2026-08-07 10 min

Fraction Calculator

Add, subtract, multiply, and divide fractions with steps. Includes mixed numbers, simplification, decimal/fraction conversion, compare, LCM, GCD, fraction-to-percent, and ordering.

Calculator

Operation Mode

Fractions

Basic Fraction Calculator

Result

11/12

Mixed: 11/12

Decimal: 0.9166666666666666 · 91.66666666666666%

Raw: 22/24 · Simplified by GCD 2

Fields above the solid black line represent the numerator, while fields below represent the denominator.

Results update automatically

Results update live as you type

Status:3/4 + 1/6= 11/12 (11/12)≈ 0.9166666666666666

Fraction Summary

Operation3/4 + 1/6

Simplified11/12

Mixed11/12

Decimal · Percent0.9166666666666666 · 91.66666666666666%

Reduction Audit

Raw cross-product:22/24

GCD used:2

LCM of denominators:12

Forms Compared

Form	Value
Fraction	11/12
Mixed	11/12
Decimal	0.9166666666666666
Percent	91.66666666666666%

fraction helper

Results

Optimal

Result (Fraction)

11/12

3/4 + 1/6 simplified

Mixed Number

11/12

Whole part plus remainder form

Decimal

0.916667

Numerator divided by denominator

Percent

91.67%

Decimal × 100

Raw (Unsimplified)

22/24

Before dividing by the GCD

GCD Used

Greatest common divisor of the raw terms

Common Denominator

b×d for addition and subtraction

LCM of Denominators

Smallest shared denominator

Fraction arithmetic is exact integer math: cross-products for +/−/×/÷, Euclid's GCD for reduction, LCM for comparison. Only the decimal and percent renderings round.

Visualizations

Decimals: Operands vs Result

0.8A

0.2B

0.9Result

Operand Magnitudes

A = 3/482%

B = 1/618%

Result as Share of One Whole

Result92%

Operand A vs Operand B

A750

Formula

Addition and subtraction scale both fractions to the shared denominator b×d before combining numerators. Multiplication goes straight across; division multiplies by the reciprocal. Every result is reduced by the GCD, then rendered as a mixed number, decimal, and percent.

+/−: (a×d ± c×b)/(b×d) · ×: (a×c)/(b×d) · ÷: (a×d)/(b×c), then ÷ GCD

Alternative Form

Compare via common denominator LCM(b,d); mixed a b/c = (a×c+b)/c; decimal = n÷d; percent = ×100

Variables

Symbol	Name	Unit	Description
a/b	First fraction	ratio	Numerator a over denominator b
c/d	Second fraction	ratio	Numerator c over denominator d
b×d	Common denominator	count	Guaranteed shared denominator for +/−
GCD	Greatest common divisor	count	Largest integer dividing raw top and bottom
LCM	Least common multiple	count	Smallest shared denominator of b and d
n÷d	Decimal value	number	Division form of the simplified fraction

Explanation

Fractions with different denominators count different-sized pieces, so +/− must first re-slice both into the same piece size (b×d or the tighter LCM). The GCD step then removes shared factors so 22/24 is reported as 11/12.

Derivation

a/b + c/d = (ad+cb)/bd. a/b − c/d = (ad−cb)/bd. a/b × c/d = ac/bd. a/b ÷ c/d = ad/bc. Simplify: (n÷g)/(d÷g) with g = gcd(n,d). Mixed: whole = ⌊|n|/d⌋, rem = |n| mod d.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write both operands with their numerators and denominators.

A = a/b, B = c/d

A:3/4

B:1/6

Operation:Add

Intermediate: 3/4 + 1/6

Result: Operands identified: 3/4 and 1/6

Step 2

Build the common denominator b×d and scale each numerator.

Common = b×d; scaled = a×d and c×b

b×d:24

a×d:18

c×b:4

Intermediate: 3×6 + 1×4 = 18 + 4 = 22

Result: Raw fraction = 22/24

Step 3

Find the GCD of the raw numerator and denominator.

GCD via Euclid: gcd(a,b) = gcd(b, a mod b)

Raw numerator:22

Raw denominator:24

GCD:2

Intermediate: gcd(22, 24) = 2

Result: Divisor = 2

Step 4

Divide top and bottom by the GCD to reach simplest form.

