How to Add, Subtract, Multiply & Divide Fractions

Fractions appear everywhere: cooking recipes, construction measurements, financial calculations, and standardized tests. Mastering the four basic operations — addition, subtraction, multiplication, and division — eliminates the guesswork and makes you faster at mental math.

This guide walks through each operation with clear steps and real numbers. For instant answers, use our Fraction Calculator to add, subtract, multiply, or divide any two fractions.

Key Vocabulary

Before jumping into the steps, make sure these terms are clear:

• Numerator: The top number (how many parts you have).
• Denominator: The bottom number (how many equal parts the whole is divided into).
• Proper fraction: Numerator is less than the denominator (3/4).
• Improper fraction: Numerator is greater than or equal to the denominator (7/4).
• Mixed number: A whole number plus a fraction (1 3/4).
• Least Common Denominator (LCD): The smallest number that both denominators divide into evenly.
• Greatest Common Divisor (GCD): The largest number that divides evenly into both the numerator and denominator. Used to simplify.

Adding Fractions

Same Denominator

When the denominators match, add the numerators and keep the denominator.

$\frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7}$

Different Denominators

When denominators differ, you need a common denominator before adding.

Step 1: Find the LCD of the two denominators. Step 2: Convert each fraction so its denominator equals the LCD. Step 3: Add the numerators. Keep the LCD as the denominator. Step 4: Simplify if possible.

Example: 1/4 + 2/3

1. LCD of 4 and 3 = 12
2. Convert: 1/4 = 3/12, and 2/3 = 8/12
3. Add: 3/12 + 8/12 = 11/12
4. 11 and 12 share no common factor, so 11/12 is already simplified.

Cooking example: A recipe calls for 1/2 cup of flour and 1/3 cup of sugar. How much total dry ingredient?

• LCD of 2 and 3 = 6
• 1/2 = 3/6, and 1/3 = 2/6
• 3/6 + 2/6 = 5/6 cup

Subtracting Fractions

The process mirrors addition. Get a common denominator, then subtract the numerators.

Example: 5/6 - 1/4

1. LCD of 6 and 4 = 12
2. Convert: 5/6 = 10/12, and 1/4 = 3/12
3. Subtract: 10/12 - 3/12 = 7/12

Construction example: You have a board that is 3/4 inch thick and need to plane off 1/8 inch. Remaining thickness:

• LCD of 4 and 8 = 8
• 3/4 = 6/8
• 6/8 - 1/8 = 5/8 inch

Multiplying Fractions

Multiplication is the simplest operation. Multiply straight across — numerator times numerator, denominator times denominator.

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Example: 2/3 x 4/5

$\frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

No common denominator needed. No conversion step.

Pro tip: Cross-cancel before multiplying. If the numerator of one fraction and the denominator of the other share a common factor, divide both by that factor first. This keeps the numbers smaller and often eliminates the need to simplify at the end.

Example with cross-canceling: 3/8 x 4/9

• The 3 (numerator) and 9 (denominator) share a factor of 3. Reduce to 1 and 3.
• The 4 (numerator) and 8 (denominator) share a factor of 4. Reduce to 1 and 2.
• Now multiply: 1/2 x 1/3 = 1/6

Without cross-canceling, you would get 12/72 and then need to simplify.

Dividing Fractions

To divide by a fraction, flip the second fraction (take its reciprocal) and multiply.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Example: 3/4 / 2/5

$\frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$

Sewing example: You have 3/4 yard of ribbon and each bow requires 1/8 yard. How many bows can you make?

$\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = \mathbf{6 \ bows}$

Working with Mixed Numbers

A mixed number like 2 1/3 must be converted to an improper fraction before performing any operation.

Conversion formula:

$Improper \ Fraction = \frac{(Whole \times Denominator) + Numerator}{Denominator}$

Example: Convert 2 1/3 to an improper fraction.

