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Math calculator

Ratio Calculator

Simplify, scale, split and compare exact ratios, solve proportions, and convert parts to percentages. See formulas, input limits, worked steps and exports.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 8/8 declared partitions (simplify, proportion, scale, split, compare, convert, three-term, invalid-domain) · Matrix
Numerical scope
All six published Ratio modes (simplify, proportion, scale, split, compare, convert), including A:B:C where those modes support it. Independent O2 expected values; exact rational BigInt; Node is runtime host only.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Simplify, scale, and compare ratios; solve proportions; split a total; convert to fraction/decimal/percent.
Scope
Finite rationals: integers, terminating decimals, or simple fractions a/b.
Verification
Engine tested · Source checked · v1.0.3 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.3 · CVP protocol 1.0.0-proposed · Evidence 2026-10-21.ratio-complete-contract-v3
Verification revision
2026-10-21.ratio-complete-contract-v3 · 7/7 property · digest 980fb61ebf20
Legacy regression
26/26 tests · Production surface contract 10/10
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
19/19 golden · 7/7 CVP boundary · 7/7 invalid · 7/7 property · 2/2 metamorphic · 2/2 round-trip · 3/3 cross-interface · 10/10 CVP contract · Manifest
Sources
Sources
Evidence
11 legacy golden · 7 legacy boundary · legacy regression suite · 19/19 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.3 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Simplifyr₁:⋯:rₙ → exact rationals; L=LCM(d₁,…,dₙ); rᵢ′=L·rᵢ; g=GCF(|r₁′|,…,|rₙ′|); (r₁′÷g):⋯:(rₙ′÷g)
ProportionA/B = C/D → A·D = B·C
Scaler₁:⋯:rₙ → (k·r₁):⋯:(k·rₙ)
Split totalpartᵢ = total × (termᵢ / Σ terms) with every term ≥ 0 and Σ terms > 0
iEach term is an exact rational. L = LCM of all denominators, rᵢ′ = L·rᵢ, then divide by g = GCF(|rᵢ′|). Same n-term definition for A:B and A:B:C. Integer inputs skip the clearing step (L=1). Zero in a denominator of a proportion (B or D when solving) is undefined. Simplify, proportion, and compare allow negative terms. Split requires every term ≥ 0 and a positive sum.

How to use

1

Choose a mode

Simplify, proportion, scale, split, compare, or convert.

2

Enter the known values

Leave the unknown blank in proportion mode, or enter a scale factor / total when needed.

3

Read the result and steps

See the reduced ratio, missing term, shares, or conversions.

Example calculations

Common configurations with formula and result.

ϟ

Simplify

12:18

÷ GCF 6
2:3
ϟ

Proportion

4:7 = x:21

x = 4×21/7
12
ϟ

Scale

1:2.5:6.5 × 3

multiply each term
3:7.5:19.5
ϟ

Split total

25 by 3:2

25×3/5, 25×2/5
15 and 10
ϟ

Concrete mix

1:2:3 with sand = 7

scale so B→7
3.5:7:10.5

Quick checks

Common values at a glance.

ProblemResult
Simplify 12:182:3
4:7 = x:21x = 12
2:5 × 48:20
Split 25 by 3:215 and 10
4:6 vs 10:15equivalent
3:5 as fractions of whole3/8 and 5/8
i Part-to-part 1:2 means of a whole of 3, the parts are 1/3 and 2/3.