Simplified = (raw ÷ GCD) / (den ÷ GCD)

Raw:22/24

GCD:2

Intermediate: 22÷2 = 11; 24÷2 = 12

Result: Simplified = 11/12

Step 5

Convert the improper result to a mixed number where applicable.

whole = ⌊|n|/d⌋, remainder = |n| mod d

Numerator:11

Denominator:12

Intermediate: whole/remainder split of 11/12

Result: Mixed = 11/12

Step 6

Express the result as a decimal and a percent.

Decimal = n÷d; Percent = decimal × 100

Fraction:11/12

Intermediate: 11÷12 = 0.9166666666666666

Result: 0.9166666666666666 = 91.66666666666666%

Final Answer

11/12

Worked Examples

See how the calculator works with different scenarios.

Scenario

Combine two unlike fractions and simplify.

Inputs

A:3/4

B:1/6

Operation:+

Calculation

Common = 4×6 = 24 Scaled: 3×6 + 1×4 = 18 + 4 = 22 Raw = 22/24, GCD = 2 22÷2 / 24÷2 = 11/12

Result

11/12 ≈ 0.9167 ≈ 91.67%

Interpretation

The LCM shortcut (12) would reach 9/12 + 2/12 = 11/12 directly; either route must end simplified, since 22/24 still shares a factor of 2.

Understanding Your Result

Your Result: Result (Fraction): 11/12

Typical range: Proper fractions sit in (−1, 1); improper results convert to mixed form. Repeating decimals (1/3, 1/6) are the norm unless denominators factor into 2s and 5s only.

What This Means

Each result is the same rational number in four costumes: the simplified fraction, the mixed number, the decimal division, and the percent. The GCD and common-denominator figures show the machinery behind the simplification.

Why It Matters

Unsimplified fractions hide equality: 22/24 and 11/12 look different but weigh the same. Reducing, comparing via LCM, and converting to decimals lets recipes, measurements, and rates actually interoperate.

Professional Interpretation

Use the raw/GCD pair to audit reductions, the LCM figure for hand-checking +/− work, and the decimal at six places for measurement transfer. For money, round the decimal explicitly — fractions are exact, currency is not.

Recommendations

•Always report the simplified form, not the raw cross-product
•Convert mixed numbers before operating, not during
•Prefer the LCM tab when denominators are large
•Check zero denominators before interpreting any output
•Carry the fraction (not the rounded decimal) into the next step
•Use compare/ordering tabs instead of eyeballing closeness

Educational Content

Introduction

A fraction a/b means a copies of a whole split into b equal parts. This calculator performs the four operations on that idea, then translates answers into the mixed, decimal, and percent forms different trades expect.

Concept Explanation

Addition and subtraction require identical piece sizes, hence the common denominator b×d. Multiplication scales pieces of pieces, division flips and scales, and the GCD strips shared factors so each rational has one canonical simplest name.

Applications

▸Scaling recipes and mixing ratios in kitchens and labs

▸Adding lumber, pipe, and sewing measurements in eighths and sixteenths

▸Combining rates, shares, and probabilities expressed as fractions

▸Converting shop math between fractions, decimals, and percents

▸Teaching equivalence, ordering, and number-line placement

▸Reducing engineering drawings to lowest-terms dimensions

Benefits

•Shows the raw cross-product so hand work can be checked line by line
•Reduces automatically via GCD — no missed simplifications
•Keeps mixed, decimal, and percent views in sync
•Handles big integers without floating-point drift on the fraction side
•Compare/LCM/GCD tabs remove the fiddliest setup steps

Limitations

•Integer numerators/denominators only; algebraic fractions are out of scope
•Very large cross-products can exceed safe-integer precision
•Repeating decimals are shown truncated, not with bar notation
•Only models exact rationals — no irrational or measured-uncertainty math

Best Practices

•Simplify before comparing or ordering
•Convert mixed numbers to improper form first
•Use LCM (not b×d) for hand-friendly denominators
•Verify denominators are nonzero before trusting output
•Keep fractions exact until the final decimal rounding

Common Misconceptions

Myth: Add across: 1/2 + 1/3 = 2/5.

