Solve for any unknown variable in direct and inverse proportions. Features cross-multiplication proofs, scale factors, and step-by-step algebraic breakdowns.
A proportion is a mathematical statement affirming that two ratios are strictly equal. Whether calculating recipe adjustments, currency exchange, map scaling, or gear ratios, follow this standard procedure:
Choose Direct Proportion (A/B = C/D) or Inverse Proportion (A×B = C×D).
Fill in the three known values and leave the unknown variable field blank.
Multiply diagonally across the equal sign to set products equal: A × D = B × C.
Divide by the remaining term to find the exact simplified value of the unknown.
Two variables are directly proportional when their ratio remains constant. If one increases, the other increases proportionally:
Example: 2 gallons cost $8.00; 5 gallons cost $20.00.
Two variables are inversely proportional when their product remains constant. If one increases, the other decreases proportionally:
Example: 4 workers finish in 6 hours; 8 workers finish in 3 hours.
A recipe calls for 2 cups of sugar to serve 6 people. How much sugar is required to serve 15 people?
On a road map, 1 inch represents 25 miles. What is the real-world distance for a measurement of 4.5 inches?
Solves for unknown variable x whether it appears as the numerator or denominator on either side of the equation (A/B = C/D) without manual algebraic isolation.
Handles both direct scaling (e.g. currency conversion, recipe portions) and inverse relationship scenarios (e.g. speed vs travel time, number of workers vs project duration).
Clearly displays intermediate products (A·D = B·C) and subsequent single-step division, reinforcing fundamental pre-algebra principles.
Calculates the multiplier ratio between terms and computes individual unit rates, verifying proportional consistency.
Switch between Direct Proportion (A/B = C/D) and Inverse Proportion (A·B = C·D) with instant recalculation.
Verifies that the product of the means equals the product of the extremes with exact arithmetic matching.
Provides exact simplified fractions, mixed numbers, and 4-decimal values alongside scale factors.
| Modality | Mathematical Equation | Cross-Rule | Behavioral Trend |
|---|---|---|---|
| Direct Proportion | A / B = C / D | A × D = B × C | Both variables increase or decrease together |
| Inverse Proportion | A × B = C × D | D = (A × B) / C | One variable increases as the other decreases |
Solve proportions, simplify ratios, and scale part-to-whole totals.
All-in-one arithmetic operations for fractions and mixed numbers.
Calculate cost per unit and compare grocery product values.
Compare fractions using butterfly cross-multiplication.
Cross-cancellation, mixed number products, and step proofs.
Reduce fractions to simplest form using Euclidean GCD.
