How do you simplify a ratio?

Find the Greatest Common Divisor (GCD) of both terms and divide each by it. Example: 12:18. GCD(12,18) = 6. Simplified: 12/6 : 18/6 = 2:3. A ratio is fully simplified when the GCD of all terms equals 1. For 3-part ratios like 6:9:12, find GCD of all three terms (GCD=3) and divide each: 2:3:4.

How do you solve a proportion A:B = C:D?

Use cross-multiplication: A x D = B x C. To find a missing value, rearrange the equation. Example: 2:5 = x:20. Cross-multiply: 2 x 20 = 5 x x. So 40 = 5x, therefore x = 8. The ratio 2:5 equals 8:20.

How do you divide an amount in a given ratio?

Add the ratio parts to get the total parts. Divide the amount by total parts to get the value per part. Multiply each ratio term by value per part. Example: Divide $300 in ratio 2:3. Total parts = 5. Value per part = 300/5 = $60. Shares: 2 x 60 = $120 and 3 x 60 = $180.

What is the difference between a ratio and a proportion?

A ratio compares two quantities (e.g. 3:4 means 3 to every 4). A proportion states that two ratios are equal (e.g. 3:4 = 6:8). A proportion is an equation involving two ratios. To check if two ratios form a proportion, cross-multiply: if A x D = B x C, then A:B = C:D.

What is a 3-part ratio?

A 3-part ratio compares three quantities simultaneously, written as A:B:C. Example: concrete mix of cement:sand:gravel = 1:2:4 means for every 1 part cement, use 2 parts sand and 4 parts gravel. To simplify, find the GCD of all three parts. To divide an amount, add all three parts: total = A+B+C, then distribute proportionally.

How do you convert a ratio to a percentage?

For ratio A:B, Part A = A/(A+B) x 100%, Part B = B/(A+B) x 100%. Example: ratio 3:7. Total = 10. Part A = 3/10 x 100 = 30%. Part B = 7/10 x 100 = 70%. Both percentages always sum to 100%.

How do you scale a ratio?

Multiply all parts of the ratio by the same scale factor. Example: scale 2:3 by a factor of 5 gives 10:15. To scale down, divide by the scale factor. Scaling preserves the proportional relationship between parts. Used in map reading, recipe scaling, and model building.

How do you compare two ratios?

Convert both ratios to decimals (divide first by second) and compare. Or cross-multiply: for A:B vs C:D, compare A x D vs B x C. If A x D > B x C, then A:B > C:D. Example: 3:5 vs 4:7. Cross-multiply: 3x7=21 vs 4x5=20. Since 21>20, ratio 3:5 is larger than 4:7.

What is an equivalent ratio?

Equivalent ratios represent the same proportional relationship. Multiply or divide both parts by the same number. Example: 2:3 is equivalent to 4:6, 6:9, 10:15. All represent the same relationship. Simplifying a ratio finds its smallest equivalent form.

What is a part-to-part vs part-to-whole ratio?

Part-to-part ratios compare one group to another group: boys to girls in a class (15:12). Part-to-whole ratios compare one group to the total: boys to all students (15:27, simplified to 5:9). Both are useful in different contexts. Part-to-whole is essentially a fraction.

How do you use ratios in recipes?

Recipe ratios var you scale up or down while maintaining proportions. Example: a recipe uses flour:sugar = 3:1. If you want to make a version with 9 cups of flour, set up proportion 3:1 = 9:x. Cross-multiply: 3x = 9, so x = 3 cups of sugar. The ratio stays constant regardless of batch size.

What is the golden ratio?

The golden ratio is approximately 1.618:1, often written as phi (phi). Two quantities are in the golden ratio if their sum to the larger part equals the larger to the smaller: (a+b)/a = a/b = 1.618. It appears in art, architecture, and nature, including the Parthenon, nautilus shells, and sunflower seed patterns.

Ratio Calculator — CalculatorCove

📊 Select Ratio Operation

A:B → simplified | supports A:B:C

Part A

First part

Enter a valid value.

Part B

Second part

Enter a valid value.

Part C (optional)

For 3-part ratio A:B:C

A:B = C:D (enter 3 values, find 4th)

A Required.

B Required.

Leave blank to solve

Leave exactly ONE field blank — the calculator will solve for it using cross-multiplication.

Split total amount in ratio A:B (:C optional)

Total Amount

Amount to split (e.g. $1000)

Enter a valid amount.

Part A

e.g. 3 in 3:2

Enter a valid value.

Part B

e.g. 2 in 3:2

Enter a valid value.

Part C (optional 3rd)

For 3-way split A:B:C

Scale A:B by factor n → (A×n):(B×n)

Part A

First ratio part

Enter a valid value.

Part B

Second ratio part

Enter a valid value.

Scale Factor

Multiply or divide by this

Enter a valid scale factor.

Compare A:B vs C:D — which is larger?

Ratio 1 — A Required.

Ratio 1 — B Required.

Ratio 2 — C Required.

