Proportion Calculator: Solve A/B = C/D, Cross-Multiplication and Similar Triangles

About this calculator

A proportion states that two ratios are equal:

\frac{A}{B}=\frac{C}{D}.

Provide any three of the four values and CalcX solves the fourth. The underlying operation is cross-multiplication: $AD=BC$ .

This pattern appears in recipes, maps, scale drawings, similar triangles, unit-rate problems and percentage relationships. The important assumption is that the relationship is truly proportional; not every pair of changing quantities is.

Formula and variables

\frac{A}{B}=\frac{C}{D}\qquad\Longleftrightarrow\qquad AD=BC

So the four solve forms are

A=\frac{BC}{D},\qquad B=\frac{AD}{C},\qquad C=\frac{AD}{B},\qquad D=\frac{BC}{A}.

Symbol	Role	Notes
$A,B$	first ratio	denominator must not be zero
$C,D$	second ratio	denominator must not be zero

If the values carry units, each ratio should compare compatible quantities and the final units should make sense dimensionally.

How to use this calculator

Enter any three of A, B, C and D.
Leave the value you want to solve blank.
Read the computed fourth term.
Check by cross-multiplying: the two products $AD$ and $BC$ should match apart from rounding.
If one denominator would be zero, the proportion is undefined.

Worked examples

Example 1: recipe scaling

A recipe uses 300 g flour for 4 servings. For 10 servings, let

Then $x=300\times10/4=750$ g.

Example 2: map scale

If 2 cm on a map represents 5 km, then 7.5 cm represents

Example 3: similar triangles

A 1.8 m person casts a 1.2 m shadow while a pole casts an 8 m shadow. Under the same sun angle,

Example 4: percent as a proportion

“What is 15% of 240?” can be written

Use the recipe example, then double both B and D. The solved value remains the same because multiplying corresponding terms by the same factor preserves the ratio.

math

\frac{300}{4}=\frac{x}{10}.

math

\frac{2}{5}=\frac{7.5}{x}\quad\Rightarrow\quad x=18.75\,km.

math

\frac{1.8}{1.2}=\frac{h}{8}\quad\Rightarrow\quad h=12\,m.

math

\frac{15}{100}=\frac{x}{240}\quad\Rightarrow\quad x=36.

Principle and derivation

Starting from $A/B=C/D$ , multiply both sides by $BD$ :

AD=BC.

Every inverse formula follows by dividing that equality by the remaining known factor. Cross-multiplication is therefore not a separate rule; it is ordinary algebra applied to equal fractions.

Similar triangles produce proportions because corresponding side lengths share the same scale factor. A map scale or recipe scale uses the same mathematical structure.

Limitations

Supported:: solve any one term in $A/B=C/D$ from the other three.
Approximate only:: many real relationships are only approximately proportional because of discounts, map distortion, nonlinear response, or changing conditions.
Not covered directly:: inverse proportion, ratio allocation, three-part ratios, and compound proportion; rewrite them into one or more ordinary proportions first.
No simplification or conversion:: the calculator does not reduce ratios, report the proportionality constant $k$ , or convert units.
Numerics:: a zero denominator can produce infinity, and displayed results are rounded.

Assumptions

The two ratios represent the same proportional relationship.
Denominators are non-zero.
Units are compatible or deliberately converted.
Real-world scale factors are assumed constant over the range being modeled.

Common pitfalls

Swapping the order in one ratio but not the other.
Treating an additive relationship as proportional.
Using a zero denominator.
Mixing centimeters, meters and kilometers without conversion.
Confusing direct and inverse proportion.

Typical use cases

Cooking and mixtures

Scale recipes, drink mixtures, paint, or concrete proportions to a different batch size.

Maps, drawings, and models

Convert between drawing and real-world dimensions or between model and full-scale dimensions.

Measurement and similar shapes

Estimate height from shadows or transfer dimensions between similar figures.

Prices and exchange rates

Check constant-unit-price purchases or simple currency conversions when a fixed rate is assumed.

Teaching

Practice cross-multiplication, distinguish direct from inverse proportion, and connect proportions with similar triangles.

FAQ

Why does cross-multiplication work?

Because multiplying $A/B=C/D$ by the common denominator $BD$ gives $AD=BC$ .

Can proportions use negative values?

Algebraically, yes, as long as the denominators are non-zero. Whether a negative value makes sense depends on the application.

Is a percentage just a proportion?

A percentage is a ratio per 100, so many percentage problems can be written as proportions such as $p/100=A/B$ .

Privacy

All values are processed locally in your browser. This static explanation does not record your inputs.

Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

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