Solve direct (A/B = C/D) and inverse (A×B = C×D) proportions using cross-multiplication with full step-by-step math breakdowns.
Leave exactly one field blank (or ?) to solve for it.
Click "Calculate" to solve the proportion.
A nurse calculates a medication dose based on a patient's body weight: if 50mg of drug is administered per 10kg of weight, how many milligrams does an 85kg patient require? A civil engineer measures map distances: if 2 centimeters on a topographic map represents 15 kilometers in reality, what distance does 7.4 centimeters represent? An operations manager estimates task completion times: if 4 workers complete a site cleanup in 12 hours, how many hours will 6 workers take?
These scenarios illustrate the core concept of proportions. A proportion is a mathematical statement asserting that two ratios or rates are equal. In a direct proportion, two quantities scale in the same direction—doubling one quantity doubles the other, expressed by the equation A / B = C / D. In an inverse proportion, two quantities scale in opposite directions—doubling one quantity halves the other, expressed by the product formula A × B = C × D.
Solving proportions relies on cross-multiplication. For direct proportions, multiplying the outer terms (extremes) and inner terms (means) yields the linear equation A × D = B × C. Isolating the missing term requires a single division step. This calculator automates cross-multiplication for both direct and inverse relationships, showing full algebraic steps. The following sections explore the underlying formulas, step-by-step algorithms, and real-world applications.
When you click "Calculate," the engine reads the four input fields for the active mode. It verifies that exactly three fields contain valid numerical values and one field is left blank or marked with a question mark. It then isolates the unknown term algebraically.
For a direct proportion A / B = C / D, cross-multiplication converts the fraction equation into an equivalent product statement: A × D = B × C. The engine isolates the missing variable using the following formulas:
// Direct Proportion Formulas
if (missing === 'D') D = (B * C) / A;
if (missing === 'C') C = (A * D) / B;
if (missing === 'B') B = (A * D) / C;
if (missing === 'A') A = (B * C) / D;For an inverse proportion A × B = C × D, the product of the first pair equals the product of the second pair. The missing term is isolated by dividing the known product by the single paired term:
// Inverse Proportion Formulas
if (missing === 'D') D = (A * B) / C;
if (missing === 'C') C = (A * B) / D;
if (missing === 'B') B = (C * D) / A;
if (missing === 'A') A = (C * D) / B;For example, in a direct proportion where A = 10, B = 25, C = 40, and D is unknown: cross-multiplication yields 10 × D = 25 × 40 = 1000. Dividing 1000 by 10 gives D = 100.
Medical dosage calculations. Pharmacists and nurses use direct proportions to calculate fluid volumes and tablet dosages. If a liquid antibiotic contains 250mg per 5ml, calculating the volume required for a 375mg dose (250 / 5 = 375 / V) returns exactly 7.5ml, preventing medication errors.
Map reading and GIS scaling. Cartographers and hikers convert map distances to ground distances using scale proportions. If a map scale is 1 : 50,000 (1cm = 500m), a measured trail distance of 8.2cm on paper converts to 4,100 meters (4.1km) in actual ground distance.
Construction material estimation. Concrete mixing requires maintaining strict water-to-cement ratios (e.g., 0.45 liters of water per 1 kg of cement). A project requiring 350 kg of cement uses direct proportion to determine that 157.5 liters of water are needed for optimal structural strength.
Workforce and staffing planning. Project managers use inverse proportions to calculate completion timelines. If 8 software developers take 15 days to build a feature module (8 × 15 = 120 worker-days), adding 4 developers (total 12) reduces the completion time to 120 / 12 = 10 days.
Physics and gas laws. Physics students use inverse proportions when solving Boyle's Law problems (P₁V₁ = P₂V₂). If a gas occupies 4.0 liters at 1.5 atmospheres of pressure, increasing the pressure to 3.0 atmospheres compresses the volume inversely to 2.0 liters.
Currency and unit conversion. International business analysts convert monetary values using live exchange rates. If $1.00 USD equals 0.92 Euros, converting $1,250 USD to Euros uses direct proportion to yield €1,150.00.
Identify whether your scenario is direct or inverse before choosing a mode. Ask yourself: "If one value increases, does the other value increase too?" If yes, choose Direct Proportion. If increasing one value decreases the other, choose Inverse Proportion.
Keep your units aligned vertically. In direct mode A / B = C / D, place the same type of quantity in numerators A and C (e.g., miles) and the corresponding quantities in denominators B and D (e.g., hours).
For simplifying complex fractional ratios before setting up a proportion, use our Ratio Calculator. For converting final decimal results into fractions or percentages, pair this tool with our Percentage Calculator.
Avoid entering zero into denominator fields. In direct proportion, setting B or D to zero creates a division-by-zero condition, which is mathematically undefined and will trigger a validation alert.
The calculation engine is written in vanilla JavaScript and runs entirely inside your web browser. No data is transmitted to external servers. Cross-multiplication executes in sub-millisecond time with 64-bit floating-point accuracy.
| Feature | This Tool | Hand Cross-Mul | Spreadsheet Solver |
|---|---|---|---|
| Algorithm | Direct & Inverse isolation | Manual multiplication/division | Goal Seek / Solver |
| Direct Proportion | Yes (A/B = C/D) | Yes | Formula setup required |
| Inverse Proportion | Yes (A×B = C×D) | Yes | Formula setup required |
| Execution Time | <1ms | 1-2 minutes | 10-30 seconds |
| Data Privacy | Client-side browser | Local paper | Local file |
| Cost | Free | Free | License required |
A proportion is direct when both variables change in the same direction (e.g., more miles driven means more gas consumed). It is inverse when variables move in opposite directions (e.g., faster speed means less travel time).
Starting with A / B = C / D, multiplying both sides by B × D clears the denominators, leaving A × D = B × C. This identity is the foundation of cross-multiplication.
Yes. The engine correctly handles negative numbers, maintaining algebraic signs through multiplication and division steps according to standard real-number rules.
All calculations use IEEE 754 double-precision floating-point arithmetic, offering up to 15–17 significant digits of mathematical precision before formatting for display.
In inverse mode (A × B = C × D), the product of the first pair remains constant. Leaving one term blank causes the engine to divide the known product by the remaining paired term.
Ratio Calculator — Simplifies multi-part ratios to lowest whole numbers and scales base ratios to target totals. Ideal for recipe expansion and aspect ratio adjustments.
Percentage Calculator — Computes percentage of numbers, percentage change, and relative variance. Useful for analyzing proportion results as percentage growth.
Fraction Calculator — Performs fraction arithmetic and reduces improper fractions to mixed numbers. Useful when proportions involve fractional terms.