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Proportion (A:B = C:D)

A proportion states that two ratios are equal: a/b = c/d. Given three values, solve for the fourth using cross-multiplication: a×d = b×c. Used for recipe scaling, map distances, similar triangles, currency conversion, and dosage.

Concept Fundamentals
a×d = b×c
a:b = c:d
12
2:3 = 8:?
3 in = 150000
1:50000 map
12.5
5:100 = ?:250

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Recipe: 2 cups flour for 3 servings → 8 cups for 12 servings (2:3 = 8:12). Map: 1 inch = 50,000 inches → 3 inches = 150,000 inches (real distance). Dosage: 5mg per 100mg weight → 12.5mg for 250mg (5:100 = 12.5:250).

Key quantities
a×d = b×c
a:b = c:d
Key relation
12
2:3 = 8:?
Key relation
3 in = 150000
1:50000 map
Key relation
12.5
5:100 = ?:250
Key relation

Ready to run the numbers?

Why: Recipes scale by proportion. Maps use scale ratios. Similar triangles have proportional sides. Currency and dosage use proportion. Whenever two ratios are equal, proportion is the tool.

How: Set up a/b = c/d. Cross-multiply: a×d = b×c. Solve for the unknown. If solving for d: d = b×c/a. Keep units consistent: if a is in cups, c must be in the same unit.

Recipe: 2 cups flour for 3 servings → 8 cups for 12 servings (2:3 = 8:12).Map: 1 inch = 50,000 inches → 3 inches = 150,000 inches (real distance).
Sources:ProportionCross-multiplication

Run the calculator when you are ready.

Solving ProportionsCross-multiplication: product of means equals product of extremes.

Real-World Examples — Click to Load

Which Value to Solve For?

Enter Your Proportion: A : B = C : D

proportion_solver
SOLVED
$ solve --proportion "2:6 = 5:15" --for=D
D =
15.0000
Scale Factor
x2.5000
Cross Product
30.00
Verified
VALID
Full Proportion
2.0000 : 6.0000 = 5.0000 : 15.0000
Share:

Value Comparison (A, B, C, D)

Ratio Equality (A/B vs C/D)

Step-by-Step Breakdown

SETUP
Proportion
A : B = C : D
Property
A × D = B × C
ext{Cross} ext{multiplication}
SOLVE
Isolate D
D = (B × C) / A
Substitute
D = (6 × 5) / 2
= 30.00 / 2
ANSWER
RESULT: D
15.0000
VERIFY
Verification
2.00 : 6.00 = 5.00 : 15.00
Cross products
30.00 = 30.00
A×D = B×C
Scale factor
×2.5000
C/A ext{or} D/B

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

In a:b = c:d, a and d are extremes, b and c are means. Product of extremes = product of means.

— Proportion property

↔️

Scale factor: if a:b = c:d, then c/a = d/b = scale factor (when a,c are same quantity).

— Scale factor

Key Takeaways

  • • A proportion states that two ratios are equal: A:B = C:D
  • • The fundamental property: A x D = B x C (cross products are equal)
  • • Proportions are used in scaling recipes, reading maps, similar triangles, currency exchange, and dosage calculations
  • • The scale factor k = C/A = D/B tells you how much bigger or smaller the second pair is
  • • Direct proportion means both increase together; inverse proportion means one increases as the other decreases

Did You Know?

🏛️The ancient Greeks used proportions to define the "Golden Ratio" (1:1.618...), found in architecture, art, and natureSource: Mathematics History
🗺️Map scales are proportions — a 1:50,000 scale means 1 cm on the map equals 50,000 cm (500 m) in real lifeSource: Cartography
🎨The Rule of Thirds in photography divides the frame in a 1:2 proportion, creating more visually appealing compositionsSource: Visual Arts
💊Pediatric drug dosing uses body-weight proportions — a child weighing half as much receives half the adult doseSource: Pharmacology
🏗️Architects use proportional scale models (typically 1:100 or 1:200) to visualize buildings before constructionSource: Architecture
🧬DNA base pairs follow Chargaff's rule — the proportion of A to T and C to G is always 1:1Source: Molecular Biology

How Proportions Work

A proportion is an equation stating that two ratios are equal. If A/B = C/D, then the cross products A*D and B*C are equal. This property — called cross multiplication — is the key to solving for any unknown.

Cross Multiplication Method

Given A:B = C:D, multiply "across": A x D = B x C. Then isolate the unknown by dividing both sides. For example, to find D: D = (B x C) / A.

Types of Proportions

Direct proportion: y = kx — as one increases, so does the other (e.g. more hours worked = more pay).

Inverse proportion: y = k/x — as one increases, the other decreases (e.g. more workers = less time to finish).

Continued proportion: A/B = B/C — B is the geometric mean of A and C.

Expert Tips

Simplify First

Reduce ratios to lowest terms before solving. 6:12 = 1:2, making calculations much easier and reducing errors.

Always Verify

After solving, check that cross products are equal (A*D = B*C). This catches arithmetic mistakes immediately.

Watch Your Units

Ensure corresponding terms have the same units. Miles with miles, dollars with dollars — mismatched units give wrong answers.

Direct vs Inverse

If doubling one value should double the other, it's direct proportion. If doubling one should halve the other, flip one ratio before solving.

Solving Methods Comparison

MethodBest ForSpeedAccuracy
Cross MultiplicationAny proportionFastExact
Unit RateRate problemsMediumExact
Scale FactorScaling problemsFastExact
Equivalent FractionsSimple ratiosFastestExact

Frequently Asked Questions

What is a proportion?

A proportion is a mathematical statement that two ratios are equal. For example, 2:3 = 4:6. It means the relationship between the first pair of numbers is identical to the relationship between the second pair.

How do I solve a proportion with a missing value?

Use cross multiplication: multiply the "diagonals" (A×D and B×C), set them equal, then solve for the unknown. For A:B = C:?, multiply A×? = B×C, so ? = (B×C)/A.

What is the difference between a ratio and a proportion?

A ratio compares two quantities (e.g., 3:4). A proportion is an equation stating that two ratios are equal (e.g., 3:4 = 6:8). A ratio is one comparison; a proportion is the equality of two comparisons.

Can proportions have negative numbers?

Yes, proportions can contain negative numbers as long as the denominators are non-zero. However, most real-world proportion problems involve positive values.

What is the scale factor in a proportion?

The scale factor is the constant multiplier that converts one ratio to another. In 2:5 = 6:15, the scale factor is 3 because you multiply both 2 and 5 by 3 to get 6 and 15.

How are proportions used in real life?

Proportions are used everywhere: scaling recipes (double the ingredients for double the servings), map reading (1 cm = 10 km), medicine dosing (mg per kg of body weight), currency conversion, construction blueprints, and many more.

Quick Reference

A×D
= B×C (Cross Products)
k
Scale Factor = C/A
1:k
Direct Proportion
k:1
Inverse Proportion

Note: This calculator solves direct proportions. For inverse proportions, flip one ratio before entering values. Results are rounded to the selected decimal places. Always verify in context.

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