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Mixed Number Calculator

Reference

Fraction and Mixed Number Formulas

Every core formula for working with mixed numbers and fractions, collected on one page — converting, simplifying, finding a common denominator, and all four operations. Try any pair of values in the calculator below to see a formula play out live, with full step-by-step working.

Live — updates as you type

Type the whole number, numerator, and denominator into their own fields. For a negative value such as −2 3/4, use the sign toggle or type −2 in the whole-number field.

Result

View step-by-step working

    Formulas

    Converting Between Mixed Numbers and Fractions

    Mixed Number to Improper Fraction

    Multiply the whole number by the denominator, add the numerator, and keep the same denominator.

    Mixed to improper formula

    whole × denominator + numerator, over the same denominator

    For 2 3/4: 2 × 4 + 3 = 11, so the improper fraction is 11/4. Every arithmetic formula below assumes both values have already been converted this way first.

    Improper Fraction to Mixed Number

    Divide the numerator by the denominator; the quotient is the whole number and the remainder is the new numerator.

    Improper to mixed formula

    numerator ÷ denominator = whole number, remainder → new numerator

    For 11/4: 11 ÷ 4 = 2 remainder 3, so the mixed number is 2 3/4 — exactly reversing the formula above, which is why the two never disagree.

    Formulas

    Simplifying and Finding a Common Denominator

    Simplifying formula

    numerator ÷ GCF, over denominator ÷ GCF

    Dividing both the numerator and the denominator by their greatest common factor shrinks a fraction to lowest terms without changing its value. 8/12 shares a GCF of 4 with itself, so 8 ÷ 4 over 12 ÷ 4 gives 2/3.

    LCD / LCM shortcut formula

    LCD(b, d) = (b × d) ÷ GCF(b, d)

    For denominators 6 and 8: 6 × 8 = 48, divided by their GCF of 2, gives an LCD of 24. This is the same number a full prime-factorization search would return, reached in a single calculation.

    Formulas

    The Four Operations

    Addition and Subtraction

    Both share the same formula shape — only the sign in the middle changes.

    Addition formula

    a/b + c/d = (ad + bc) / bd

    Subtraction formula

    a/b − c/d = (ad − bc) / bd

    Multiplying each fraction by the other's denominator rewrites both over the same shared denominator, bd, before the numerators combine. This formula requires a matching denominator to work — it is the one precondition every addition and subtraction formula on this page shares.

    Multiplication

    No shared denominator needed — multiply straight across.

    Multiplication formula

    a/b × c/d = (a×c) / (b×d)

    Numerators multiply together, denominators multiply together, and that's the whole formula. Unlike addition and subtraction, multiplication never requires matching denominators first — mixing the two up is one of the most common errors on this page.

    Division

    Flip the second fraction into its reciprocal, then multiply straight across.

    Division formula

    a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)

    Only the second fraction — the divisor — gets turned upside down into its reciprocal. The first fraction stays exactly as it is. Once the second fraction is flipped, the rest of the formula is identical to multiplication.

    Formulas

    Decimal and Percent Conversion

    Fraction to decimal formula

    numerator ÷ denominator = decimal

    For 5/8: 5 ÷ 8 = 0.625. A mixed number converts the same way after being rewritten as an improper fraction, or by converting just the fraction part and adding it to the whole number.

    Decimal to percent formula

    decimal × 100 = percent

    0.625 × 100 = 62.5%. Reversing it — a percent to a decimal — divides by 100 instead, so 62.5% ÷ 100 returns to 0.625 with nothing lost.

    Formula in Action

    The Universal Operate Formula, Worked Through

    Worked Example — 1 2/3 + 2 3/4

    Every operation follows the same four-stage chain: convert to improper fractions, apply the matching formula, simplify, then convert back.

    1. Convert to improper fractions

      1 2/3 → 5/3, and 2 3/4 → 11/4

    2. Apply the addition formula

      5/3 + 11/4 = (5×4 + 11×3) / (3×4) = 53/12

    3. Simplify with the GCF

      53/12 → GCF 1 → already lowest terms

    4. Convert back to a mixed number

      53/12 = 4 5/12

    1 2/3 + 2 3/4 = 4 5/12

    1 2/3, 2 3/4 MIXED NUMBERS 5/3, 11/4 IMPROPER FORM 53/12 OPERATE 4 5/12 SIMPLIFY & CONVERT

    Why It Matters

    Why Formulas Matter

    A formula is a compressed procedure — every worked example on this site, however different its numbers look, follows the exact shape written out in algebra on this page. Once the shape of a/b + c/d = (ad + bc)/bd is genuinely familiar, a new addition problem with unfamiliar denominators stops looking unfamiliar at all, because the letters were never standing for specific numbers in the first place. They were always standing for "whatever numerator and denominator this particular problem happens to hand you."

    That's the real value of memorizing the formula rather than only the worked steps for one example: a step-by-step solution shows what happened for 2 1/3 + 1 1/4 specifically, while the formula shows what happens for every addition problem that will ever exist. Recognizing the pattern is what turns a calculator page from a one-off answer into something that actually teaches the method underneath it.

    Formulas also make it easy to see how the four operations relate to one another, rather than treating each as an unrelated set of rules to memorize from scratch. Subtraction is addition with a different sign in the middle. Division is multiplication after one fraction has been flipped into its reciprocal. Once that relationship is visible, learning a fifth operation — like converting to a percent — is just one more short formula layered on top of ideas that are already familiar, not a brand-new topic.

    None of this replaces working through examples by hand. Formulas describe the shape of a correct answer; the worked steps and the live calculator on this page are what confirm that a specific formula was applied correctly on a specific problem.

    Common Mistakes

    Where Formula Use Goes Wrong

    Questions

    Frequently Asked Questions

    Questions about reading and applying this formula reference, not the mechanics already covered on each operation's own page.

    How do I know which formula on this page applies to my problem?

    Start from the operation, not the numbers. If the problem says "add" or "subtract," reach for the addition/subtraction formula and remember it needs a shared denominator first. If it says "multiply," the straight-across formula applies with no denominator matching required at all. "Divide" means flip the second fraction, and any conversion between forms (mixed number, improper fraction, decimal, percent) uses the conversion formulas rather than an operation formula.

    Why do some formulas use a/b and c/d while others use words like "whole" and "numerator"?

    Algebraic notation such as a/b + c/d = (ad + bc)/bd is exact and works for any fraction, so it's used for the four operations, where the letters simply stand in for whatever numbers are involved. Conversion formulas are written as short phrases — like "whole × denominator + numerator" — because they describe a sequence of arithmetic actions to carry out on specific parts of a mixed number, which reads more clearly as a short instruction than as bare algebra.

    Do these formulas work for negative fractions too?

    Yes, every formula on this page holds for negative values without any change in shape. The sign travels with the whole quantity through the conversion to an improper fraction, and from there the addition, subtraction, multiplication, and division formulas apply exactly as written — negative numbers don't need a separate set of rules.