Fraction Calculator

About the Fraction Calculator

A fraction $\frac{a}{b}$ describes $a$ equal parts out of $b$ . Adding and subtracting fractions needs a common denominator, while multiplying and dividing work directly on numerators and denominators.

This calculator reads a whole expression such as $1\tfrac12 + \tfrac{3}{4} \times \tfrac{2}{5}$ , respects the order of operations, and keeps every intermediate value as an exact fraction, so there is no rounding error.

What is a fraction?

A fraction is a number written as one integer over another, $\frac{a}{b}$ with $b \neq 0$ . The bottom number, the denominator, says how many equal parts a whole is cut into; the top number, the numerator, says how many of those parts you have. So $\frac34$ is three pieces of a whole cut into four.

A fraction is also a division: $\frac34$ is the same number as $3 \div 4 = 0.75$ . When the numerator is smaller than the denominator the fraction is proper and lies between 0 and 1. When it is larger, the fraction is improper, such as $\frac{7}{4}$ , and can also be written as the mixed number $1\tfrac34$ . Many different fractions name the same value; $\frac{2}{4}$ , $\frac{3}{6}$ and $\frac{50}{100}$ all equal $\frac12$ , and the one with no common factor left is said to be in lowest terms.

How to use the Fraction Calculator

Type the expression using / for each fraction bar, for example 2/3 + 5/6.
Write a mixed number with a space between the whole part and the fraction: 1 1/2.
Use +, −, * or ×, and ÷ between fractions. Parentheses are allowed.
Press Calculate to see the least common denominator, the combined numerator and the reduced result.

Formulas

Adding fractions

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

In practice the least common denominator keeps the numbers small.

Multiplying fractions

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Dividing fractions

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Multiply by the reciprocal of the divisor.

Simplifying

\frac{a}{b} = \frac{a \div g}{b \div g},\quad g = \gcd(a, b)

Worked example: 2/3 + 5/6

The worked example adds $\frac23$ and $\frac56$ . The denominators differ, so the calculator first finds their least common denominator, 6, and rewrites $\frac23$ as $\frac46$ . Adding the numerators gives $\frac96$ , which shares a factor of 3 top and bottom and reduces to $\frac32$ , or $1\tfrac12$ as a mixed number. The decimal check, 1.5, confirms the result.

The solution below is generated by the same engine that powers the calculator above.

Final answer

\frac{3}{2}

How we read your problem

Exact arithmetic with fractions (results kept in lowest terms).

Step-by-step solution

Rewrite both fractions with the least common denominator
The LCD of 3 and 6 is 6.
$\frac{2}{3} + \frac{5}{6} = \frac{4}{6} + \frac{5}{6}$
Add the numerators and keep the denominator
$\frac{4 + 5}{6} = \frac{9}{6}$
Simplify
Divide numerator and denominator by their GCF, 3.
$\frac{9}{6} = \frac{3}{2}$
Final answer in lowest terms
$\frac{3}{2} = 1\tfrac{1}{2}$

Formulas used

Addition
$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
Multiplication
$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
Division
$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Fractions are combined over a common denominator for addition and subtraction, multiplied straight across, and divided by multiplying by the reciprocal. The result is reduced by the greatest common factor.

Verification

Decimal check: 3/2 ≈ 1.5
As a mixed number: 1 1/2

Detailed explanation

Why addition needs a common denominator

You can only add pieces of the same size. Thirds and sixths are different sizes, so $\frac23 + \frac56$ cannot be added until both are expressed in sixths. Multiplying the top and bottom of a fraction by the same number does not change its value, which is what makes the rewriting legal. The least common denominator is used rather than simply the product of the denominators because it keeps the numbers small and usually reduces the simplifying work at the end.

Multiplying and dividing

Multiplication does not need a common denominator: you multiply straight across, numerator by numerator and denominator by denominator. Taking $\frac23$ of $\frac34$ means two thirds of three quarters, which is $\frac{6}{12} = \frac12$ . Division asks how many times the divisor fits into the dividend, and that is the same as multiplying by its reciprocal. For $\frac34 \div \frac25$ the calculator rewrites the problem as $\frac34 \times \frac52 = \frac{15}{8}$ , which is $1\tfrac78$ .

Mixed numbers, decimals and longer expressions

Mixed numbers are converted to improper fractions before any arithmetic, and decimals are turned into exact fractions, so $0.25 + \frac13$ becomes $\frac14 + \frac13 = \frac{7}{12}$ rather than a rounded decimal. In an expression such as $1\tfrac12 - \frac34 \times \frac25$ the multiplication is done first, giving $\frac{3}{10}$ , and the subtraction then gives $\frac{6}{5}$ . Keeping every intermediate value exact is what lets the final answer be exact as well.

Common mistakes

Adding denominators: $\frac12 + \frac13$ is $\frac56$ , not $\frac25$ .
Forgetting to flip the second fraction when dividing.
Treating $1\tfrac12$ as $1 \times \tfrac12$ ; a mixed number means $1 + \tfrac12$ .
Leaving the answer unreduced, such as $\frac{6}{8}$ instead of $\frac34$ .
Cross-multiplying when adding; cross-multiplication is a shortcut for comparing or solving proportions, not for sums.
Simplifying across an addition sign, for example cancelling the 3s in $\frac{3 + 1}{3}$ .

Tips

Cancel common factors before multiplying to keep numbers small.
Convert mixed numbers to improper fractions before multiplying or dividing.
Check a result by converting both sides to decimals.
To compare two fractions, compare $ad$ with $bc$ for $\frac{a}{b}$ and $\frac{c}{d}$ with positive denominators.
Estimate first: $\frac23 + \frac56$ is a bit less than $\frac12 + 1$ , so an answer near 1.5 is reasonable.
Reduce each fraction before you start; smaller numbers make the common denominator easier to find.

Frequently asked questions

How do I add fractions with different denominators?

Rewrite both fractions over their least common denominator, add the numerators, and reduce. For $\frac23 + \frac56$ the LCD is 6, giving $\frac46 + \frac56 = \frac96 = \frac32$ .

How do I enter a mixed number?

Type the whole number, a space, then the fraction, for example 2 3/4. The calculator converts it to $\frac{11}{4}$ before calculating.

Can I mix decimals and fractions?

Yes. A decimal such as 0.25 is converted to the exact fraction $\frac14$ first, so the answer stays exact.

Why is my answer shown as an improper fraction and a mixed number?

Both describe the same value. Improper fractions are easier to use in further algebra; mixed numbers are easier to picture.

What happens if I divide by zero?

Division by a zero fraction is undefined, so the calculator stops and tells you which part of the expression caused it.

Does it follow the order of operations?

Yes. Multiplication and division are done before addition and subtraction, and parentheses are evaluated first.

How do I divide a fraction by a fraction?

Keep the first fraction, change division to multiplication and flip the second. $\frac34 \div \frac25 = \frac34 \times \frac52 = \frac{15}{8} = 1\tfrac78$ .

How do I turn an improper fraction into a mixed number?

Divide the numerator by the denominator. The quotient is the whole part and the remainder sits over the original denominator: $\frac{15}{8}$ is 1 remainder 7, so $1\tfrac78$ .

How do I convert a fraction to a decimal or a percent?

Divide the numerator by the denominator to get the decimal, then multiply by 100 for the percent. $\frac{3}{8} = 0.375 = 37.5\%$ . The calculator shows the decimal value of every answer as a check.

Calculate with fractions now

Type any fraction expression above, including mixed numbers and decimals, to get an exact answer in lowest terms with every step explained.

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