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Ratio & Proportion Calculator

Solve for the missing value in a direct or inverse proportion with cross multiplication, work compound proportion problems, and simplify a ratio to its percent, decimal, and fraction form — all in one tool, with instant, step-by-step results.

4 modes in one tool Step-by-step solutions Free, no sign-up Updated: Jul 19, 2026

Set up the proportion: A / B = C / D. Leave the box you want to solve for blank (or type ?); fill in the other three.

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In an inverse proportion the products are equal: A × B = C × D. Leave the box you want to solve for blank; fill in the other three. (E.g. workers–days, speed–time.)

×
=
×

Use this when a result depends on more than one factor at once. Enter each factor's old → new value and its relationship (direct/inverse); the tool computes the new result from the known one.

FactorOldNewRelationship

Reduce a ratio to its simplest form and see its percent, decimal, and fraction equivalents. Separate terms with : or / (2 or 3 terms).

Use a period for decimals: 1.5 : 2, for example.
Leave blank to just simplify. Fill it in to check whether the two ratios are equal, and which is larger.
Quick answer A proportion is the equality of two ratios: A/B = C/D. To find a missing value, use cross multiplication: A × D = B × C, then divide to isolate the unknown — for example, if D is unknown, D = (B × C) / A. In an inverse proportion, the products are equal instead: A × B = C × D.
A/B = C/DDirect proportion
A×B = C×DInverse proportion
A×D = B×CCross multiplication
φ ≈ 1.618Golden ratio
⚙️ Note: All calculations run in your browser, algorithmically and exactly — no data is sent to a server. Direct proportion uses cross multiplication (A×D = B×C); inverse proportion uses equal products (A×B = C×D). Results are rounded to at most 6 decimal places for very large or very small numbers, and this tool is for informational and educational purposes only.
📐 Ratio & Proportion Calculator — ToolPico ↗

What is a ratio, and how do you solve a proportion?

A complete guide — with formulas and examples — to direct proportion, inverse proportion, compound proportion, cross multiplication, and the fourth proportional.

A ratio is a comparison of two quantities of the same kind, obtained by dividing one by the other, written as a : b or a/b. A proportion is the equality of two ratios: a/b = c/d. Finding the fourth term from three known terms in this equality is called solving a proportion. This calculator solves direct, inverse, and compound proportions, and shows the full step-by-step solution for every result.

How do you solve a direct proportion? (cross multiplication)

Quick answerIn a direct proportion, two ratios are equal: A/B = C/D. To find the unknown, use cross multiplication: A × D = B × C. Then isolate the unknown — for example, if D is unknown, D = (B × C) / A. If two quantities rise together and fall together, the relationship is a direct proportion.
  • Example: 2 lb of apples cost $10 — how much do 5 lb cost? → 2/10 = 5/x → x = (10 × 5) / 2 = $25.
  • Ratio constant (k): In a direct proportion, A/B = C/D = k; k is the value per unit (in this example, 1 lb = $5).
  • Speed–distance (at a constant time), workers–output, and scaling a recipe are all direct proportions.

What is an inverse proportion? With an example

Quick answerIn an inverse proportion, as one quantity increases, the other decreases at the same rate; their product stays constant: A × B = C × D. Example: if 6 workers finish a job in 12 days, how long will 8 workers take? 6 × 12 = 8 × x → x = 72 / 8 = 9 days. More workers means less time.

The most common mix-up with inverse proportion is cross multiplication: cross multiplication does not apply to inverse proportion — the two products are set equal directly instead. Tap–fill time, speed–travel time, and worker–days problems are all inverse proportions.

What is the fourth proportional?

Quick answerThe fourth proportional is the fourth term calculated in A/B = C/D from three known terms. The formula depends on which term is missing: A=(B×C)/D, B=(A×D)/C, C=(A×D)/B, D=(B×C)/A. In this tool, simply leave the relevant box empty.

How do you solve a compound proportion?

Quick answerIn a compound proportion, the result depends on more than one factor. You multiply the ratio for each factor: new/old for a directly proportional factor, old/new for an inversely proportional one. Example: if 4 workers produce 120 units in 6 hours, how many units do 6 workers produce in 8 hours? 120 × (6/4) × (8/6) = 240 units.

In the "Compound proportion" tab you choose each factor's old–new value and its direct/inverse relationship; the tool shows every multiplier and computes the new result. Most worker–day–output and cost problems are solved this way.

Real-world ratio & proportion examples

Common problems, which type of proportion they are, and how they're solved. Try your own numbers with the calculator above.

