Simplify ratios, solve proportions, find percentages, scale values, and compare ratios, all with full step-by-step solutions.
A ratio expresses the relationship between two quantities and is one of the most versatile tools in applied math. In cooking, a vinaigrette is typically 3 parts oil to 1 part acid (3:1). Concrete mixes are specified as cement:sand:gravel ratios. Understanding ratios lets you scale any formula reliably without recalculating every ingredient. In finance, ratios are the primary language of business analysis. The P/E ratio tells you how much investors pay per dollar of earnings. The debt-to-income ratio tells lenders whether you can afford a loan. The S&P 500's average P/E ratio over the long term is approximately 15-16x, values above 25x historically signal overvaluation. In construction, standard concrete for footings is typically a 1:2:3 ratio (1 part cement, 2 parts sand, 3 parts aggregate). Getting this wrong affects structural strength, concrete that's too watery is significantly weaker than properly mixed concrete.
A ratio is a comparison of two or more quantities that expresses how much of one thing there is relative to another. Written as a : b (spoken "a to b"), ratios appear in virtually every field: cooking uses ingredient ratios, finance uses debt-to-equity ratios, engineering uses gear ratios, chemistry uses mixture ratios, and art uses aspect ratios. Understanding how to work with ratios (simplify them, solve for unknowns, scale them up or down, and compare them) is a foundational quantitative skill that pays dividends far beyond mathematics class.
Ratios differ from fractions in a subtle but important way. The fraction 2/5 says "2 out of 5 total parts." The ratio 2:3 says "2 parts of A for every 3 parts of B", which represents 5 total parts but focuses on the relationship between the parts rather than one part's share of the whole. A ratio can also extend to three or more terms (2:3:5) describing three quantities simultaneously, something fractions cannot naturally express.
Simplifying a ratio means reducing it to its lowest terms so both parts share no common factor greater than 1. The algorithm finds the Greatest Common Divisor (GCD) using the Euclidean method, then divides both parts by it.
GCD(24, 36) = 12 | 24 ÷ 12 = 2 | 36 ÷ 12 = 3 | Simplified: 2 : 3
This calculator also handles decimal inputs. For 1.5 : 2.25, it multiplies both by 100 (converting to 150 : 225), finds GCD(150, 225) = 75, and returns 2 : 3. This makes it useful for real-world measurements that don't naturally come as whole numbers.
A proportion states two ratios are equal: a : b = c : d. Given any three values, the fourth can be found using the means-extremes property, more commonly known as cross-multiplication.
Rearranging for each possible unknown: if c is unknown, c = (a × d) / b. This is the fundamental tool for solving word problems like "if 3 workers complete a job in 5 days, how many days do 7 workers take?" or scaling ingredient quantities.
Cross multiply: 5 × 24 = 8 × x ⇒ 120 = 8x ⇒ x = 15. Check: 5/8 = 15/24 = 0.625 ✓
This mode converts a part-and-whole pair into the three most useful representations: simplified ratio, fraction, and percentage. It answers questions like "3 students failed out of 8 total, what fraction is that, and what percentage?"
Fraction: 3/8 | Decimal: 0.375 | Percentage: 37.5% | Ratio: 3 : 8 (already simplified)
Scaling multiplies both parts of a ratio by the same factor, producing an equivalent ratio at a different magnitude. A scale factor greater than 1 enlarges; less than 1 (a decimal) reduces. The proportional relationship is preserved regardless of scale, 2 : 3 scaled by 5 becomes 10 : 15, which simplifies back to 2 : 3.
Factor = 3 | New ratio = (2×3) : (5×3) = 6 : 15 | Simplifies to 2 : 5 (same proportions, tripled amounts)
Comparing two ratios determines which is larger, or whether they're equal. This calculator uses two parallel methods for maximum clarity. The cross-multiplication method avoids floating-point imprecision for exact integer ratios, while the decimal conversion method gives an intuitive sense of magnitude difference.
Cross products: 5×4 = 20 vs 3×8 = 24 | 20 < 24 ⇒ 5:8 < 3:4 | Decimals: 0.625 < 0.75 ✓
| Context | Example Ratio | What It Means |
|---|---|---|
| Cooking | 1 : 2 (rice : water) | 1 cup rice needs 2 cups water |
| Map Scale | 1 : 25,000 | 1 cm on map = 250 m in reality |
| Finance | 3 : 1 (debt : equity) | $3 owed for every $1 owned |
| Aspect Ratio | 16 : 9 | Screen width is 16/9 ≈ 1.78× its height |
| Gear Ratio | 2 : 5 | Input turns twice for every 5 output turns |
| Bleach dilution | 1 : 10 | 1 part bleach per 10 parts water (≈ 9%) |
| Scale model | 1 : 48 | Model is 1/48th of real object size |
| Probability odds | 7 : 3 | 7 chances of success, 3 of failure (70%) |
| Mortar mix | 1 : 3 (cement : sand) | 1 part cement to every 3 parts sand |
| Photography | 3 : 2 | Classic 35mm frame, 6×4 inch print |
The Euclidean algorithm is one of the oldest and most elegant algorithms in mathematics, attributed to Euclid around 300 BCE. It finds the greatest common divisor of two integers efficiently by repeatedly applying the division remainder until a remainder of zero is reached. The last non-zero remainder is the GCD.
So 48 : 18 simplifies to (48÷6) : (18÷6) = 8 : 3. The algorithm runs in O(log min(a,b)) time, making it extremely fast even for very large numbers. For decimal inputs, the calculator first multiplies both values by a power of 10 sufficient to make them integers, applies integer GCD, then divides back down, handling cases like 0.4 : 0.6 → 2 : 3 correctly.
These three representations are closely related but serve different communicative purposes. A ratio (2 : 3) emphasizes the relationship between two distinct quantities, it's the natural language for mixing, comparing, and scaling. A fraction (2/5) places one quantity in the context of a total and is ideal for part-of-whole thinking: two slices out of five. A percentage (40%) is a fraction scaled to base-100, universally intuitive for comparisons and communication.
Converting between them is straightforward: ratio a : b → fraction a/(a+b) for part-to-total, or a/b for part-to-part as a fraction. Multiply by 100 for percentage. The Part-to-Whole mode performs these conversions automatically, and the Simplify mode expresses the a/b form's decimal equivalent alongside the simplified ratio.
Essential for ratio-based cooking, baking, and material mixing.
View on Amazon →A classic book teaching you to cook by ratio rather than recipe.
View on Amazon →Covers all fractional sizes for accurate ratio measurements in cooking and mixing.
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