Ratio 2 — D Required.

Result

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⚠️ Disclaimer: Results are mathematically exact for integer inputs. For decimal inputs, rounding may occur. Always verify critical financial or professional calculations.

Sources & Methodology

✓GCD computed using Euclidean algorithm (exact integer arithmetic). Proportion solving uses cross-multiplication. Ratio-to-percentage uses exact fractional conversion. All operations verified against Khan Academy and NIST standards.

📘

Khan Academy — Ratios, Rates & Proportions

Khan Academy’s complete curriculum on ratio and proportion covering simplification, equivalent ratios, proportional reasoning, and real-world applications. The global standard for math education used by millions of students.

🏛️

NIST Digital Library of Mathematical Functions

U.S. government authoritative reference for mathematical definitions including the Euclidean algorithm for GCD computation, which underpins all ratio simplification in this calculator.

Formulas used:
Simplify: GCD = Euclidean algorithm; simplified = A/GCD : B/GCD Solve: Cross-multiply A×D = B×C; solve for unknown Divide: Share = Total × (Part / Sum of Parts) Scale: Scaled A = A×n, Scaled B = B×n Compare: Decimal A:B = A/B, Decimal C:D = C/D, compare values

Last reviewed: April 2026

Complete Guide to Ratios & Proportions — Every Calculation Explained

A ratio compares two or more quantities by showing their relative sizes. Written as A:B, A to B, or the fraction A/B — all three notations mean the same thing. Ratios are one of the most practical mathematical tools in daily life: adjusting recipes, mixing paint or concrete, reading maps, interpreting financial reports, scaling blueprints, calculating odds, and splitting money fairly all rely on ratio arithmetic.

Worked Example: Dividing Prize Money

Three people split a $1,500 prize in ratio 2:3:5.
Step 1: Total parts = 2 + 3 + 5 = 10
Step 2: Value per part = $1,500 ÷ 10 = $150
Person A: 2 × $150 = $300
Person B: 3 × $150 = $450
Person C: 5 × $150 = $750
Check: $300 + $450 + $750 = $1,500 ✓

How to Simplify a Ratio

To simplify a ratio, divide all parts by their GCD (Greatest Common Divisor). The GCD is the largest number that divides all parts evenly. Example: simplify 24:36. GCD(24,36) = 12. Simplified: 24/12 : 36/12 = 2:3. A ratio is fully simplified when the GCD of all terms equals 1. For a 3-part ratio like 12:18:24, GCD(12,18,24) = 6. Simplified: 2:3:4. Simplified ratios are easier to work with and more meaningful for comparison.

How to Solve Proportions (Finding Missing Values)

A proportion states two ratios are equal: A:B = C:D. To find a missing value, use cross-multiplication: A × D = B × C. Rearrange for the unknown. Example: 3:4 = x:20. Cross-multiply: 3 × 20 = 4 × x. So 60 = 4x, giving x = 15. This is the proportion 3:4 = 15:20. Real-world applications include scaling recipes, reading maps, converting currencies, and calculating shadow lengths in geometry.

How to Divide an Amount by a Ratio

Dividing an amount in a given ratio is the most practical ratio calculation. Used for splitting profits, inheritances, bills, and resources. Formula: Share = Total × (Part / Sum of All Parts). Example: divide $800 in ratio 3:5. Total parts = 8. Share A = $800 × 3/8 = $300. Share B = $800 × 5/8 = $500. For a 3-part ratio, the same logic applies with 3 values. The shares always sum to the original total — use this as your verification check.

💡 Gap Competitors Miss — 3-Part Ratio Division: Most online ratio calculators only handle 2-part ratios. Our calculator supports 3-part ratio splitting (A:B:C), which is essential for business partnerships, inheritance disputes, concrete mixing (cement:sand:gravel), and recipe scaling with three ingredients. Example: concrete mix 1:2:4 using 35 kg total. Cement = 35×1/7 = 5 kg. Sand = 35×2/7 = 10 kg. Gravel = 35×4/7 = 20 kg.

How to Scale a Ratio

To scale a ratio up or down, multiply or divide all parts by the same scale factor. The ratio stays equivalent as long as the factor is applied uniformly. Example: scale 2:3 by factor 4 gives 8:12 (equivalent ratio). Scale 15:10 down by factor 5 gives 3:2. Used in: map reading (1:50,000 scale), model building (scale factor), recipe scaling (double or halve), and engineering drawings.

Ratio to Percentage, Decimal, and Fraction Conversion

Ratio	As Fraction	As Decimal	Part A %	Part B %
1:1	1/2 and 1/2	1.000	50%	50%
1:3	1/4 and 3/4	0.333	25%	75%
2:3	2/5 and 3/5	0.667	40%	60%
3:7	3/10 and 7/10	0.429	30%	70%
1:4	1/5 and 4/5	0.250	20%	80%

Part-to-Part vs Part-to-Whole Ratios

A part-to-part ratio compares one group to another group directly. Example: a class has 15 boys and 12 girls. The boy-to-girl ratio is 15:12 = 5:4. A part-to-whole ratio compares one group to the total. The boy-to-total ratio is 15:27 = 5:9. Both are valid ways to express the same data. Part-to-whole ratios are essentially fractions and can directly represent probability, concentration, or market share.