Example problems and how to solve them
ProblemTypeSolution
2 lb of apples cost $10 — 5 lb?Direct2/10=5/x → $25
3 pens cost $15 — 7 pens?Direct3/15=7/x → $35
80 mph for 3 hours — how many miles?Direct1/80=3/x → 240 mi
6 workers finish a job in 12 days — 8 workers?Inverse6×12=8×x → 9 days
4 taps fill a pool in 3 hours — 6 taps?Inverse4×3=6×x → 2 hours
120 mph for 2 hours — at 80 mph?Inverse120×2=80×x → 3 hours
4 workers make 120 units in 6 hrs — 6 workers, 8 hrs?Compound120×(6/4)×(8/6) → 240 units

Popular calculations

Related calculators: scale, mixing ratio & golden ratio

Three everyday uses of ratio: a map scale, splitting an amount by ratio, and the golden ratio. All work instantly in your browser.

🗺️Scale / map ratio
If the scale is 1 : N, real distance = map distance × N. Works in both directions.
🧪Mix / split by ratio
Split a total amount according to a given ratio. E.g. split $6,000 in a 2 : 3 : 5 ratio.
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🌀Golden ratio (φ)
Split a length into a long and short part by the golden ratio (φ ≈ 1.618).
🖥️Aspect ratio / resolution
Find a missing screen/video dimension that keeps the same aspect ratio (e.g. 16:9).

Ratio & proportion reference tables

Citable quick reference: proportion types and formulas, common map scales, ratio-to-percent conversion, and golden ratio values.

Proportion types and their core formulas
TypeDefinitionFormula
Direct proportionOne increases as the other increasesA/B = C/D
Cross multiplicationHow you solve a direct proportionA×D = B×C
Inverse proportionOne increases as the other decreasesA×B = C×D
Compound proportionThe result depends on multiple factorsNew = Old × Π(ratio)
Ratio constant (k)Constant in a direct proportionk = A/B = C/D
Common map scales — what 1 cm on the map equals in reality
Scale (1 : N)1 cm equalsTypical use
1 : 1,00010 mSite / parcel plan
1 : 25,000250 mTopographic map
1 : 100,0001 kmRegional map
1 : 500,0005 kmState / highway map
1 : 1,000,00010 kmCountry map

Real distance = map distance (cm) × N. Since 1 km = 100,000 cm, larger scales simplify neatly into kilometers.

Ratio → fraction → decimal → percent (share of a within a:b)
Ratio (a:b)Fraction a/(a+b)DecimalPercent
1 : 41/50.2020%
1 : 31/40.2525%
1 : 11/20.5050%
2 : 32/50.4040%
3 : 13/40.7575%
16 : 916/250.6464%

The "Simplify & Convert" tab performs these conversions automatically for any ratio you enter.

Golden ratio (φ) values
QuantityValue
φ (golden ratio)1.6180339887…
1/φ0.6180339887…
Long part's share61.8%
Short part's share38.2%
φ²2.6180339887…

When a line segment is divided by the golden ratio, long/short = whole/long = φ. Used in art, architecture, and design as an aesthetically pleasing proportion.

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Ratio & proportion terms glossary

Short definitions of the core terms used in ratio and proportion problems.

RatioA comparison of two quantities of the same kind; written as a : b or a/b. Has no units.
ProportionThe equality of two ratios: a/b = c/d. Its parts are called the terms of the proportion.
Direct proportionTwo quantities rise together and fall together; their ratio stays constant (k).
Inverse proportionAs one quantity rises the other falls; their product stays constant (A×B = C×D).
Cross multiplicationIn a/b = c/d, the equality a×d = b×c; the method used to solve a direct proportion.
Fourth proportionalThe fourth term calculated from three known terms of a proportion.
Ratio constant (k)The shared value of the ratios in a direct proportion; the amount per unit.
Compound proportionA result that is proportional to more than one quantity at the same time.
ScaleThe ratio of a map or model length to the real length; written as 1 : N.
Golden ratio (φ)A special ratio of about 1.618, satisfying long/short = whole/long.

In-depth guides

Step-by-step explanations of the most commonly confused ratio and proportion questions.

Direct or inverse proportion? How to tell the difference

The rule is simple: if one quantity increasing makes the other increase too, it's a direct proportion; if one increasing makes the other decrease, it's an inverse proportion. "More workers → less time" is an inverse (opposite-direction) relationship. "More pounds → more cost" is a direct (same-direction) relationship.

Practical tip: ask yourself, "if I double one quantity, does the other double too (direct), or get cut in half (inverse)?" Direct proportion is solved with cross multiplication; inverse proportion is solved by setting the two products equal.

Why does cross multiplication work?

