Ratio Calculator

To simplify a ratio, find the Greatest Common Divisor (GCD) of both numbers and divide each by it. GCD of 12 and 18 is 6, so 12:18 = 2:3. Steps: (1) Find all common factors of A and B; (2) Divide both by the largest one. For large numbers, use the Euclidean algorithm: GCD(48,36) = GCD(36,12) = GCD(12,0) = 12, so 48:36 = 4:3. A ratio is fully simplified when A and B have no common factors other than 1.

Use cross-multiplication: if A:B = C:D, then A×D = B×C. To find D: D = (B×C)/A. Example: 3:4 = 9:D → D = (4×9)/3 = 12. So 3:4 = 9:12. To find C: C = (A×D)/B. Example: 3:4 = C:12 → C = (3×12)/4 = 9. This method works because equivalent ratios are equivalent fractions — 3/4 = 9/12 — and the cross products of equal fractions are always equal.

Equivalent ratios express the same relationship between quantities at different scales. 1:2, 2:4, 3:6, 4:8, and 10:20 are all equivalent — they all simplify to 1:2. You create equivalent ratios by multiplying or dividing both terms by the same non-zero number. Equivalent ratios are like equivalent fractions: 1/2, 2/4, and 3/6 are all the same value. When comparing ratios, convert them to their simplified form to check equivalence.

To scale a ratio by factor k: multiply both terms by k. Ratio 3:5 scaled by 4 = 12:20. To scale down, divide by k. A recipe calling for flour:sugar in ratio 3:2 — to make 5x the recipe, scale by 5: 15:10 (or simplify to 3:2). To scale to a specific total: if you want A+B = 100 and the simplified ratio is 3:7, then A = 30 and B = 70. Scaling preserves the relationship between the quantities while changing their absolute values.

Convert both ratios to decimals (divide A by B) and compare. 3:4 = 0.75; 5:7 ≈ 0.714. Since 0.75 > 0.714, the ratio 3:4 is larger. Alternatively, find a common second term: 3:4 = 21:28 and 5:7 = 20:28. Since 21 > 20, 3:4 > 5:7. Or cross-multiply: 3:4 vs 5:7 → compare 3×7=21 vs 4×5=20 → 21 > 20 → 3:4 is larger. All three methods give the same result.

A ratio compares two quantities of the same kind (boys to girls, apples to oranges). A rate compares quantities of different kinds, typically involving time or another reference unit: 60 miles per hour (miles per hour), $12 per kilogram (price per weight), 2000 calories per day. A unit rate has a denominator of 1: "per hour," "per person," "per dollar." All rates are ratios, but not all ratios are rates.

Recipes use ratios to maintain flavor and texture regardless of batch size. A basic bread ratio is flour:water ≈ 5:3 by weight. To double the recipe, multiply both by 2. To make half, multiply by 0.5. The ratio stays constant. Baking ratios (by weight) for common items: pound cake 1:1:1:1 (flour:butter:sugar:eggs); pancakes roughly 2:2:1 (flour:liquid:egg). Understanding ratios lets you scale any recipe precisely without losing the balance of ingredients.

Map scales are ratios between distance on the map and actual distance on the ground. A scale of 1:50,000 means 1 cm on the map = 50,000 cm (500 m) in reality. A scale of 1:100,000 means 1 cm = 1 km. Large-scale maps (1:10,000) show more detail over a smaller area. Small-scale maps (1:1,000,000) show less detail over a larger area. To find actual distance: multiply map distance × scale denominator. A 4 cm measurement on a 1:25,000 map = 4 × 25,000 = 100,000 cm = 1 km.

The golden ratio (φ ≈ 1.618) is a special ratio where A:B = (A+B):A. It appears throughout nature (spiral shells, flower petals), art (Parthenon proportions), and design. The ratio of consecutive Fibonacci numbers approaches φ: 1:1, 1:2, 2:3, 3:5, 5:8, 8:13 → 8/13 ≈ 0.615, 13/8 ≈ 1.625. A golden rectangle has sides in ratio 1:1.618. Many designers use golden ratio proportions for aesthetically pleasing layouts, though its widespread presence in art is partly overstated by popular accounts.

Financial ratios compare two related financial figures to assess a company's performance and health. Common examples: P/E ratio (price-to-earnings) — stock price ÷ earnings per share; a P/E of 20 means you pay $20 for each $1 of earnings. Debt-to-equity ratio — total debt ÷ shareholders' equity; a ratio of 2:1 means $2 in debt for every $1 of equity. Current ratio — current assets ÷ current liabilities; above 1.5 generally indicates good short-term liquidity. Ratios allow comparison across companies of different sizes.