Ratio calculator specification

Version 1.0.3 · Engine tested

Calculation status

Review policy · Evidence

Definition
A ratio compares relative sizes of two or more quantities (part-to-part), written A:B or A:B:C. Equivalent ratios keep the same relationship when every term is multiplied or divided by the same nonzero factor. Proportions equate two ratios: A:B = C:D.
What it calculates
Simplify, scale, and compare ratios; solve proportions; split a total; convert to fraction/decimal/percent.
Inputs
  • Ratio terms (2 or 3), optional missing term for proportions
  • Scale factor or total amount depending on mode
Outputs
  • Simplified or scaled ratio
  • Missing proportion term
  • Shares of a total
  • Equivalence check
  • Fraction, decimal, and percent forms
Formula
A:B = C:D ↔ A/B = C/D; simplify n terms via exact rationals, L=LCM(dᵢ), rᵢ′=L·rᵢ, g=GCF(|rᵢ′|); scale rᵢ → k·rᵢ; share = total × term/Σ for nonnegative terms
Assumptions
  • Finite rationals: integers, terminating decimals, or simple fractions a/b.
  • Proportion denominators in A/B or C/D must be nonzero when used.
  • Simplify / proportion / compare allow negative terms. Split does not.
Units
  • Dimensionless (or consistent units across terms)
Boundary conditions
  • Cannot simplify an all-zero ratio → ALL_ZERO
  • Proportion needs exactly one blank among A,B,C,D → PROPORTION_ONE_BLANK
  • Split requires every ratio term ≥ 0 → SPLIT_NEGATIVE_TERM
  • Split requires Σ terms > 0 → SPLIT_ZERO_SUM
  • Scale divide-by-zero → DIVISION_BY_ZERO
Numerical precision
  • Terms are exact rationals. Simplify multiplies by L = LCM of all denominators, then divides by GCF of the resulting integers — the same n-term definition for A:B and A:B:C.
  • Integer inputs have L = 1, so denominator clearing is skipped and the terms are divided directly by their GCF. Decimal or fraction inputs show the LCM clearing step.
  • Shareable URLs (?aInput=12&bInput=18&mode=simplify) restore inputs and are server-rendered with the same deterministic engine. Query `v` is ignored — it is not a calculation_version pin. Pin a version with REST/MCP `CalculatorX-Spec-Version`.
  • REST and SSR return { value, mode, formula, steps, … } plus mode-specific fields (terms, solved, shares, equivalent).
Example
12:18 → 2:3; 1.5:2.25 → 2:3; 3/4:5/6 → 9:10; 4:7 = x:21 → x=12
Validation cases

12 published on this page · 26/26 tests · Production surface contract 10/10 · View evidence

  • simplify 12:18 → 2:3
  • 4:7 = x:21 → 12
  • 2:5 × 4 → 8:20
  • split 25 by 3:2 → 15, 10
  • 4:6 vs 10:15 → equivalent
  • 1.5:2.25 → 2:3
  • 3/4:5/6 → 9:10
  • split −2:3 of 100 → error SPLIT_NEGATIVE_TERM
  • simplify 6:9:12 → 2:3:4
  • 1:2:3 × 4 → 4:8:12
  • split 30 by 1:2:3 → 5, 10, 15
  • convert 2:3 → A/B = 2/3; 40% | 60% of whole
Sources
  • Encyclopedia of Mathematics — Greatest common divisor — Pairwise Euclidean GCF used after clearing denominators
    Supports: Computation of the GCF of the integerized ratio terms after clearing denominators; multiple-term GCF is obtained iteratively (GCF(a,b,c)=GCF(GCF(a,b),c))
  • CalculatorX split convention — Nonnegative parts only
    Supports: Split requires every term ≥ 0 and Σ terms > 0, so shares are nonnegative allocations of a total
  • Margaret Stevens — The Maths Workbook (University of Oxford) — Chapter 1, §1: Algebraic Expressions and Fractions; §3: Equations
    Supports: Algebraic addition/subtraction, multiplication/division, distributive rules, equivalent fractions, common denominators and reciprocals. Ratios are interpreted as quotients. Equality of ratios with nonzero denominators permits cross multiplication, and solving a proportion uses the chapter’s equation transformations. A zero divisor is excluded; integer quotient/remainder uses the separately cited Euclidean-division reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.0.3

Background

Interpretation and common distinctions.

What does this ratio calculator compute?

A ratio compares relative sizes of two or more quantities (part-to-part), written A:B or A:B:C. Equivalent ratios keep the same relationship when every term is multiplied or divided by the same nonzero factor. Proportions equate two ratios: A:B = C:D.

The published formula or method is A:B = C:D ↔ A/B = C/D; simplify n terms via exact rationals, L=LCM(dᵢ), rᵢ′=L·rᵢ, g=GCF(|rᵢ′|); scale rᵢ → k·rᵢ; share = total × term/Σ for nonnegative terms. The following references support the stated mathematics or method:

  • Margaret Stevens — The Maths Workbook (University of Oxford): Algebraic addition/subtraction, multiplication/division, distributive rules, equivalent fractions, common denominators and reciprocals. Ratios are interpreted as quotients. Equality of ratios with nonzero denominators permits cross multiplication, and solving a proportion uses the chapter’s equation transformations. A zero divisor is excluded; integer quotient/remainder uses the separately cited Euclidean-division reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses 12:18 with ÷ GCF 6. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.ratio v1.0.3 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

What is a ratio?

A ratio compares the relative size of two or more numbers. Common writings:

  • A:B or “A to B”
  • as a fraction A/B (part-to-part)
  • three-term forms such as A:B:C (recipes, mixes)

Part-to-part compares parts to each other (e.g. cucumber:tomato = 5:4). Part-to-whole compares a part to the total (e.g. cucumber:vegetables = 5:50 = 1:10).

