Proportion Calculator
How to Solve Proportions Using Cross Multiplication
A proportion is a mathematical statement establishing that two ratios or fractions are completely equivalent: A / B = C / D. Proportions are widely utilized in map scaling, baking recipe adjustments, model car scale modeling, medicine dosage scaling, and unit conversions.
The fundamental law of proportions states that the product of the means equals the product of the extremes: A × D = B × C (known colloquially as cross-multiplication).
To solve for any single unknown variable among the four terms, multiply the two known numbers that lie diagonally opposite each other, and divide that product by the third remaining known number.
For example: If solving for D, D = (B × C) / A. If solving for A, A = (B × C) / D. If solving for B, B = (A × D) / C. If solving for C, C = (A × D) / B.
Important boundary condition: The divisor term cannot be zero. If cross-multiplication requires dividing by zero, the equation is mathematically undefined.
Worked Recipe Scaling Example
Worked Example: Baking Recipe Scaling
A bakery recipe for 4 servings calls for 6 tablespoons of butter. You are catering a dinner party for 14 guests. How many tablespoons of butter are required?
- Set up the proportion matching servings to butter: Servings1 / Butter1 = Servings2 / Butter2.
- Substitute known quantities: 4 / 6 = 14 / X (where X is required butter).
- Apply cross-multiplication: 4 × X = 6 × 14.
- Multiply the known diagonal: 6 × 14 = 84.
- Divide by the remaining term: X = 84 ÷ 4 = 21 tablespoons.
Scaling the recipe from 4 to 14 guests requires exactly 21 tablespoons of butter.
Cross-Multiplication Formulas by Target Variable
Reference the algebraic solutions below to isolate and solve for any unknown position in the proportion.
| Unknown Variable | Proportion Form | Cross-Product Step | Isolated Solution Formula | Division Restriction |
|---|---|---|---|---|
| D (Bottom Right) | A / B = C / D | A × D = B × C | D = (B × C) / A | A ≠ 0 |
| C (Top Right) | A / B = C / D | B × C = A × D | C = (A × D) / B | B ≠ 0 |
| B (Bottom Left) | A / B = C / D | A × D = B × C | B = (A × D) / C | C ≠ 0 |
| A (Top Left) | A / B = C / D | A × D = B × C | A = (B × C) / D | D ≠ 0 |
Common Proportion Pitfalls
- Cross-multiplying when fractions are added or subtracted: Cross-multiplication applies ONLY across an equal sign (A/B = C/D), NEVER across addition or subtraction (A/B + C/D).
- Mixing up the units: If placing servings on top on the left side, servings must also be on top on the right side. Inverting units yields incorrect answers.
- Dividing by zero: If the diagonally isolated coefficient equals zero, solving is impossible.
- Rounding intermediate cross-products: Always maintain exact numbers until the final division step.
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Frequently Asked Questions About Proportions
What is cross multiplication?
Cross multiplication is an algebraic technique used to solve proportions: multiplying the top of each fraction by the bottom of the other across the equal sign (A × D = B × C).
How do you solve a:b = c:x?
Multiply b by c, and divide by a: x = (b × c) / a. For example, in 2:5 = 6:x, x = (5 × 6) / 2 = 15.
What is a proportion in real life?
Proportions are used whenever two quantities scale at the same rate, such as converting currency exchange rates, calculating vehicular fuel mileage, or enlarging photographs.
Can proportion numbers be negative or decimals?
Yes. Our calculator handles positive numbers, negative numbers, and high-precision decimal values.
How do you check if two ratios form a true proportion?
Cross-multiply: if A × D equals B × C, the ratios form a true proportion. For example, 3/4 = 9/12 is true because 3 × 12 = 36 and 4 × 9 = 36.
What is the difference between direct and inverse proportion?
In direct proportion, as one variable increases, the other increases proportionally (A/B = constant). In inverse proportion, as one variable increases, the other decreases (A × B = constant).