Truth: Pieces differ in size. Convert first: 3/6 + 2/6 = 5/6 — visibly larger than 2/5.

Myth: Multiplying always makes numbers bigger.

Truth: Multiplying by a proper fraction shrinks: 8 × 1/2 = 4. Only factors above 1 grow a value.

Myth: Bigger denominators mean bigger fractions.

Truth: Denominator is piece size: 1/8 is smaller than 1/4. Compare numerators only after denominators match.

Myth: 22/24 is a finished answer.

Truth: It still shares GCD 2. The finished form is 11/12 — one rational, one simplest name.

Practical Tips

•Find the LCM first when denominators are large
•Turn mixed numbers improper before touching an operator
•Read the raw line to check hand cross-multiplication
•Use Fraction→% for instant percent equivalents
•Compare via common denominator, never by numerator alone
•Keep exact fractions until the last step, then round once
•Treat Undefined as a stop sign: fix the zero denominator

Common Mistakes

Adding numerators and denominators separately

Scale to a common denominator first: a/b + c/d = (ad+cb)/bd, never (a+c)/(b+d).

Forgetting to simplify

Always divide by the GCD shown; 22/24 must become 11/12 before reporting.

Flipping the wrong fraction in division

Keep the first, change ÷ to ×, flip the second: a/b ÷ c/d = a/b × d/c.

Operating on mixed numbers in place

Convert to improper fractions first — 1½ × 2¼ means 3/2 × 9/4, not 2 and 2/8.

Entering a zero denominator

Any denominator of 0 is Undefined. Correct the input; no simplification can rescue it.

Assumptions

This calculator makes the following assumptions to provide estimates:

Integer Parts: Numerators and denominators are integers; decimal inputs belong in the conversion tabs.

Nonzero Denominators: Valid results require b ≠ 0 and d ≠ 0; zeros yield Undefined.

Exact Rationals: Fraction arithmetic is exact integer math until the decimal/percent rendering.

Positive Denominators: Signs normalize so the denominator stays positive in simplest form.

Safe-Integer Range: Cross-products are assumed within JavaScript safe-integer precision.

Limitations

Be aware of these limitations when using this calculator:

No Algebra: Fractional expressions with variables (e.g. (x+1)/(x−2)) are not supported.

Big-Integer Ceiling: Extremely large numerators/denominators can lose integer precision.

Truncated Repeats: Repeating decimals display truncated rather than with exact repetend notation.

Single Operation: Each basic calculation combines exactly two fractions; chain via the result.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a fraction?

A fraction represents a part of a whole with a numerator (top) and denominator (bottom). In 3/4, the numerator 3 counts taken parts out of 4 equal parts total.

How do I add fractions?

Find a common denominator, convert, then add numerators: a/b + c/d = (a×d + c×b)/(b×d). Example: 3/4 + 1/6 = (18+4)/24 = 22/24 = 11/12.

How do I subtract fractions?

Use the same common-denominator step, then subtract: a/b − c/d = (a×d − c×b)/(b×d). Example: 3/4 − 1/6 = (18−4)/24 = 14/24 = 7/12.

How do I multiply fractions?

Multiply straight across with no common denominator: a/b × c/d = (a×c)/(b×d). Example: 3/4 × 1/6 = 3/24 = 1/8.

How do I divide fractions?

Multiply by the reciprocal: a/b ÷ c/d = (a×d)/(b×c). Example: 3/4 ÷ 1/6 = 3/4 × 6/1 = 18/4 = 9/2 = 4½.

How do I simplify a fraction?

Divide numerator and denominator by their GCD. 24/36 has GCD 12, so it reduces to 2/3. A fraction is simplest when GCD is 1.

What is a mixed number?

A whole number plus a proper fraction, e.g. 1¾. Convert with (whole×denominator + numerator)/denominator: 1¾ = 7/4.

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

Divide: the quotient is the whole part and the remainder is the new numerator. 7/4 = 1¾; 17/5 = 3⅖.

What are LCM and GCD?

The LCM is the smallest number divisible by both denominators (used to add/subtract); the GCD is the largest shared divisor (used to simplify). For 4 and 6: LCM = 12, GCD = 2.