Modes in this calculator

Mode Use when
Simplify Reduce A:B or A:B:C to lowest terms
Proportion Solve A:B = C:D for one missing value
Scale Multiply or divide every term by a factor
Split Divide a total using the ratio parts
Compare Check whether two ratios are equivalent
Convert Fraction, decimal, %, and part-of-total %

Input precision, limits and data paths

Use at most48 digits per integer/decimal, or per numerator and denominator of a fraction; signs do not count, leading zeros count, and each token is limited to128 characters. Two or three terms are supported in simplify, scale and split; other modes use A/B and, when needed, C/D. Matching aliases must name the same rational value; conflicting aliases, unknown fields and nonempty inactive fields are rejected.

Decimal and fraction strings preserve the accepted rational exactly. JSON numbers have already passed through binary64 parsing; unsafe integer numbers are rejected, and decimals use their shortest normalized decimal text. Send exact source values as strings. Exponents, mixed numbers and zero fraction denominators are rejected. Exactness applies to accepted tokens, not information rounded before the request.

A proportion requires exactly one blank and a unique solution with both B and D nonzero, including a solved denominator. Degenerate cross-product equations do not establish a valid quotient. Scale factors must be nonzero; split terms and totals must be nonnegative, with a positive sum of terms. Convert leaves undefined A/B or part-of-whole percentages as null and explains the condition. Bars show approximate part-of-whole shares; tiny parts may be invisible, so exact values remain in the text legend. Negative ratios have no part-of-whole bar.

Calculations run locally in the browser. Shared URLs send parameters to the server; REST/MCP also send inputs. Browser history may retain the most recent20 local snapshots; Clear inputs does not delete that history. Optional analytics depends on site configuration; a retention period, deletion guarantee or processing region is not established by this review. Share version labels identify the current-engine calculation and do not replay a historical engine; use REST/MCP version pins for strict version matching.

Other calculators in this family: Percent Error Calculator, Percentage Calculator, Percentage Change Calculator, Proportion Calculator . Explore all Percent, Ratio & Proportion.

→ Percent Error Calculator Calculate absolute or signed percent error, solve for observed or accepted values, and inspect every inverse branch, numerical limit, formula and export. › → Percentage Calculator Calculate percentages, percentage of a number, and “what percent is X of Y” with clear arithmetic steps. Runs locally in your browser. Try it free. › → Percentage Change Calculator Calculate percentage increase or decrease and reverse-solve initial or final values. See complete signed solutions, formulas, and numerical limits. › → Proportion Calculator Solve a proportion for one missing term using exact rational arithmetic. Enter integers, decimals or fractions; check nonzero denominators and worked steps. ›

Frequently asked questions

Key distinctions behind the calculation.

What is a ratio?

A comparison of relative sizes of quantities, often written A:B (“A to B”). Part-to-part ratios compare parts to each other; part-to-whole compares a part to the total.

How do you simplify a ratio?

Divide every term by their greatest common factor (GCF). Example: 12:18 ÷ 6 → 2:3.

What is an equivalent ratio?

Two ratios that express the same relationship, such as 2:3 and 4:6. You get them by multiplying or dividing all terms by the same nonzero number.

How do you solve a proportion A:B = C:D?

Write A/B = C/D and cross-multiply: A·D = B·C. Solve for the missing letter. Example: 4:7 = x:21 → x = 4×21/7 = 12.

Can I split a total by a ratio?

Yes, when every ratio term is ≥ 0. Add the parts, then each share is total × (that part / sum). For 3:2 and total 25: shares are 15 and 10. Negative terms are rejected in Split.

How do I convert a ratio to a fraction of the whole?

Add the terms for the whole. Each term becomes a numerator over that sum. For 1:2, whole = 3 → 1/3 and 2/3.

Are three-term ratios supported?

Yes for simplify, scale, and split (A:B:C). Proportion mode solves two-term pairs A:B = C:D.

How are decimals and fractions simplified?

Each term is an exact rational. The calculator multiplies every term by the LCM of all denominators, then divides by the GCF of those integers. Example: 1.5:2.25 → 3/2:9/4 → multiply by 4 → 6:9 → GCF 3 → 2:3. Same n-term rule for 6:9:12 → 2:3:4. Integer inputs skip the clearing step.

Can a split use negative ratio terms?

No. Split requires every term ≥ 0 and a positive sum, so shares are ordinary nonnegative allocations. Simplify, proportion, and compare still allow negatives (for example −12:18 → −2:3).

Is 2:3 the same as 66.67%?

No. 2:3 is part-to-part. Of a whole of 5 parts, the shares are 40% and 60%. 2 as a percent of 3 is a different reading (A of B), shown separately in Convert.