Mixed Number Calculator | Add & Divide
Add, subtract, multiply, and divide mixed numbers. Converts between mixed numbers, improper fractions, and decimals with step-by-step working.
First number
Second number
What Is the Mixed Number Calculator | Add & Divide?
The Mixed Number Calculator performs addition, subtraction, multiplication, and division on mixed numbers, numbers like 2¾ or 1½ that combine a whole part and a fractional part. It shows the full step-by-step working: converting to improper fractions, finding the Lowest Common Denominator for add/subtract, operating, simplifying with GCD, and converting back to mixed number form.
- ›All four operations: add (+), subtract (−), multiply (×), and divide (÷), switch operations using the dropdown between the two numbers.
- ›Three result formats: the answer is shown as a mixed number, an improper fraction, and a decimal simultaneously.
- ›Full working shown: every step from conversion through LCD/GCD computation to the final mixed number is numbered and displayed.
- ›Stacked fraction inputs: numerator and denominator are entered in proper stacked form to match the visual appearance of fractions on paper.
- ›Handles negatives and zero whole parts: enter 0 as the whole number for proper fractions, or negative whole numbers for negative mixed numbers.
Formula
| Operation | Method | Key step |
|---|---|---|
| Addition | (a/b) + (c/d) = (a·LCD/b + c·LCD/d) / LCD | Find LCD, rewrite both fractions |
| Subtraction | (a/b) − (c/d) = (a·LCD/b − c·LCD/d) / LCD | Find LCD, subtract numerators |
| Multiplication | (a/b) × (c/d) = (a·c) / (b·d) | Multiply numerators and denominators |
| Division | (a/b) ÷ (c/d) = (a/b) × (d/c) | Multiply by reciprocal of divisor |
| Simplify | Divide numerator and denominator by GCD | GCD found by Euclidean algorithm |
| Convert back | Improper n/d → whole n÷d remainder n mod d | whole = floor(|n|/d), rem = |n| mod d |
How to Use
- 1Enter the whole number, numerator, and denominator for the first mixed number (use 0 for whole if entering a proper fraction).
- 2Select the operation (+, −, ×, ÷) from the dropdown between the two inputs.
- 3Enter the whole number, numerator, and denominator for the second mixed number.
- 4Click "Calculate", or press Enter, to compute the result.
- 5Read the answer in three formats: mixed number, improper fraction, and decimal.
- 6Follow the numbered steps to see the full working including LCD and GCD.
- 7Click "Clear" to reset all fields and start a new calculation.
Example Calculation
Example 1, Addition: 2¾ + 1½
Example 2, Division: 3½ ÷ 1¼
Tip: entering negative mixed numbers
Understanding Mixed Number | Add & Divide
What is a Mixed Number?
A mixed number combines a whole number and a proper fraction into a single quantity. For example, 2¾ means 2 whole units plus ¾ of another unit. Mixed numbers are a natural way to express quantities between whole numbers: 2¾ cups of flour, 3½ hours of work, 1¼ metres of fabric.
Every mixed number can be converted to an improper fraction (where the numerator is larger than or equal to the denominator) and vice versa. To convert a b/c to an improper fraction: multiply the whole number by the denominator and add the numerator: (a·c + b)/c. This conversion is the first step in all arithmetic operations with mixed numbers.
- ›Proper fraction: numerator less than denominator (e.g. 3/4).
- ›Improper fraction: numerator greater than or equal to denominator (e.g. 11/4).
- ›Mixed number: whole number plus a proper fraction (e.g. 2¾ = 11/4).
- ›Every improper fraction can be written as a mixed number; every mixed number as an improper fraction.
Adding and Subtracting Mixed Numbers
To add or subtract mixed numbers, convert both to improper fractions, find the Lowest Common Denominator (LCD), rewrite both fractions with that denominator, add or subtract the numerators, then simplify and convert back to a mixed number.
The LCD of two denominators a and b is found using the GCD: LCD = |a·b| / GCD(a, b). The GCD is computed efficiently with the Euclidean algorithm, repeatedly replacing (a, b) with (b, a mod b) until b = 0, at which point a is the GCD.
- ›Example: 1/3 + 1/4 → LCD = 12 → 4/12 + 3/12 = 7/12.
- ›Shortcut when denominators share no factors (coprime): LCD = product of denominators.
- ›Always simplify the result by dividing numerator and denominator by their GCD.
- ›For subtraction, the same LCD method applies, just subtract numerators instead.
Multiplying and Dividing Mixed Numbers
Multiplication and division of mixed numbers do not require a common denominator. For multiplication, convert both to improper fractions and multiply numerators together and denominators together: (a/b) × (c/d) = (a·c)/(b·d). Then simplify.