How do I compare two fractions?

Convert to a common denominator and compare numerators: 3/4 = 9/12 beats 2/3 = 8/12. Equivalently cross-multiply: 3×3 > 2×4.

How do I convert a fraction to a decimal?

Divide numerator by denominator: 1/4 = 0.25 (terminating), 1/3 = 0.333… (repeating). Denominators with only prime factors 2 and 5 terminate.

How do I convert a decimal to a fraction?

Write the decimal over 10ⁿ for n decimal places, then simplify: 0.75 = 75/100 = 3/4.

How do I convert a fraction to a percent?

Divide and multiply by 100: 3/4 = 0.75 = 75%. Use the Fraction→% tab for instant conversion.

What are proper, improper, and unit fractions?

Proper fractions are below 1 (3/4); improper are 1 or more (5/3); unit fractions have numerator 1 (1/3).

What are equivalent fractions?

Different writings of the same value: 1/2 = 2/4 = 3/6. Multiply or divide top and bottom by the same nonzero number.

Why can't a denominator be zero?

Division by zero is undefined — no number times 0 gives a nonzero numerator. This calculator shows "Undefined" for zero denominators.

What is a reciprocal?

Swap numerator and denominator: the reciprocal of 3/4 is 4/3, and their product is 1. Reciprocals power fraction division.

How do I add or multiply mixed numbers?

Convert to improper fractions first, operate, then convert back. For 1½ × 2¼: 3/2 × 9/4 = 27/8 = 3⅜.

What is a complex fraction?

A fraction containing fractions, e.g. (1/2)/(3/4). Simplify via the reciprocal: 1/2 × 4/3 = 2/3.

How do I order fractions least to greatest?

Convert all to a common denominator and sort numerators; compare whole parts first for mixed numbers.

Where are fractions used in real life?

Cooking (1/2 cup), construction (3/8 inch), finance (interest increments), medicine (dosages), time (quarter-hour), and sports statistics.

What is the difference between a fraction and a ratio?

A fraction is part-to-whole (3/4 of a pizza); a ratio relates two quantities (3:4 boys to girls). Ratios can be written as fractions but mean something different.

How do I solve fraction word problems?

Identify the fractions, choose the operation from the wording (altogether = add, left = subtract, of = multiply, shared = divide), compute, simplify, and restate in context.

What is the difference between terminating and repeating decimals?

Terminating decimals end (0.25); repeating decimals cycle forever (0.1666…). Only denominators built from 2s and 5s terminate.

How are fractions used in finance?

Rate increments, stock-point moves, return ratios, and exchange values are all fractional. Adding and comparing fractions is daily work in budgeting and investing.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Adding Unlike Fractions Without Fear

Common denominators via b×d and LCM.

Mixed Numbers, Clean Conversions

Improper-first workflow for every operation.

Simplify Like a Pro with GCD

Euclid's algorithm on real reductions.

Fractions, Decimals, Percents: One Number

Switching forms without losing exactness.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Numerator: Top number: how many equal parts are taken.
Denominator: Bottom number: how many equal parts make the whole; never zero.
GCD: Greatest common divisor: largest integer dividing both parts, used to simplify.
LCM: Least common multiple: smallest shared denominator for adding or subtracting.
Mixed number: Whole number plus proper fraction, e.g. 3⅞.
Improper fraction: Fraction ≥ 1 with numerator at least the denominator.
Reciprocal: Flipped fraction d/c; the key to division.
Equivalent fractions: Different writings of the same value, e.g. 1/2 = 2/4.
Simplest form: Form with GCD 1 — the canonical name of a rational.
Common denominator: Shared piece size (b×d or LCM) enabling +/−.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Arithmetic: fractions Wolfram MathWorld — Fraction Wikipedia — Fraction Math is Fun — Fractions Calculator.net — Fraction Calculator Desmos — Graphing Calculator WolframAlpha NCES — Mathematics resources

Applicable Standards

NCTM Principles to Actions: fraction equivalence and ordering expectationsCommon Core State Standards for Mathematics, grades 3–6 fractions domain

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Page Information

Last Updated:

2026-08-07

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.