$\frac{(2 \times 3) + 1}{3} = \frac{7}{3}$

Adding Mixed Numbers: 1 1/2 + 2 2/3

1. Convert: 1 1/2 = 3/2, and 2 2/3 = 8/3
2. LCD of 2 and 3 = 6
3. 3/2 = 9/6, and 8/3 = 16/6
4. 9/6 + 16/6 = 25/6
5. Convert back: 25/6 = 4 1/6

Subtracting Mixed Numbers: 5 1/4 - 2 3/4

1. Convert: 5 1/4 = 21/4, and 2 3/4 = 11/4
2. Same denominator already: 21/4 - 11/4 = 10/4
3. Simplify: 10/4 = 5/2 = 2 1/2

How to Simplify Fractions

A fraction is in simplest form when the numerator and denominator share no common factor other than 1. To simplify, find the GCD and divide both parts.

Example: Simplify 18/24

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• GCD = 6
• 18/6 = 3, and 24/6 = 4
• Simplified: 3/4

Quick method: If both numbers are even, divide by 2 repeatedly. If both are divisible by 3, divide by 3. Continue until no common factor remains.

Common Mistakes to Avoid

1. Adding denominators. 1/3 + 1/4 isn’t 2/7. You must find a common denominator first. The correct answer is 7/12.
2. Forgetting to simplify. An answer of 6/8 is correct but incomplete. Always reduce to 3/4.
3. Not converting mixed numbers. You can’t multiply 2 1/2 x 3 by doing “2 x 3 = 6 and 1/2 x 3 = 3/2.” Convert to 5/2 first, then multiply.
4. Flipping the wrong fraction when dividing. Only flip the second fraction (the divisor). The first stays unchanged.
5. Losing the sign in subtraction. When subtracting, pay attention to which fraction is larger. If the result is negative, write it as a negative fraction.

Real-World Applications

Scenario	Operation	Example
Doubling a recipe	Multiplication	3/4 cup x 2 = 1 1/2 cups
Splitting pizza evenly	Division	3/4 pizza / 3 people = 1/4 each
Combining board lengths	Addition	5 1/2 ft + 3 3/4 ft = 9 1/4 ft
Cutting fabric	Subtraction	2 1/3 yd - 7/8 yd = 1 11/24 yd
Calculating discounts	Multiplication	1/3 off $45 = $15 savings

Frequently Asked Questions

How do I find the least common denominator quickly?

List the multiples of the larger denominator until you find one that the smaller denominator divides into evenly. For 4 and 6: multiples of 6 are 6, 12, 18… 12 is divisible by 4, so LCD = 12. For larger numbers, use the formula: LCD = (a x b) / GCD(a, b). For 12 and 18, GCD is 6, so LCD = (12 x 18) / 6 = 36.

Can I add fractions without finding a common denominator?

The common denominator is mathematically required for addition and subtraction. A shortcut is to cross-multiply: for a/b + c/d, the answer is (ad + bc) / bd, then simplify. For 1/3 + 1/4: (1x4 + 1x3) / (3x4) = 7/12. This always works but may produce larger numbers that need more simplifying.

What is the difference between a fraction and a decimal?

A fraction represents a ratio of two integers (3/4). A decimal represents the same value in base-10 notation (0.75). To convert a fraction to a decimal, divide the numerator by the denominator: 3 / 4 = 0.75. Some fractions produce repeating decimals (1/3 = 0.333…), which is why fractions are often more precise for exact calculations. Use our Percentage Calculator to convert between fractions, decimals, and percentages.

How do I convert a decimal back to a fraction?

Write the decimal as a fraction over a power of 10, then simplify. For 0.625: that is 625/1000. The GCD of 625 and 1000 is 125. Divide both: 625/125 = 5 and 1000/125 = 8. The fraction is 5/8. For repeating decimals like 0.333…, recognize that 1/3 produces this pattern, so 0.333… = 1/3.

Run any fraction problem through our Fraction Calculator to verify your work and see the fully simplified result.