock in these skills is steady practice with quick feedback. Work a problem by hand, then verify each step with our fraction calculator to see exactly where your reasoning is solid. For more clear, example-driven lessons across arithmetic, algebra, and beyond, explore the Math Solver AI blog and keep building your confidence one problem at a time.
