Ratio Calculator
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Ratios and Proportions Explained
To simplify a ratio, divide both terms by their greatest common divisor: 12:18 becomes 2:3 because both divide by 6. To find a missing term in A:B = C:D, cross-multiply, so D = B × C ÷ A and 3:4 = 9:12. The calculator above simplifies, solves, scales and compares ratios and shows each step of the working.
A ratio compares two quantities and expresses how much of one thing there is relative to another. Written as A:B or A/B, ratios can be simplified, scaled, and used to solve proportions where one value is unknown.
Ratios are simplified by dividing both terms by their Greatest Common Divisor (GCD). A ratio of 12:18 simplifies to 2:3 because GCD(12,18) = 6. Equivalent ratios, like 2:3, 4:6, 6:9, all express the same relationship at different scales.
Simplify: Divide A and B by GCD(A,B)
Solve A:B = C:? D = B × C / A (cross-multiplication)
Scale: A×k : B×k
Compare: Convert to same denominator or decimal
Solve A:B = C:? D = B × C / A (cross-multiplication)
Scale: A×k : B×k
Compare: Convert to same denominator or decimal
Ratios vs Fractions
A ratio A:B and a fraction A/B look similar but mean different things. A:B compares two separate quantities (3 boys to 4 girls). The fraction of people who are boys is A/(A+B) = 3/7, while A/B = 3/4 is just the ratio written as a division. Both representations are useful: choose based on whether you're comparing parts-to-parts or parts-to-whole.
Cross-Multiplication
To solve A:B = C:D for an unknown: A×D = B×C (cross-multiply). If A=3, B=4, C=9, then D = (B×C)/A = (4×9)/3 = 12. This works because ratios are equivalent fractions: the cross products of equal fractions are always equal.
Real-World Ratio Uses
Cooking (recipe scaling), maps (scale 1:50,000), currency exchange (dollars to euros), mixing (concrete mix 1:2:3), finance (P/E ratio, debt-to-equity), photography (aspect ratio 16:9), and any problem comparing two related quantities.
Part-to-Part vs Part-to-Whole
A:B is part-to-part (boys to girls). A:(A+B) is part-to-whole (boys to total students). If the ratio of boys to girls is 3:4, there are 3 boys out of every 7 students total, a part-to-whole ratio of 3:7. This distinction matters for probability and statistics problems.
Common Ratios Reference Table
Each ratio in lowest terms, the first part as a fraction of the whole, A divided by B, and how a total splits between the two parts.
| Ratio A:B | Simplest form | A as part of whole | A ÷ B | Split A / B |
|---|---|---|---|---|
| 1:1 | 1:1 | 1/2 | 1 | 50% / 50% |
| 1:2 | 1:2 | 1/3 | 0.5 | 33.33% / 66.67% |
| 2:3 | 2:3 | 2/5 | 0.6667 | 40% / 60% |
| 3:4 | 3:4 | 3/7 | 0.75 | 42.86% / 57.14% |
| 3:5 | 3:5 | 3/8 | 0.6 | 37.5% / 62.5% |
| 4:5 | 4:5 | 4/9 | 0.8 | 44.44% / 55.56% |
| 5:8 | 5:8 | 5/13 | 0.625 | 38.46% / 61.54% |
| 3:2 | 3:2 | 3/5 | 1.5 | 60% / 40% |
| 4:3 (screen) | 4:3 | 4/7 | 1.3333 | 57.14% / 42.86% |
| 16:9 (screen) | 16:9 | 16/25 | 1.7778 | 64% / 36% |
| 21:9 (screen) | 7:3 | 7/10 | 2.3333 | 70% / 30% |
Screen sizes such as 21:9 are marketing names; in lowest terms that ratio is 7:3.
Four Worked Examples
Sharing an amount in a ratio
Split $360 in the ratio 2:3:4. Add the parts: 2 + 3 + 4 = 9, so one part is $360 ÷ 9 = $40. The shares are $80, $120 and $160, and they add back to $360.
Scaling a recipe
A pancake recipe uses 3 cups of flour to 2 cups of milk and makes 12 pancakes. For 30 pancakes the scale factor is 30 ÷ 12 = 2.5, so you need 7.5 cups of flour and 5 cups of milk. The ratio stays 3:2.
Reading a map scale
USGS 7.5-minute topographic maps use a scale of 1:24,000. A trail that measures 3.5 inches on the map is 3.5 × 24,000 = 84,000 inches on the ground, which is 7,000 feet or about 1.33 miles.
Simplifying a ratio with decimals
For 1.5:2.5, multiply both terms by 10 to clear the decimals, giving 15:25. The greatest common divisor is 5, so the ratio is 3:5. The calculator does the same scaling for you.
Common Ratio Mistakes
- Mixing part to part with part to whole. If boys to girls is 3:4, boys are 3/7 of the class, not 3/4.
- Swapping the order. 2:3 and 3:2 are different ratios. Keep the terms in the order the question names them.
- Mixing units. Convert first: 2 feet to 8 inches is 24:8, which is 3:1, not 2:8.
- Adding instead of multiplying when scaling. Adding 2 to both terms of 3:5 gives 5:7, which is a different ratio. Multiply or divide both terms by the same number.
- Using zero. A ratio with a 0 term cannot be simplified or scaled in a useful way, and a proportion with a 0 in the divisor has no solution. The calculator asks for positive numbers in these modes.
Ratios and fractions are two views of the same numbers; the fraction calculator handles the fraction side, and the GCF calculator shows the greatest common factor in detail.
Method and sources. Simplifying divides both terms by their greatest common divisor, found with Euclid's algorithm (decimals are first scaled to whole numbers). Missing terms use the cross product rule A × D = B × C. Map scale: USGS standard 1:24,000 scale for 7.5-minute topographic quadrangles. Every figure in the table and examples above was computed with these rules.
Frequently Asked Questions
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.
Add the parts of the ratio, divide the total by that sum to get one part, then multiply. To split 450 in the ratio 4:5, the parts add to 9, one part is 450 ÷ 9 = 50, and the shares are 200 and 250. The same works for three or more parts.
Clear the decimals or fractions first, then divide by the greatest common divisor. For 0.75:1.25, multiply by 100 to get 75:125, then divide by 25 to get 3:5. For 1/2 : 3/4, multiply both by 4 (the common denominator) to get 2:3.
Divide each part by the total of the parts and multiply by 100. For 3:5, the total is 8, so the first part is 3 ÷ 8 = 37.5% and the second is 62.5%. If you want A as a percent of B instead, divide A by B: 3 ÷ 5 = 60%.