Fraction Calculator
A fraction calculator adds, subtracts, multiplies and divides fractions and mixed numbers, such as 1/3 + 1/4 = 7/12, and reduces any answer to lowest terms. This free tool also simplifies fractions, converts between fractions and decimals, and splits rational expressions into partial fractions. Every result comes with a step-by-step solution and a check of the answer, and needs no signup.
Calculation
Add fractions
Add two fractions or mixed numbers.
Formula
Leave Whole empty for a simple fraction. To enter 2 1/3, type 2 in Whole, 1 on top and 3 on the bottom. For a negative mixed number, put the minus sign on the whole number. Your numbers stay in your browser and are never sent anywhere.
Step-by-Step Solution
Given
Formula
Step-by-step working
Check your answer
Final summary
What Is a Fraction?
A fraction represents a part of a whole, one number written over another, separated by a fraction bar. The top number is the numerator (how many parts you have) and the bottom is the denominator (total equal parts in the whole). For a visual introduction to how fractions work, Math is Fun covers the basics of fractions clearly.
Types of Fractions
| Fraction type | Definition and example |
|---|---|
| Proper fraction | Numerator is smaller than denominator. Value is less than 1. Example: 3/4, 2/7, 5/9 |
| Improper fraction | Numerator is equal to or greater than denominator. Value is 1 or more. Example: 7/4, 9/3, 11/5 |
| Mixed number | A whole number combined with a proper fraction. Example: 2¾ (means 2 + 3/4 = 11/4) |
| Equivalent fractions | Different fractions that represent the same value. Example: 1/2 = 2/4 = 4/8 = 50/100 |
| Proper decimal fraction | A fraction whose denominator is a power of 10. Example: 3/10 = 0.3, 45/100 = 0.45 |
How to Use This Fraction Calculator
- Pick a tab: Add, Subtract, Multiply, Divide, Simplify, To decimal, To fraction, or Partial fractions.
- Type the numerator (top) and denominator (bottom) of each fraction. The optional Whole box lets you enter a mixed number: whole = 2, top = 3, bottom = 4 means 2¾. Leave it empty for a simple fraction.
- Click Calculate or press Enter. The Step-by-Step Solution box opens directly under the inputs.
- Read the simplified fraction, mixed number, decimal and percent at the top, then follow the numbered steps to see how the answer was reached.
- Use Check your answer to confirm the result, then Share result or Download PDF to keep the working as a PDF.
- Click Reset to clear the inputs and start again.
The calculator handles positive and negative fractions, improper fractions and mixed numbers, using exact whole-number arithmetic. To combine three fractions, calculate the first two and use that answer with the third.
What the Step-by-Step Solution Shows
After you press Calculate, the answer box lists everything needed to follow the method and verify the result, not just the final number.
- The answer: the simplified fraction, the mixed number, the decimal and the percent, so you can use whichever form your homework asks for.
- Given and formula: your inputs and the rule used, with your own numbers substituted in.
- Numbered steps: mixed numbers converted to improper fractions, the least common denominator (LCD) found with the GCD formula, the numerators combined, and the result simplified, with a one-line reason for each step.
- Check your answer: the reverse operation, such as subtracting the second fraction from a sum, which should return the first fraction.
- Share result and Download PDF: Share sends the answer and a link through your apps, and Download PDF saves the full working as a file to print.
Limits and Rounding
Fraction arithmetic is exact, so sums, differences, products and quotients never lose accuracy. Each number can have up to 12 digits. A decimal that repeats is shown with the repeating digits in brackets, such as 5/12 = 0.41[6], and the rounded six-place value is given next to it. The Decimal to fraction tab accepts terminating decimals with up to 9 decimal places. Partial fractions need a denominator x² + bx + c with two different real roots, so a negative or zero discriminant is explained instead of solved.
Who This Calculator Is For
Built for students checking homework, teachers preparing worked examples, and anyone who needs a fraction result they can trust without redoing the arithmetic by hand.
What This Calculator Assumes
- The two fractions you enter are exactly as intended, double-check numerator/denominator placement before reading the result.
- Results are exact fraction arithmetic, not decimal approximation, and are always simplified to lowest terms.
- Mixed numbers are converted to improper fractions internally before any calculation.
- A denominator of zero is never valid and will return an error rather than a result.
