Ratio Calculator: Simplify, Solve, and Scale Ratios

A ratio expresses the relative size of two or more quantities. A ratio of 3:2 means that for every 3 units of the first quantity, there are 2 units of the second. Ratios appear in cooking (recipe scaling), mixing (paint, concrete, fuel), finance (debt-to-equity), photography (aspect ratios), and map scales.

Ratios can be written in three ways: with a colon (3:2), as a fraction (3/2), or as a decimal (1.5). All three express the same relationship. The colon notation is most common for comparative ratios; fraction notation is more common in mathematical calculations; decimal notation is useful for quick comparisons.

To simplify a ratio, divide both numbers by their greatest common divisor (GCD). For a ratio of 12:8, the GCD is 4. Dividing both by 4 gives 3:2. The simplified ratio expresses the same relationship with the smallest possible whole numbers.

Finding the GCD: list the factors of each number and identify the largest one they share. For 12 and 8: factors of 12 are 1, 2, 3, 4, 6, 12; factors of 8 are 1, 2, 4, 8. The largest shared factor is 4. Alternatively, use the Euclidean algorithm: divide the larger by the smaller, take the remainder, repeat until the remainder is zero.

A proportion states that two ratios are equal: a/b = c/d. If three values are known, the fourth can be found by cross-multiplication: a × d = b × c, therefore d = (b × c) ÷ a. This is the fundamental operation behind scaling.

Example: a recipe for 4 people uses 300 g of flour. For 7 people, how much flour? Set up the proportion: 300/4 = x/7. Cross-multiply: 4x = 2100. x = 525 g. The same logic applies to any scaling problem where the relationship between two quantities is proportional.

Recipe scaling multiplies all ingredients by the same ratio factor. To convert a recipe from 4 servings to 6, multiply every ingredient by 6/4 = 1.5. Ingredient A at 200 g becomes 300 g; ingredient B at 100 ml becomes 150 ml.

Mixing ratios work similarly. If a concrete mix requires cement, sand, and aggregate in a 1:2:4 ratio by volume, and you need 28 units total: cement = 28 × (1/7) = 4 units, sand = 28 × (2/7) = 8 units, aggregate = 28 × (4/7) = 16 units. The total parts in the ratio (1+2+4=7) serve as the denominator.

Financial ratios measure relationships between balance sheet or income statement values. Debt-to-equity ratio = total debt ÷ total equity. A ratio of 0.5:1 means $0.50 of debt for every $1 of equity. Price-to-earnings (P/E) ratio = share price ÷ earnings per share. These ratios are standardized expressions that allow comparison across companies of different sizes.

Aspect ratios in photography and video are familiar examples: 16:9 means 16 units wide for every 9 units tall. A 1920×1080 screen and a 3840×2160 screen both have a 16:9 aspect ratio — the ratio is the same even though the pixel dimensions differ significantly. Aspect ratio calculations are a direct application of ratio simplification.