If you multiply both sides of a/b = c/d by b × d, the denominators cancel out and you're left with a × d = b × c. In other words, cross multiplication is a shortcut that clears the fractions from the equation so it can be solved in one step.

That's why cross multiplication only applies to the equality of two ratios (direct proportion). In an inverse proportion the ratios aren't equal — you write A×B = C×D directly instead.

How do you split an amount according to a ratio?

Divide the total by the sum of the ratio terms to get the value of one share, then multiply by each term. For example, to split $6,000 in a 2 : 3 : 5 ratio: total shares = 2+3+5 = 10; one share = 6,000 / 10 = $600. The parts are $1,200, $1,800, and $3,000.

Just enter the total and the ratio into the "Mix / split by ratio" tool above — it returns every part and its percentage automatically. Inheritance splits, profit sharing, and mixing problems are all solved this way.

Frequently asked questions

How do you solve a proportion?
A proportion is the equality of two ratios: A/B = C/D. To find the missing value from three known terms, use cross multiplication: A × D = B × C. Then isolate the unknown; for example, if D is unknown, D = (B × C) / A. In an inverse proportion, the products are equal instead: A × B = C × D.
What is a direct proportion?
A direct proportion is a relationship where two quantities increase together and decrease together at the same rate; their ratios stay constant (A/B = C/D = k). For example, if 2 lb of apples costs $10, then 4 lb costs $20 — weight and price are directly proportional.
What is an example of inverse proportion?
In an inverse proportion, as one quantity increases the other decreases; their product stays constant (A × B = C × D). Example: if 6 workers finish a job in 12 days, how long will 8 workers take? 6 × 12 = 8 × D → D = 72 / 8 = 9 days.
How does cross multiplication work?
In A/B = C/D, cross multiplication multiplies the terms diagonally: A × D = B × C. Solving this equation for the unknown term gives the missing value. Cross multiplication only applies to direct proportion (the equality of two ratios).
What is the fourth proportional, and how do you find it?
The fourth proportional is the fourth term calculated from three known terms in A/B = C/D. The formula depends on which term is missing: A=(B×C)/D, B=(A×D)/C, C=(A×D)/B, D=(B×C)/A. Simply leave the relevant box empty (or type ?) in this tool.
What is a compound proportion?
A compound proportion is when a result depends on more than one quantity at once. Multiply the ratio for each factor: new/old for a direct relationship, old/new for an inverse one. Example: if 4 workers produce 120 units in 6 hours, 6 workers produce 120 × (6/4) × (8/6) = 240 units in 8 hours.
How do you simplify a ratio?
Divide every term by their greatest common divisor (GCD). For example, the GCD of 12:18 is 6, so dividing each term by 6 gives 2:3. For decimal ratios, first multiply every term by a power of 10 to make them whole numbers. The "Simplify & Convert" tab also shows the percent, decimal, and fraction equivalents.
How do you calculate a map scale?
If the scale is 1:N, one unit on the map equals N units in reality. Real distance = map distance × N. For example, on a 1:100,000 scale map, 3 cm represents 300,000 cm = 3 km in reality. Conversely, divide the real distance by N to find the length on the map.
How do you find a missing width or height for an aspect ratio like 16:9?
Set up a direct proportion between the aspect ratio and the pixel dimensions: ratio width / ratio height = pixel width / pixel height. For example, for a 16:9 screen with a width of 1920 px, height = 1920 × (9/16) = 1080 px. The "Aspect ratio / resolution" tool does this automatically — enter one dimension and it solves for the other.
How do you tell if two ratios are equal?
Two ratios A:B and C:D are equal exactly when their cross products match: A × D = B × C — equivalently, when A/B and C/D reduce to the same decimal value. If they aren't equal, whichever decimal is larger is the larger ratio. The "Simplify & Convert" tab's optional second ratio field checks this automatically.

Methodology & sources

ToolPico's Ratio & Proportion Calculator is a free, independent tool. All calculations run in your browser — algorithmic and exact; no data is sent to a server, and results don't depend on data that needs regular updates (evergreen).

Method: direct proportion is solved with cross multiplication (A×D = B×C), inverse proportion with equal products (A×B = C×D), and compound proportion by multiplying the ratio for each factor. Ratio simplification scales terms to whole numbers and reduces them using the GCD (greatest common divisor, via the Euclidean algorithm). The scale, mix, and golden ratio tools are based on standard mathematical definitions (golden ratio φ = (1+√5)/2 ≈ 1.618).

Basis: Standard mathematics (ratio and proportion). Last updated: July 19, 2026. Results are rounded to at most 6 decimal places and are for informational and educational purposes only.
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