Fraction Formulas Explained
Adding and Subtracting Fractions
To add or subtract fractions, both fractions must have the same denominator, called the common denominator. If they already share a denominator, add or subtract the numerators directly. If not, find the least common denominator (LCD) first.
Multiplying Fractions
Multiplying fractions is the simplest operation, multiply the numerators together and multiply the denominators together. No common denominator is needed. If your result is a percentage you need to interpret, the Percentage Calculator can help convert it directly.
Dividing Fractions
To divide fractions, flip the second fraction (take its reciprocal) and multiply. This is called the "keep, change, flip" method.
Simplifying Fractions (Reducing to Lowest Terms)
A fraction is in its simplest form (lowest terms) when the numerator and denominator share no common factors other than 1. To simplify, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by it.
For more methods, including prime factorization and a chart of common simplified fractions, read our guide on how to simplify fractions.
Worked Examples, Step by Step
Example 1: Add 1/3 + 1/4
These fractions have different denominators, so we need to find the least common denominator (LCD) first.
- Write the fractions to add: 1/3 + 1/4
- Find the LCD of 3 and 4: LCM(3, 4) = 12
- Convert 1/3 to twelfths: 1/3 = 4/12 (multiply top and bottom by 4)
- Convert 1/4 to twelfths: 1/4 = 3/12 (multiply top and bottom by 3)
- Add the numerators: 4 + 3 = 7 (keep the denominator: 12)
- Result: 7/12
- Check if simplifiable: GCD(7, 12) = 1 → already in simplest form
Example 2: Divide 3/4 ÷ 2/5
Use the keep, change, flip method, keep the first fraction, change division to multiplication, flip the second fraction.
- Write the division: 3/4 ÷ 2/5
- Keep the first fraction: 3/4
- Change ÷ to ×
- Flip the second fraction: 2/5 becomes 5/2
- Multiply: 3/4 × 5/2 = (3×5) / (4×2) = 15/8
- Check if simplifiable: GCD(15, 8) = 1 → already in simplest form
- Convert to mixed number: 15/8 = 1 and 7/8 (1 whole + 7/8 remaining)
Example 3: Add Mixed Numbers, 2 1/3 + 1 3/4
Convert mixed numbers to improper fractions first, then add. Once you have your result, the Standard Deviation Calculator can help if you need to average multiple fraction results across a dataset.
- Convert 2 1/3 to an improper fraction: (2×3 + 1)/3 = 7/3
- Convert 1 3/4 to an improper fraction: (1×4 + 3)/4 = 7/4
- Now add: 7/3 + 7/4
- Find LCD of 3 and 4: LCM(3, 4) = 12
- Convert 7/3 → 28/12 and 7/4 → 21/12
- Add numerators: 28 + 21 = 49
- Result: 49/12
- Convert to mixed number: 49 ÷ 12 = 4 remainder 1 → 4 1/12
Working with Mixed Numbers and Whole Numbers
A mixed number contains a whole number and a fraction together, for example, 2¾ means 2 whole units plus three-quarters of another unit. To perform arithmetic with mixed numbers, convert them to improper fractions first.
To multiply a fraction by a whole number, treat the whole number as a fraction with 1 as its denominator. For example: 4 × 2/3 = (4/1) × (2/3) = 8/3 = 2 2/3.
When to Use Mixed Numbers vs Improper Fractions
| Form | Best used when |
|---|---|
| Mixed number (2 3/4) | Communicating a measurement or quantity, more intuitive to read. Preferred in everyday contexts: cooking, construction, measuring. |
| Improper fraction (11/4) | Performing arithmetic, easier to multiply and divide. Always convert to improper fractions before calculating. |
Equivalent Fractions and Comparing Fractions
Equivalent fractions are different fractions that represent the same value. For example, 1/2 = 2/4 = 4/8 = 50/100. To find equivalent fractions, multiply or divide both the numerator and denominator by the same non-zero number. To determine whether two fractions are equivalent, use cross multiplication: a/b = c/d if and only if a×d = b×c.
Partial Fraction Decomposition
Partial fraction decomposition breaks a complex rational expression, a single fraction with a polynomial denominator, into a sum of simpler fractions. This technique is widely used in calculus (for integration) and in engineering (for signal processing and Laplace transforms).
