Fraction calculator with step-by-step work
Pick an operation, enter two fractions (with a whole number part for mixed numbers) and the answer appears at once, simplified and with its decimal value. Every step is written out: finding the least common denominator, combining the numerators, then reducing with the greatest common factor using the Euclidean algorithm.
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📌 Key facts
- To add two fractions, use a common denominator: a/b + c/d = (ad + bc) / bd.
- To multiply fractions, multiply the numerators together and the denominators together.
- Dividing by a fraction means multiplying by its reciprocal: a/b / (c/d) = a/b x d/c.
- A fraction is in lowest terms when its numerator and denominator share no factor other than 1.
- To reduce a fraction, divide both parts by their greatest common factor: 48/36 = 4/3.
How to add and subtract fractions
To add or subtract, the fractions need the same denominator. Use the least common denominator (LCD), which is the least common multiple of the denominators. Then add or subtract the numerators and keep the denominator.
\[ \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a \times k + c \times m}{\operatorname{LCM}(b,\,d)} \quad \text{where } k = \dfrac{\operatorname{LCM}}{b} \text{ and } m = \dfrac{\operatorname{LCM}}{d} \]
| Problem | LCD | Answer |
|---|---|---|
| \( \dfrac{1}{2} + \dfrac{1}{3} \) | 6 | \( \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6} \) |
| \( \dfrac{3}{4} + \dfrac{5}{6} \) | 12 | \( \dfrac{9}{12} + \dfrac{10}{12} = \dfrac{19}{12} \) |
| \( \dfrac{5}{6} - \dfrac{3}{4} \) | 12 | \( \dfrac{10}{12} - \dfrac{9}{12} = \dfrac{1}{12} \) |
| \( \dfrac{7}{10} + \dfrac{1}{5} \) | 10 | \( \dfrac{7}{10} + \dfrac{2}{10} = \dfrac{9}{10} \) |
Multiplying, dividing and simplifying fractions
To multiply, multiply the numerators together and the denominators together. To divide, multiply by the reciprocal of the second fraction (flip it). You cannot divide by a fraction that equals zero.
\[ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d} \qquad \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \qquad (c \neq 0) \]
Then simplify by dividing the numerator and denominator by their greatest common factor (GCF). The Euclidean algorithm finds it: divide the larger number by the smaller, then the smaller by the remainder, and repeat until the remainder is 0. The last nonzero remainder is the GCF.
| Fraction | Euclidean algorithm | GCF | Lowest terms |
|---|---|---|---|
| 48/36 | \( 48 = 36 \times 1 + 12 \); \( 36 = 12 \times 3 + 0 \) | 12 | 4/3 |
| 84/126 | \( 126 = 84 \times 1 + 42 \); \( 84 = 42 \times 2 + 0 \) | 42 | 2/3 |
| 17/5 | \( 17 = 5 \times 3 + 2 \); \( 5 = 2 \times 2 + 1 \); \( 2 = 1 \times 2 + 0 \) | 1 | 17/5 |
Mixed numbers, decimals and negative fractions
Enter a mixed number such as 2 1/3 with 2 in the whole box, 1 as the numerator and 3 as the denominator. It equals \( \dfrac{2 \times 3 + 1}{3} = \dfrac{7}{3} \). For a negative mixed number, put the minus sign on the whole number: \( - 2 \dfrac{1}{3} = - \dfrac{7}{3} \).
The decimal value is computed exactly by long division: 1/4 = 0.25 (a terminating decimal), but 1/3 = 0.(3) and 1/7 = 0.(142857), where the digits in parentheses repeat forever. A fraction in lowest terms terminates only if its denominator has no prime factors other than 2 and 5.
The sign of a fraction always goes in the numerator: 3/−4 is written −3/4, and −3/−4 is 3/4.
Fraction calculator: classic mistakes
Many students make the same error. They add the numerators and the denominators separately. For example, 1/2 + 1/3 is not 2/5, but 5/6.
| Mistake | Correct work |
|---|---|
| 1/2 + 1/3 = 2/5 | 3/6 + 2/6 = 5/6 |
| 2/3 x 3/4 = 6/7 | 6/12 = 1/2 |
| (1/2) / (1/4) = 1/8 | 1/2 x 4 = 2 |
So always find a common denominator before you add or subtract.
Fraction calculator: mixed numbers and reducing
A mixed number, such as 1 3/4, turns into an improper fraction: 1 x 4 + 3 = 7, so 7/4. Then you calculate as usual.
To reduce, look for the greatest common factor. For example, for 84/126, the GCF is 42. Divide both numbers by 42 and you get 2/3.
Do not reduce before you choose the operation, though. It is better to calculate first, then simplify the result. Finally, check the decimal value: 2/3 is about 0.667.
Fractions in everyday life
Fractions show up everywhere. In the kitchen, 3/4 cup of flour mixes with 1/2 cup of sugar. In the workshop, a 5/16 inch bolt needs a different wrench than a 3/8 inch one.
- To split a pizza into unequal slices, add the fractions.
- To scale a recipe, multiply each fraction by your factor.
- To see how many portions fit in a whole, divide.
Also, a fraction converts to a percentage: 3/4 = 75%.
Fractions: tools to go further
To find a common denominator, use the GCF and LCM calculator: the LCM of the denominators gives the smallest common denominator. To factor a large number before reducing, try the prime factorization calculator.
Next, solve an equation with fractions in the equation solver. Finally, turn the result into a rate with the percent change calculator.
3 real-life examples
Add 3/4 + 5/6
GCF(4, 6) = 2, so LCM = 4 × 6 ÷ 2 = 12. Rewrite \( \dfrac{3}{4} = \dfrac{9}{12} \) and \( \dfrac{5}{6} = \dfrac{10}{12} \). The sum is 19/12, already in lowest terms because GCF(19, 12) = 1. As a mixed number: 1 7/12, about 1.5833.
Divide 2/3 by 4/9
Dividing by 4/9 means multiplying by 9/4: \( \dfrac{2}{3} \times \dfrac{9}{4} = \dfrac{18}{12} \). The GCF of 18 and 12 is \( 6 (18 = 12 \times 1 + 6 \,;\, 12 = 6 \times 2 + 0) \), so the answer is 3/2, or 1.5.
Simplify 48/36
Euclid: \( 48 = 36 \times 1 + 12 \), then \( 36 = 12 \times 3 + 0 \). The GCF is 12. Divide top and bottom by 12: \( \dfrac{48}{12} = 4 \) and \( \dfrac{36}{12} = 3 \). In lowest terms the fraction is 4/3.