Fraction Calculator
Do all four operations with fractions, compare two fractions or simplify one, with mixed numbers and decimals accepted. The result comes in its simplest form, with the work explained.
Educational use: results are estimates made for study and reference. Checking the answer and deciding what to do with it is the sole responsibility of whoever uses the tool. Read the legal notice.
On this page
- How to use the calculator
- Worked examples, one per operation
- Classic mistakes with fractions
- The rules behind each operation
- Limits of this calculator
- Seeing the addition of fractions
- Reference tables
- Questions that come up while calculating
- Fraction vocabulary
- Where fractions show up outside of school
- Where fractions come from
- Curiosities that help it stick
- Keep exploring
How to use the calculator
The tool handles six types of calculation. You choose one in the first field, and the form shows only what that operation needs.
- Choose the operation: add, subtract, multiply, divide, compare or simplify.
- Enter the fractions. Use a slash for fractions (
3/4), a space between the whole part and the fraction in mixed numbers (1 1/2) and a decimal point in decimals (0.75). Negative numbers take the minus sign in front (-2/3). - Click Calculate. The result appears already simplified, along with its mixed number, decimal and percentage forms.
- Open the step-by-step if you want to see how the calculation was done, from the LCM to the final simplification.
What each option does
- Add and subtract: rewrite the fractions with the same denominator, do the calculation and simplify.
- Multiply: multiplies numerators and denominators and simplifies the result.
- Divide: multiplies by the reciprocal fraction, and warns you when the divisor is zero.
- Compare: tells you which of the two is larger and by how much they differ.
- Simplify or convert: reduces a fraction to its simplest form and turns decimals into fractions. Typing
0.375returns3/8.
You can mix formats in the same calculation. Adding 1 1/2 and 0.25 works without any trouble.
Worked examples, one per operation
1. Adding different denominators
What is 1/2 + 1/3?
Result: 5/6, which as a decimal is 0.8333… (the 3 repeats forever).
2. A recipe with fractional measurements
One recipe step calls for 3/4 cup of sugar and another step calls for 2/3 cup more. How much sugar in total?
That is one cup and 5/12, a bit more than one and a third cups.
3. Subtraction
What is 3/4 − 1/6?
Result: 7/12. It can't be simplified, because 7 and 12 have no common divisor other than 1.
4. Multiplication with simplification
What is 2/3 × 9/10?
Result: 3/5, or 0.6, or 60%. You can make it easier by cancelling first: the 3 in the denominator cancels with the 9 in the numerator, and the 2 cancels with the 10.
5. Dividing by the reciprocal
What is 5/6 ÷ 10/9?
Result: 3/4. "Dividing by 10/9" means "how many times does 10/9 fit into 5/6", and the answer is less than once, 0.75.
6. Which is larger?
Which is larger, 5/8 or 7/12?
5/8 is 0.625 and 7/12 is 0.5833…, a difference of 1/24. With cross-multiplication, the larger product points to the larger fraction.
7. From decimal to fraction
How do you write 0.375 as a fraction?
Result: 3/8. Three decimal places become a denominator of 1000, and simplifying does the rest.
Classic mistakes with fractions
Adding numerator to numerator and denominator to denominator
The number one mistake. It is tempting to write 1/2 + 1/3 = 2/5, but that would be 0.4, less than a half, and we already know that 1/2 alone is 0.5. Adding fractions requires a common denominator, and only the numerators are added.
Forgetting to flip when dividing
Dividing 1/2 by 1/4 is not 1/8. It is 1/2 × 4/1 = 2: a quarter fits into a half twice. The fraction on the right side of the ÷ is the one that gets turned upside down.
Thinking a bigger denominator means a bigger fraction
With whole numbers, 8 is greater than 3. With fractions that have the same numerator, it is the opposite: 1/8 is less than 1/3, because the pizza was cut into smaller slices. When comparing, use cross-multiplication or decimals.
"Cancelling" terms of a sum
In (2 + 3) ÷ 2, you can't cancel the 2 and be left with 3. Only factors can be cancelled, meaning numbers that are being multiplied. Cancelling terms of a sum changes the result.
Getting tangled up with mixed numbers
In 2 1/2 × 3, multiplying only the whole part gives 6 1/2, which is wrong. The right way is to turn it into an improper fraction: 5/2 × 3 = 15/2 = 7 1/2. The calculator does this conversion for you.