Real-World Ratio Applications

Cooking & baking: Recipe scaling while maintaining flavor ratios (flour:sugar:butter)
Construction: Concrete mix ratios (cement:sand:aggregate = 1:2:4), mortar (1:3), plaster
Finance: Debt-to-income ratio, price-to-earnings (P/E) ratio, current ratio for liquidity
Maps & scales: Map scale 1:50,000 means 1 cm on map = 500 m in reality
Photography & screens: Aspect ratios (16:9 for HD video, 4:3 for old TV, 1:1 for Instagram)
Chemistry: Solution concentrations, mixing ratios for paint, fertilizer, and cleaning products
Splitting money: Business profit sharing, splitting bills, inheritance division
Sports: Win-loss ratios, odds in betting (5:1 means 5 losses expected per 1 win)

The Golden Ratio — Nature’s Most Famous Ratio

The golden ratio (approximately 1.618:1, denoted by φ phi) is a special proportion where the sum of two quantities to the larger quantity equals the larger to the smaller: (a+b)/a = a/b = 1.618. It appears in the Parthenon’s dimensions, nautilus shell spirals, sunflower seed patterns, and the human face proportions. In design, the “golden rectangle” (sides in ratio 1.618:1) is considered the most aesthetically pleasing shape. The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13…) approaches the golden ratio as consecutive terms are divided.

Common Ratio Mistakes to Avoid

Confusing ratio with fraction: 3:5 as a fraction is 3/8 (part of whole), not 3/5 (ratio value)
Not simplifying first: Always simplify before solving proportions to work with smaller numbers
Wrong order: In ratio A:B, A always refers to the first quantity mentioned. Order matters.
Adding instead of proportioning: In a 2:3 split, the total is 5 parts, not 6
Percentage confusion: Ratio 3:7 means Part A is 30%, not 3%, of the total

Frequently Asked Questions

Find the GCD of all parts and divide each by it. Example: 24:36. GCD = 12. Simplified: 2:3. For 3-part ratio 6:9:12, GCD = 3. Simplified: 2:3:4. A ratio is fully simplified when GCD = 1.

Use cross-multiplication: A×D = B×C. Enter 3 known values, leave the unknown blank. Example: 2:5 = x:20. Cross-multiply: 2×20 = 5×x, so x = 40/5 = 8. The missing value is 8.

Add ratio parts to get total. Divide amount by total to get value per part. Multiply each part by value-per-part. Example: $300 in ratio 2:3. Total parts = 5. Per part = $60. Shares: $120 and $180.

A 3-part ratio compares three quantities: A:B:C. Example: concrete mix 1:2:4 (cement:sand:gravel). To divide $700 in ratio 1:2:4: total = 7 parts. Shares: $100, $200, $400. Most calculators only handle 2-part — ours supports 3-part.

A ratio compares two values (3:4). A proportion states two ratios are equal (3:4 = 6:8). A proportion is an equation; a ratio is just a comparison. To check if two ratios form a proportion, cross-multiply: if A×D = B×C, they are proportional.

For ratio A:B: % of A = A/(A+B)×100, % of B = B/(A+B)×100. Example: ratio 3:7. A% = 3/10×100 = 30%. B% = 7/10×100 = 70%. Both always sum to 100%.

Convert both to decimals (divide first by second) or cross-multiply. Example: 3:5 vs 4:7. Cross-multiply: 3×7=21 vs 4×5=20. Since 21>20, ratio 3:5 is larger than 4:7.

Multiply all parts by the scale factor. Scale 2:3 by 4 gives 8:12. Divide to scale down. Scale factor preserves the proportional relationship. Used for recipe scaling, map reading, and engineering drawings.

Ratios that represent the same relationship: 2:3, 4:6, 10:15 are all equivalent. Multiply or divide all parts by the same number. Simplifying finds the smallest equivalent form.

Set up a proportion with the original recipe and desired batch size. Recipe uses flour:sugar = 3:1 (3 cups flour, 1 cup sugar). For 9 cups flour: 3:1 = 9:x. Cross-multiply: x = 3 cups sugar. Ratios let you scale any recipe up or down precisely.

For every 16 units of width, the height is 9 units. This is the standard aspect ratio for HD TVs, monitors, and YouTube videos. A 1920px wide screen at 16:9 has height = 1920 × 9/16 = 1080px. This is why 1920×1080 is called Full HD or 1080p.

Approximately 1.618:1 (phi, φ). The ratio of two lengths where the longer to the shorter equals the sum to the longer: (a+b)/a = a/b = 1.618. Appears in the Parthenon, nautilus shells, sunflower patterns, and Fibonacci sequence. Considered aesthetically perfect in art and architecture.

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