How to calculate with fractions
A fraction has a numerator on top and a denominator underneath. The denominator says how many equal parts the whole is split into; the numerator says how many of those parts you have. Every operation comes down to working with those two numbers separately, and every answer is finished by dividing both by their greatest common divisor.
Comparing two quantities or solving a proportion such as 3 : 4 = 9 : x is a ratio question; the ratio calculator handles that.
Adding and subtracting
- Find the least common multiple of the denominators. For 1/4 + 1/6 it is 12.
- Rewrite each fraction over it: 1/4 = 3/12 and 1/6 = 2/12.
- Add or subtract the numerators: 3 + 2 = 5, giving 5/12.
- Simplify if possible. 5/12 is already in lowest terms.
Multiplying
Multiply the numerators together and the denominators together, then simplify: 2/3 × 9/4 = 18/12 = 3/2 = 1 1/2. No common denominator is needed.
Dividing
Dividing by a fraction is multiplying by its reciprocal. 3/4 ÷ 3/8 becomes 3/4 × 8/3 = 24/12 = 2. A fraction can be divided by anything except zero.
Common fraction mistakes
- Adding straight across. 1/2 + 1/3 is not 2/5 — 2/5 is smaller than the 1/2 you started with. Over a common denominator it is 3/6 + 2/6 = 5/6.
- Multiplying mixed numbers part by part. 2 1/2 × 2 1/2 is not 4 1/4. As improper fractions it is 5/2 × 5/2 = 25/4, which is 6 1/4.
- Flipping the wrong fraction. In 3/4 ÷ 3/8 only the divisor turns over, giving 3/4 × 8/3. Flipping the first fraction instead gives 4/3 × 3/8 = 1/2, not the correct 2.
When you work by hand, cancelling before you multiply keeps the numbers small: in 4/9 × 3/8, the 4 cancels into the 8 and the 3 into the 9, leaving 1/3 × 1/2 = 1/6 — the same answer as 12/72 without the large numbers. Scaling a recipe is the everyday version: one and a half times a 3/4-cup measure is 3/4 × 3/2 = 9/8, or 1 1/8 cups.
Mixed numbers and negative fractions
A mixed number such as 2 1/3 is converted to an improper fraction before any arithmetic: 2 × 3 + 1 = 7, so 2 1/3 = 7/3. When a mixed number is negative, the sign belongs to the whole of it — −2 1/3 is −7/3. The answer is converted back at the end, so 7/2 is also shown as 3 1/2.
Fractions as decimals
A simplified fraction ends as a decimal only when its denominator has no prime factors other than 2 and 5. 3/8 = 0.375 ends; 1/3 and 22/7 repeat forever. The calculator marks the repeating block in brackets — 1/6 = 0.1(6) — and finds it exactly by long division rather than from a rounded float.
Worked examples
- 1/2 + 1/3. The least common multiple of 2 and 3 is 6. 1/2 = 3/6 and 1/3 = 2/6, so the sum is 3/6 + 2/6 = 5/6, which is already in lowest terms and equals 0.8(3) as a decimal.
- 3/4 − 5/6. The least common multiple of 4 and 6 is 12. 3/4 = 9/12 and 5/6 = 10/12, so 9/12 − 10/12 = −1/12, a negative result because 5/6 is the larger fraction.
- 2 1/2 ÷ 3/4. 2 1/2 converts to the improper fraction 5/2. Dividing by 3/4 multiplies by its reciprocal: 5/2 × 4/3 = 20/6, which simplifies to 10/3, or 3 1/3 as a mixed number.
- 0.75 + 1/8. The decimal 0.75 converts exactly to 3/4. Over a common denominator of 8, that is 6/8 + 1/8 = 7/8.
Edge cases and blocked input
- A denominator of 0, in either fraction, is refused before any arithmetic runs — a fraction cannot be divided into zero parts.
- Dividing by a fraction whose numerator is 0, such as ÷ 0/5, is the same as dividing by zero and is also blocked.
- A negative sign on the whole-part box applies to the entire mixed number, not just the whole-number part: −2 1/3 means −(2 + 1/3) = −7/3. Put the minus on the numerator instead if you want a positive whole part with a negative fraction.
- Numbers over 15 digits in any numerator, denominator or whole part are rejected as too long for the input boxes.
- A result that is a whole number, such as 3/4 ÷ 3/8 = 2, is shown as a plain whole number rather than as a fraction with 1 underneath.
Reading the result
Every answer is shown three ways: as a simplified fraction, as a mixed number when the value is greater than 1 (or less than −1), and as a decimal. If the decimal repeats forever, the repeating digits are wrapped in brackets, such as 0.1(6) for 1/6, rather than cut off or rounded. The step-by-step list below the answer shows the common denominator used, the rewritten fractions, and the operation applied to the numerators.
Limitations
This calculator works with two fractions or mixed numbers and one operation at a time — it does not chain several operations in one expression, and it does not handle algebraic fractions with letters in them. Numerators, denominators and whole parts are limited to 15 digits each. All the arithmetic runs in your browser; nothing you type is sent to a server.
Related calculators
To simplify a ratio down to whole numbers, or to solve a proportion such as 3 : 4 = 9 : x, use the ratio calculator. To round a decimal result to a fixed number of places, try the rounding calculator. To find the greatest common divisor used when simplifying a fraction by hand, see the GCD and LCM calculator.
Frequently asked questions
How do you add fractions with different denominators?
Rewrite both fractions over a common denominator — the least common multiple of the two denominators — then add the numerators. 1/4 + 1/6 becomes 3/12 + 2/12 = 5/12. The calculator shows the common denominator and both rewritten fractions.
How do you divide fractions?
Multiply by the reciprocal of the second fraction: flip it upside down and multiply. 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2.
How do I enter a mixed number?
Type the whole part in the first box and the fraction in the numerator and denominator boxes. For 2 1/3, enter 2, 1 and 3. For a plain fraction leave the whole-part box empty.
How are negative mixed numbers handled?
A minus sign on the whole part applies to the whole mixed number: −2 1/3 means −(2 + 1/3) = −7/3, not −2 + 1/3. That is how mixed numbers are written in textbooks, and the calculator follows it. For a negative plain fraction, put the minus on the numerator.
Can I enter decimals?
Yes, in the whole-part box with the fraction boxes left empty. 0.75 is converted to 3/4 exactly. Numerators and denominators themselves must be whole numbers.
Why does the result show a decimal in brackets?
Some fractions never end as decimals. 1/3 is 0.333… with the 3 repeating forever, written 0.(3). The calculator finds the repeating block exactly by long division rather than rounding.
What happens if I divide by zero?
Division by zero is undefined, so the calculator stops and says so instead of printing an answer. A denominator of 0 is refused for the same reason.
Is there a limit on the size of the numbers?
Each box accepts up to 15 digits. The arithmetic itself is exact at any size — nothing is rounded — so large denominators do not lose accuracy.