Common Mistakes When Working with Fractions
Mistake 1: Adding the denominators instead of finding a common denominator
The most frequent error. Students write 1/3 + 1/4 = 2/7, adding both numerators and both denominators. This is always wrong. The denominator represents the total equal parts in the whole, it only changes when you convert to an equivalent fraction, not when you add.
Mistake 2: Forgetting to simplify the result
After any operation, check whether the result can be simplified. A fraction is not in its final form until it is in lowest terms. The calculator always simplifies automatically, but when working by hand, always find the GCD of the result and divide both parts by it.
Mistake 3: Flipping the wrong fraction when dividing
In the keep, change, flip method, only the second fraction (the divisor) gets flipped, never the first. Flipping the first fraction instead gives a completely different result.
Mistake 4: Not converting mixed numbers before calculating
Mixed numbers cannot be multiplied or divided directly. Always convert to improper fractions first. For addition and subtraction you can work with the whole number and fraction parts separately, but converting first is the safer habit.
Mistake 5: Averaging percentages without accounting for group size
When converting a fraction result to a percentage, a simple average of two percentages is only valid if the denominators (group sizes) are equal. If they differ, the weighted average gives the correct result. Use the percentage calculator to check.
For a full walkthrough of all four operations with more mixed-number examples, read our guide on how to add, subtract, multiply and divide fractions.
Frequently Asked Questions
- How do you add fractions with different denominators?
- Find the least common denominator (LCD) of the two fractions, this is the smallest number that both denominators divide into evenly. Convert each fraction to an equivalent fraction with that LCD as the denominator. Then add the numerators and keep the common denominator. For example, to add 1/3 + 1/4: the LCD is 12, so 1/3 = 4/12 and 1/4 = 3/12, giving 7/12.
- How do you multiply fractions?
- Multiply the numerators together to get the new numerator, then multiply the denominators together to get the new denominator. No common denominator is needed. For example: 2/3 × 3/5 = (2×3)/(3×5) = 6/15, which simplifies to 2/5. You can also simplify before multiplying (cross-cancellation) to keep numbers smaller.
- How do you divide fractions?
- Use the keep, change, flip method: keep the first fraction unchanged, change the division sign to multiplication, and flip the second fraction (swap numerator and denominator to get the reciprocal). Then multiply the two fractions and simplify. For example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
- How do you simplify a fraction?
- Find the greatest common divisor (GCD) of the numerator and denominator, the largest number that divides both without a remainder. Then divide both the numerator and denominator by the GCD. For example, to simplify 12/18: GCD(12, 18) = 6, so 12÷6 = 2 and 18÷6 = 3, giving the simplified form 2/3.
- How do you convert a fraction to a decimal?
- Divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375. Some fractions produce terminating decimals (like 1/4 = 0.25) while others produce repeating decimals (like 1/3 = 0.333...). The calculator shows both the exact fraction and the decimal approximation simultaneously.
- How do you convert a decimal to a fraction?
- Count the decimal places in the number. Write the decimal as a fraction with 1 followed by that many zeros as the denominator, then simplify. For example, 0.375 has 3 decimal places, so it becomes 375/1000. GCD(375, 1000) = 125, giving 3/8. For repeating decimals, use algebra: let x = 0.333..., then 10x = 3.333..., subtract: 9x = 3, so x = 3/9 = 1/3.
- How do you add fractions with whole numbers?
- Convert the mixed number to an improper fraction first. Multiply the whole number by the denominator and add the numerator to get the new numerator, keep the same denominator. For example, 2 1/3 = (2×3+1)/3 = 7/3. Then add the fractions normally. The calculator's optional Whole number fields handle this conversion automatically.
- What is an improper fraction and how do you convert it to a mixed number?
- An improper fraction has a numerator greater than or equal to its denominator, for example 11/4. To convert to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the original denominator. For 11/4: 11 ÷ 4 = 2 remainder 3, giving 2 3/4.
- What is the least common denominator (LCD)?
- The least common denominator is the smallest number that is a multiple of both denominators in a pair of fractions. It is the same as the least common multiple (LCM) of the denominators. You need the LCD when adding or subtracting fractions with different denominators. For example, the LCD of 1/3 and 1/4 is 12, because 12 is the smallest number divisible by both 3 and 4.