How to use the fraction calculator
- Enter the first fraction. Type the numerator and denominator; for a mixed number such as 1 1/2, also fill in the whole number.
- Choose the operation: add, subtract, multiply or divide.
- Enter the second fraction the same way. A whole number on its own, such as 5, works too.
- Read the answer in lowest terms, as a mixed number, a decimal and a percent, and follow the numbered steps to see how it was found.
How to add and subtract fractions
Fractions can only be added or subtracted when they have the same denominator. If they do not, rewrite each one over a common denominator. The smallest choice is the least common multiple (LCM) of the two denominators, which keeps the numbers small.
a/b ± c/d = (a × L/b ± c × L/d) / L, where L = LCM(b, d)
- a, c
- numerators
- b, d
- denominators (never zero)
- L
- least common denominator, the least common multiple of b and d
Worked example: 1 1/2 + 2/3
- Write as improper fractions: 1 1/2 = (1 × 2 + 1)/2 = 3/2.
- Least common denominator: LCM(2, 3) = 6.
- Rewrite over the common denominator: 3/2 = 9/6; 2/3 = 4/6.
- Add the numerators: 9 + 4 = 13, so 13/6.
- Simplify: 13/6 is already in lowest terms.
- Convert to a mixed number: 13 ÷ 6 = 2 remainder 1, so 2 1/6.
The answer is 13/6, or 2 1/6 as a mixed number. Subtraction works the same way: 3/4 − 5/6 = 9/12 − 10/12 = −1/12. A negative answer just means the second fraction was larger.
How to multiply and divide fractions
Multiplying is simpler than adding: multiply straight across, numerator times numerator and denominator times denominator. No common denominator is needed.
a/b × c/d = (a × c)/(b × d); a/b ÷ c/d = a/b × d/c (c ≠ 0)
- d/c
- the reciprocal of c/d: the fraction turned upside down
- Multiply: 2/3 × 9/4 = (2 × 9)/(3 × 4) = 18/12 = 3/2 = 1 1/2.
- Divide: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2. Dividing by one half doubles the number.
- Mixed numbers: convert first. 2 1/3 × 1 1/2 = 7/3 × 3/2 = 21/6 = 7/2 = 3 1/2. Multiplying the whole numbers and fractions separately (2 × 1 and 1/3 × 1/2) gives the wrong answer.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator: 2 1/3 = (2 × 3 + 1)/3 = 7/3. For a negative mixed number, the minus sign applies to the whole value, so −2 1/3 = −7/3.
How to simplify fractions
A fraction is simplified (in lowest terms) when its numerator and denominator share no common factor other than 1. To get there in one step, divide both by their greatest common divisor. For 18/24, the GCD is 6, so 18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4. You can also divide by any shared factor repeatedly (18/24 → 9/12 → 3/4); you will reach the same answer.
The GCD can be found with Euclid’s method: divide the larger number by the smaller, keep the remainder, and repeat with the smaller number and the remainder until the remainder is 0. For 24 and 18: 24 ÷ 18 leaves 6, and 18 ÷ 6 leaves 0, so the GCD is 6.
Converting fractions to decimals and percents
A fraction is a division problem, so divide the numerator by the denominator to get the decimal, and multiply the decimal by 100 for the percent. Some decimals end (1/8 = 0.125) and others repeat forever (1/3 = 0.3333…). In lowest terms, a fraction ends only when its denominator’s prime factors are 2s and 5s. This table writes repeating digits in parentheses, so 0.1(6) means 0.1 followed by 6s that never end.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.(3) | 33.3333% |
| 2/3 | 0.(6) | 66.6667% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/6 | 0.1(6) | 16.6667% |
| 5/6 | 0.8(3) | 83.3333% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/10 | 0.1 | 10% |
| 3/10 | 0.3 | 30% |
| 1/12 | 0.08(3) | 8.3333% |
| 1/16 | 0.0625 | 6.25% |
| 1/32 | 0.03125 | 3.125% |
Percents are rounded to four decimal places. For other percent problems, try the percentage calculator.
Fraction tips and common mistakes
- Never add denominators. 1/2 + 1/3 is not 2/5; it is 3/6 + 2/6 = 5/6.
- Simplify before multiplying to keep numbers small: in 2/3 × 9/4, cancel the 3 with the 9 and the 2 with the 4 to get 1/1 × 3/2 = 3/2 directly.
- A zero denominator is undefined. 5/0 has no value, and dividing by a fraction equal to 0 is not allowed.
- Measurements: inch rulers use halves, quarters, eighths and sixteenths, so 5/8 in. + 3/16 in. = 13/16 in., or 0.8125 in. as a decimal.
For decimals, powers and roots, use the scientific calculator. To average scores given as fractions, such as 17/20 on a quiz, the grade calculator turns them into a weighted class grade.
Frequently asked questions
How do you add fractions with different denominators?
Rewrite both fractions over a common denominator, ideally the least common multiple of the denominators, then add the numerators and simplify. For 1/6 + 1/4, the least common denominator is 12, so the sum is 2/12 + 3/12 = 5/12. The denominator stays the same when you add; only the numerators are added.
How do you multiply fractions?
Multiply the numerators together and the denominators together, then simplify. For 2/3 × 9/4, that is 18/12, which reduces to 3/2 or 1 1/2. No common denominator is needed. With mixed numbers, first convert each one to an improper fraction: 2 1/3 × 1 1/2 becomes 7/3 × 3/2 = 7/2.
How do you divide fractions?
Keep the first fraction, change division to multiplication, and flip the second fraction (use its reciprocal). For 3/4 ÷ 1/2, that gives 3/4 × 2/1 = 6/4 = 3/2, or 1 1/2. You cannot divide by a fraction equal to zero, because zero has no reciprocal.
How do you simplify a fraction?
Divide the numerator and denominator by their greatest common divisor (GCD). For 18/24 the GCD is 6, so 18/24 = 3/4. A fraction is in lowest terms when the only whole number that divides both parts is 1. The calculator always simplifies its answer and shows the GCD it used.
How do you convert a fraction to a decimal?
Divide the numerator by the denominator. 3/8 is 3 ÷ 8 = 0.375, which ends, while 2/3 is 0.6666… and repeats forever; this page writes that as 0.(6), with the repeating digits in parentheses. A fraction in lowest terms gives a terminating decimal only when its denominator has no prime factors other than 2 and 5.
What is an improper fraction?
An improper fraction has a numerator equal to or larger than its denominator, such as 13/6. It can be written as a mixed number by dividing: 13 ÷ 6 = 2 remainder 1, so 13/6 = 2 1/6. Improper fractions are easier to calculate with, and mixed numbers are easier to read, so the calculator shows both.