Adding and Subtracting Fractions Calculator
Add or subtract ordinary fractions and mixed numbers, reduce the exact answer, and see the common-denominator steps.
Inputs
1st fraction
2nd fraction
Live result
Reduce inputs if possible: 3/4 and 5/6.
Use LCD 12: 9/12 and 10/12.
Add numerators: 9 + 10 = 19, giving 19/12.
19/12 is already in lowest terms.
Common-denominator detail
| Input | Original value | Reduced form | At the LCD | Multiplier |
|---|
How to use this adding and subtracting fractions calculator
What this calculator does. This tool combines two rational numbers by addition or subtraction and returns an exact simplified fraction, a mixed-number form when useful, a decimal approximation, the least common denominator used by the working method, and a compact step-by-step derivation. It handles ordinary fractions and mixed numbers. It is an arithmetic aid: it does not decide whether a fraction model is appropriate for a real-world situation, and it does not replace the need to interpret units in a word problem.
When to use it. Use it to check homework involving unlike denominators, verify a hand calculation before submitting work, convert mixed-number addition into an exact simplified result, or inspect why a subtraction produces a negative fraction. The method follows the standard common-denominator approach described in the NCETM guidance on adding and subtracting with common denominators.
How to calculate. The calculator opens with a complete demonstration: 3/4 + 5/6 = 19/12, so the results and example Excel workbook are available immediately.
- Choose Operation: Addition (+) combines the two values, while Subtraction ( – ) subtracts the second value from the first.
- Choose Number form. Fractions uses numerator/denominator pairs. Mixed numbers adds a whole-number field for each value. When the current entries are valid, switching form converts them rather than changing their value.
- Replace the demonstration numbers with your own integers. Results update live; no Calculate button is required.
- Read the exact result first, then the mixed-number and decimal forms, the LCD, the four working steps, and the common-denominator table.
- Select Download Excel to export the current validated inputs and outputs to an XLSX workbook. Reset clears the demonstration and all numeric inputs. After Reset, Excel export is disabled until a complete valid pair of numbers is entered again.
Input guide. Operation is required and accepts Addition (+) or Subtraction ( – ); changing it reverses only the arithmetic operation, not either input. Number form is required and accepts Fractions or Mixed numbers. Numerator (n₁) and Numerator (n₂) are required integers. In fraction mode they may be negative; for example – 3 is valid. Do not enter decimals, grouping commas, percent signs, or scientific notation. Denominator (d₁) and Denominator (d₂) are required positive integers from 1 through 100,000; zero is invalid because division by zero is undefined. Whole number (w₁) and Whole number (w₂) appear in mixed-number mode and accept integers from – 100,000 through 100,000. The fractional numerator in mixed mode must have absolute value smaller than its denominator. For a conventional negative mixed number such as – 2 1/3, enter whole number – 2 and numerator 1; if the whole number is 0, a negative numerator can represent a negative proper fraction. A common mistake is entering 2.5 as a whole number or writing 5/6 into one field; each field accepts one integer only.
Output guide. Simplified result is the exact reduced rational result and is the primary answer. Mixed-number result rewrites an improper result such as 19/12 as 1 7/12; values between – 1 and 1 remain proper fractions. Decimal approximation shows the same value rounded to at most six decimal places and is useful for magnitude checks, not for replacing the exact fraction. Least common denominator is the LCM of the reduced input denominators; a value of 1 means both inputs reduce to whole numbers. The summary pills repeat Operation, Form, LCD, and Reduced so the current state is visible at a glance. The four step outputs show input reduction, conversion to the LCD, numerator addition/subtraction, and final simplification. The note below the table confirms that both reduced denominators divide the LCD exactly. In the table, Input identifies the first or second value, Original value shows what you entered, Reduced form shows the equivalent lowest-term fraction, At the LCD shows the equivalent fraction with the shared denominator, and Multiplier is the integer factor used to reach that denominator.
Worked example. The startup example uses 3/4 and 5/6 with Addition. Both fractions are already reduced. The least common denominator of 4 and 6 is 12. Multiplying 3/4 by 3/3 gives 9/12; multiplying 5/6 by 2/2 gives 10/12. The numerators are then added: 9 + 10 = 19, so the exact result is 19/12. Because 19 and 12 share no factor greater than 1, that fraction is already simplified. Dividing 19 by 12 gives the displayed decimal 1.583333, and separating the whole part gives the mixed number 1 7/12.
Learn more. For a detailed lesson on why equal denominators are needed before combining numerators, see BYU-Pathway Worldwide's addition and subtraction of fractions lesson. If you want to inspect the simplification step, its simplifying fractions lesson explains reducing numerator and denominator by common factors.
Why the least common denominator works
Fractions can be added or subtracted directly only when their denominators describe the same-sized parts. Rewriting each fraction with a common denominator changes the representation, not the value, because the numerator and denominator are multiplied by the same nonzero integer. Using the least common denominator keeps those equivalent numbers smaller than using the raw product when the original denominators share factors. A dedicated least common multiple lesson shows how the LCM provides the smallest shared multiple.
For reduced fractions a/b and c/d, first choose L = lcm(b,d). Rewrite them as a(L/b)/L and c(L/d)/L. Then add or subtract the new numerators and reduce the resulting fraction by its greatest common divisor.
Common mistakes and interpretation
- Do not add denominators. For 1/3 + 1/4, the result is not 2/7; the fractions first need equivalent denominators.
- A denominator cannot be zero. A zero numerator is valid and represents zero.
- A negative subtraction result is valid. It simply means the second rational number is larger than the first.
- Mixed-number notation is a presentation choice. The exact improper fraction and its mixed-number form represent the same rational value.
- When checking with decimals, remember that repeating decimals are rounded on screen. The simplified fraction remains the exact result.