Stopping halfway through simplifying
6/8 and 3/4 are worth the same, but 6/8 can still be reduced. A fraction is in its simplest form only when the GCF of the numerator and denominator is 1.
Rounding the decimal too early
Using 0.33 in place of 1/3 in a calculation with many steps accumulates error. Keep the fraction until the end and convert only in the result.
The rules behind each operation
Addition and subtraction
Multiplication
Division
Comparison
Simplification
The idea that ties it all together is that of equivalent fractions: multiplying or dividing the numerator and denominator by the same number (other than zero) does not change the value. That is why 1/2 = 2/4 = 50/100, and it is what lets you rewrite two fractions with the same denominator in order to add them.
Mixed number and improper fraction
A mixed number like 2 3/5 equals (2 × 5 + 3) ÷ 5 = 13/5. Going back, you divide the numerator by the denominator: 13 ÷ 5 is 2 with 3 left over, so 2 3/5.
Decimal and fraction
A decimal with n places becomes a fraction by writing it over 10ⁿ: 0.25 = 25/100 = 1/4. In the opposite direction, you divide the numerator by the denominator. If the division doesn't end, you get a repeating decimal, as in 1/3 = 0.333…
Limits of this calculator
The calculator works with exact arithmetic on large integers, without rounding the fractions, and accepts up to 15 digits in each part and up to 12 decimal places. The conversion to decimal shows the repeating block (in parentheses) when the division doesn't end, and the percentage of a repeating decimal appears rounded, marked with "≈".
The result depends on you having typed the right numbers: check the numerator and denominator before calculating. This tool is for studying and checking calculations. Do not use the result to dose medication, size parts or structures, or for decisions where an error would have serious consequences. In those cases, consult a professional and check the calculation another way. Checking and using the answers is entirely the user's responsibility.
Bottom line: this tool is educational. Results do not replace your own checking, and how you use them is entirely your responsibility. More details in the legal notice.
Seeing the addition of fractions
The sum 1/2 + 1/3 becomes clear when both bars are cut into sixths. A half is 3 sixths, a third is 2 sixths, and together they make 5 sixths:
Reference tables
Some fractions show up so often in everyday life that it is worth knowing their decimal and percentage by heart:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33% |
| 2/3 | 0.666… | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
To simplify faster, divisibility rules help. If the numerator and denominator pass the same test, you can divide both by the same number:
| Divisible by | How to tell |
|---|---|
| 2 | the last digit is even |
| 3 | the sum of the digits is divisible by 3 |
| 5 | ends in 0 or 5 |
| 9 | the sum of the digits is divisible by 9 |
| 10 | ends in 0 |
And a useful rule about decimals: once simplified, a fraction has an exact (terminating) decimal only if the denominator has just the prime factors 2 and 5. That is why 3/8 and 7/20 terminate, but 1/3 and 5/6 become repeating decimals.
Questions that come up while calculating
How do you add fractions with different denominators?
Rewrite both with the same denominator, preferably the LCM of the denominators, and then add only the numerators. For example, 1/2 + 1/3: the LCM of 2 and 3 is 6, so 3/6 + 2/6 = 5/6.
How do you simplify a fraction?
Divide the numerator and denominator by the GCF (also called the GCD), the largest number that divides both. In 18/24, the GCF is 6, and the fraction becomes 3/4. If the GCF is 1, it is already in its simplest form.
How do you divide one fraction by another?
Multiply the first by the reciprocal of the second. In 5/6 ÷ 10/9, the reciprocal of 10/9 is 9/10, so it becomes 5/6 × 9/10 = 45/60 = 3/4.
How do you turn a decimal into a fraction?
Write the decimal over a power of 10 with as many zeros as there are decimal places, and simplify. 0.375 is 375/1000, which simplifies to 3/8. In the calculator, use the simplify or convert option.
What is a repeating decimal?
It is an infinite decimal in which a group of digits repeats, like 0.333… (from 1/3) or 0.1666… (from 1/6). The calculator shows the repeating group in parentheses, for example 0.1(6).
How do you turn a mixed number into an improper fraction?
Multiply the whole part by the denominator and add the numerator. In 2 3/5: 2 × 5 + 3 = 13, so 13/5. You can type the mixed number directly, like 2 3/5.
How do you compare two fractions?
Cross-multiply: the numerator of one times the denominator of the other. The larger product points to the larger fraction. You can also convert both to decimals, or put both over the same denominator.