Division is performed by multiplying by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c). The reciprocal of a fraction simply swaps numerator and denominator. A common memory aid is "keep, change, flip", keep the first fraction, change division to multiplication, flip the second fraction.
- ›Cross-cancellation before multiplying reduces numbers early and avoids large intermediate values.
- ›Division by a mixed number is equivalent to multiplication by its reciprocal (as an improper fraction).
- ›Multiplying a mixed number by a whole number: convert whole number to n/1 and apply the same rule.
- ›For 2¾ × 1½: 11/4 × 3/2 = 33/8 = 4 1/8.
Simplifying Fractions with the GCD
A fraction is in simplest form (lowest terms) when the numerator and denominator share no common factor other than 1. To simplify, divide both by their Greatest Common Divisor. The Euclidean algorithm finds the GCD efficiently in O(log min(a, b)) steps:
Frequently Asked Questions
How do you add two mixed numbers?
To add two mixed numbers, follow these steps:
- ›Convert each mixed number to an improper fraction: a b/c → (a·c + b)/c
- ›Find the Lowest Common Denominator (LCD) of the two denominators
- ›Rewrite both fractions with the LCD as denominator
- ›Add the numerators, keep the denominator
- ›Simplify the result using GCD
- ›Convert back to a mixed number
Example: 1½ + 2¼ = 3/2 + 9/4 = 6/4 + 9/4 = 15/4 = 3¾.
How do you convert a mixed number to an improper fraction?
To convert a mixed number a b/c to an improper fraction:
The denominator stays the same. You multiply the whole number by the denominator to find how many fraction pieces fit in the whole part, then add the numerator (extra pieces). For negative mixed numbers like −2¾, the improper fraction is −11/4.
How do you multiply mixed numbers?
Do not multiply the whole and fractional parts separately, that gives the wrong answer. Instead:
- ›Convert both mixed numbers to improper fractions
- ›Multiply numerator × numerator and denominator × denominator
- ›Simplify the result using GCD
- ›Convert back to a mixed number
Example: 2½ × 1⅓ = 5/2 × 4/3 = 20/6 = 10/3 = 3⅓. Note that 2×1 + ½×⅓ = 2 + 1/6 ≠ 3⅓, this is why converting to improper fractions first is essential.
How do you divide mixed numbers?
To divide two mixed numbers, use "keep, change, flip":
- ›Convert both mixed numbers to improper fractions
- ›Keep the first fraction unchanged
- ›Change the ÷ sign to ×
- ›Flip (take the reciprocal of) the second fraction
- ›Multiply numerator × numerator and denominator × denominator
- ›Simplify and convert back to mixed number form
Example: 3½ ÷ 2¼ = 7/2 ÷ 9/4 = 7/2 × 4/9 = 28/18 = 14/9 = 1 5/9.
What is the Lowest Common Denominator and why is it needed for addition?
The LCD (Lowest Common Denominator) is the smallest number that is a multiple of both denominators. It is needed for addition and subtraction because fractions can only be combined when they measure the same-sized pieces, i.e., have the same denominator.
The LCD equals LCM(a, b) = |a·b| / GCD(a, b). For example, to add 1/4 + 1/6: GCD(4,6) = 2, LCD = 4×6/2 = 12. Then 1/4 = 3/12 and 1/6 = 2/12, so 1/4 + 1/6 = 5/12.
- ›For multiplication and division, no LCD is needed, just multiply or use the reciprocal.
- ›The LCD is always the LCM (Least Common Multiple) of the denominators.
- ›If the denominators are coprime (GCD = 1), LCD = product of denominators.
How do you simplify a fraction to lowest terms?
Divide both the numerator and denominator by their Greatest Common Divisor (GCD). The GCD is found using the Euclidean algorithm:
A fraction is in lowest terms when GCD(numerator, denominator) = 1. This calculator applies simplification automatically at each step.
Can this calculator handle improper fractions and negative mixed numbers?
Yes, you can enter any combination of proper fractions, improper fractions, and negative values:
- ›Proper fraction (e.g. ¾): enter whole = 0, numerator = 3, denominator = 4.
- ›Improper fraction (e.g. 7/4): enter whole = 1, numerator = 3, denominator = 4 (equivalent to 1¾ = 7/4).
- ›Negative mixed number (e.g. −2¾): enter whole = −2, numerator = 3, denominator = 4.
- ›Zero: enter whole = 0, numerator = 0, and any nonzero denominator.
The calculator normalises all inputs to improper fractions before operating, so the sign is handled correctly regardless of which fields you use.