Mixed Number Calculator

Add, Subtract, Multiply or Divide Mixed Numbers

Mixed Numbers Formula

Formula

\[ w_1 + \frac{n_1}{d_1} = \frac{w_1 d_1 + n_1}{d_1}, \qquad w_2 + \frac{n_2}{d_2} = \frac{w_2 d_2 + n_2}{d_2} \\ \frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}, \qquad \frac{a}{b} - \frac{c}{d} = \frac{ad - cb}{bd}, \qquad \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}, \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc} \]

Where:

\(\frac{a}{b}\), \(\frac{c}{d}\) — the two mixed numbers converted to improper fractions
\(w_1\), \(w_2\) — the whole parts; \(n_1\), \(n_2\) — the numerators; \(d_1\), \(d_2\) — the denominators

The first line converts each mixed number to an improper fraction; the second line shows the arithmetic for the four supported operations. After combining, the result is reduced by the greatest common divisor and converted back to a mixed number.

Mixed Numbers – Calculation Example

Example: add the mixed numbers \(1\frac{1}{2}\) and \(2\frac{1}{3}\) (the calculator's default values).

\[ 1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}, \qquad 2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3} \]

\[ \frac{3}{2} + \frac{7}{3} = \frac{9}{6} + \frac{14}{6} = \frac{23}{6} = 3\frac{5}{6} \]

Result: \(3\frac{5}{6}\) as a mixed number, \(\frac{23}{6}\) as an improper fraction, and \(3.8\overline{3}\) (about 3.8333) as a decimal.

How to use the calculator

Enter the First Mixed Number: type its Whole part, Numerator and Denominator. For a plain fraction such as \(\frac{3}{4}\), set the whole part to 0.
Enter the Second Mixed Number the same way.
For negative numbers, put the minus sign on the whole part (for example, −1 and 3/4 mean \(-(1\frac{3}{4})\)).
Choose the Operation from the dropdown: + Add, − Subtract, × Multiply or ÷ Divide.
Read the result as a mixed number, an improper fraction and a decimal, and check the Step-by-Step Solution to see how the answer was obtained.

What is a mixed number?

A mixed number combines a whole number and a proper fraction, written side by side: \(2\frac{1}{3}\) means 2 whole parts plus one third, i.e. two full pies and a third of another. Mixed numbers are common in everyday life — recipes call for \(1\frac{1}{2}\) cups of flour, a board may measure \(3\frac{3}{4}\) feet — because they show the size of a quantity at a glance: you immediately see there are 2 wholes plus a bit.

The same quantity can also be written as an improper fraction, whose numerator is larger than its denominator: \(2\frac{1}{3} = \frac{7}{3}\). Improper fractions are what the math actually runs on — to add, subtract, multiply or divide, you first convert every mixed number to an improper fraction, which is exactly what the calculator does in step 1 of its solution.

When the denominator does not divide evenly into 10, 100, 1000 and so on, the decimal form of a fraction repeats forever. The calculator detects repeating digits and marks them with an overline, so \(\frac{23}{6}\) is shown as \(3.8\overline{3}\) (the 3 repeats). When the fraction terminates, you simply get the exact decimal, such as \(\frac{7}{4} = 1.75\). If you only need a rounded value for practical work, the Rounding Calculator can help.

How the four operations work

Addition and subtraction require a common denominator. The calculator uses the least common denominator of the two denominators (the LCM), rewrites both fractions over it, and adds or subtracts the numerators.

Multiplication is straightforward: multiply the numerators together and the denominators together, then simplify.

Division means multiplying by the reciprocal — flip the second fraction and then multiply. Division by a mixed number equal to zero is undefined, and the calculator will tell you so instead of returning a wrong answer.

Finally the answer is reduced to lowest terms by dividing out the greatest common factor (GCF) of the numerator and denominator, then converted back to a mixed number by dividing the numerator by the denominator: the quotient is the whole part and the remainder becomes the new numerator.