Does the calculator accept negative fractions?
Yes. Put the minus sign before the number, as in -3/4. In the result, the sign stays on the numerator.
Why can't a fraction have a denominator of zero?
Dividing by zero has no defined result. There is no number that, multiplied by zero, gives back 5, for example. That is why the calculator reports an error instead of calculating.
Can I type fractions with decimals, like 1.5/2?
Not in that format. Write decimals on their own (1.5) or fractions with whole numbers (3/4). To mix them, use one entry for each in the same operation, such as 1.5 and 3/4.
Fraction vocabulary
- Numerator
- The top number. It tells you how many parts are being considered.
- Denominator
- The bottom number. It tells you how many equal parts the whole was divided into, and it can never be zero.
- Proper fraction
- A fraction in which the numerator is smaller than the denominator, like 3/4. It is worth less than 1.
- Improper fraction
- A fraction in which the numerator is greater than or equal to the denominator, like 7/4. It is worth 1 or more.
- Mixed number
- A whole part followed by a proper fraction, like 1 3/4.
- Equivalent fractions
- Fractions written in different ways that are worth the same, like 1/2, 2/4 and 50/100.
- LCM
- Least common multiple: the smallest number that is a multiple of all the numbers considered. It serves as the common denominator.
- GCF
- Greatest common factor (also called the greatest common divisor, GCD): the largest number that divides all the numbers considered. It is used to simplify.
- Irreducible fraction
- A fraction in its simplest form, in which the GCF of the numerator and denominator is 1.
- Reciprocal
- The fraction with the numerator and denominator swapped. The reciprocal of 2/3 is 3/2.
Where fractions show up outside of school
- Cooking. Recipes in cups and spoons live on 1/4, 1/2 and 3/4. Doubling or scaling down a recipe is multiplying fractions.
- Measuring and woodworking. Rulers and tape measures in inches are divided into eighths and sixteenths, and adding measurements is adding fractions.
- Music. Note lengths are a sequence of fractions: the whole note is worth 1, the half note 1/2, the quarter note 1/4. Time signatures like 3/4 and 6/8 are real fractions.
- Splitting chores and bills. "Each person takes 2/5 of the work" only works out when the parts add up to the whole.
- Probability. The chance of an event is a fraction: favorable outcomes over possible outcomes. Flipping heads on a coin is 1/2.
- Scales and maps. A 1:50,000 scale says that each unit on paper is worth 50,000 on the ground, a ratio in fraction form.
- Math studies. Algebra, equations and proportions use fractions all the time, and people who master this foundation struggle less in the later stages.
Where fractions come from
The word "fraction" comes from the Latin fractio, linked to "breaking". A fraction is, literally, a broken whole. The need to break things up is ancient: dividing harvests, land and inheritances called for numbers smaller than one.
In ancient Egypt, scribes wrote almost all fractions as sums of fractions with a numerator of 1, known as unit fractions. The Rhind papyrus, copied by the scribe Ahmes around 1650 BC, contains tables for breaking fractions down this way. It was laborious, but it worked for dividing up bread and beer.
The horizontal bar separating numerator and denominator is credited to Arab mathematicians of the Middle Ages and reached Europe in the book Liber Abaci (1202), by Leonardo of Pisa, known as Fibonacci, who helped spread the Hindu-Arabic numeral system. Decimals, for their part, were strongly championed by the Flemish mathematician Simon Stevin in 1585, in the short treatise De Thiende.
Curiosities that help it stick
- 0.999… equals 1. If 1/3 = 0.333…, then 3 × 1/3 = 0.999…, which has to be 1. It is strange, but it is exact.
- 1/7 has a six-digit cycle. 1/7 = 0.142857142857…, and the multiples of 1/7 reappear with the same digits rotated: 2/7 = 0.285714…
- Between two fractions there is always another. Between 1/2 and 2/3 fits 3/5, and between 1/2 and 3/5 fits 4/7, and so on without end. There is no such thing as "the next fraction".
- The "in-between" fraction by adding. Adding numerators and denominators (1/2 and 2/3 → 3/5) is not an addition of fractions, but it gives a fraction that always lies between the two. Mathematicians call this the mediant.
- A mixed number is a hidden sum. 2 1/2 means 2 + 1/2. Without the plus sign written out, that is easy to lose sight of.