Things to keep in mind

Each part must be a whole number. You cannot type 0.5 in the numerator field; enter 0 or 1 and adjust the denominator instead.
The denominator cannot be 0 — division into zero parts is undefined.
Negative numbers: the minus sign belongs on the whole part. Entering −1, numerator 3, denominator 4 is read as \(-1\frac{3}{4} = -(1 + \frac{3}{4})\), that is \(-\frac{7}{4}\). Do not also make the numerator negative, or the signs will cancel unexpectedly.
A negative denominator is allowed but discouraged — the calculator moves the sign to the numerator automatically, but writing \(\frac{5}{-4}\) as \(\frac{-5}{4}\) avoids confusion.
The fraction part is not re-checked. A numerator larger than the denominator (for example whole part 1, fraction \(\frac{5}{3}\)) is accepted and treated as \(1 + \frac{5}{3} = \frac{8}{3}\); the result is always returned as a properly reduced mixed number or plain fraction.
Values are limited to whole numbers from −9,999 to 9,999 in each field, which is plenty for homework, cooking, and woodworking problems.

To work with plain fractions only, or to check each individual step by hand, the site also offers a dedicated Ratio Calculator for ratio-style problems. For converting in the other direction, simplifying an improper fraction into whole-and-remainder form uses the same divide-and-take-the-remainder logic shown in the step-by-step solution above.

Frequently Asked Questions

How do I add or subtract mixed numbers by hand?

First convert each mixed number to an improper fraction: multiply the whole part by the denominator and add the numerator. For example, \(2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}\). Then find a common denominator, rewrite both fractions over it, and add or subtract the numerators. Finally, simplify and convert back: \(1\frac{1}{2} + 2\frac{1}{3} = \frac{3}{2} + \frac{7}{3} = \frac{23}{6} = 3\frac{5}{6}\).

Do I need a common denominator when multiplying mixed numbers?

No — a common denominator is only needed for addition and subtraction. To multiply, convert both numbers to improper fractions, multiply the numerators together and the denominators together, then simplify. For example, \(1\frac{1}{2} \times 2\frac{1}{3} = \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} = 3\frac{1}{2}\). Skipping the common-denominator step saves a lot of work.

How do I convert an improper fraction back to a mixed number?

Divide the numerator by the denominator using long division. The quotient becomes the whole part, and the remainder becomes the new numerator over the same denominator. For example, \(\frac{23}{6}\): dividing 23 by 6 gives 3 remainder 5, so the answer is \(3\frac{5}{6}\). If the remainder is 0, the result is simply a whole number.

How do I divide one mixed number by another?

Convert both numbers to improper fractions, then multiply the first by the reciprocal of the second — flip the second fraction and multiply. For example, \(3\frac{1}{2} \div 1\frac{1}{4} = \frac{7}{2} \times \frac{4}{5} = \frac{28}{10} = 2\frac{4}{5}\). Remember to simplify at the end. Dividing by zero is undefined, so the second number must not equal 0.

Why does my answer show a repeating decimal like 3.83?

A fraction produces a repeating decimal whenever its denominator has prime factors other than 2 and 5 — the decimal never terminates. The overline marks the digits that repeat forever, so \(3.8\overline{3}\) means 3.83333… Only denominators made of 2s and 5s (like 4, 8, 10, 20) give terminating decimals such as \(\frac{7}{4} = 1.75\).

What are the most common mistakes with mixed number arithmetic?

The biggest one is treating the whole parts and fractions separately when multiplying or dividing — you cannot just multiply whole parts and multiply fractions. Another is adding fractions without a common denominator, for example writing \(\frac{1}{2} + \frac{1}{3} = \frac{2}{5}\) instead of \(\frac{5}{6}\). Also watch the order in subtraction: it is not commutative, so \(2\frac{1}{3} - 1\frac{1}{2} \neq 1\frac{1}{2} - 2\frac{1}{3}\).

Can this calculator handle negative numbers and plain fractions?

Yes. To enter a plain fraction like \(\frac{3}{4}\), set the whole part to 0. For negative values, put the minus sign on the whole part only: entering −1 with fraction 3/4 means \(-(1 + \frac{3}{4}) = -\frac{7}{4}\). Do not also mark the numerator negative, or the signs will cancel unexpectedly and you will get a positive result.