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Math & School
Ratio Calculator — Simplify, Solve Proportions & Scale Ratios
Reduce ratios, find the missing term in a proportion, and scale recipes, maps and mixes
A ratio compares two quantities — parts of cement to sand, boys to girls in a class, miles on a map, ingredients in a recipe. It's written A:B and read "A to B." This calculator does the three things people actually need from ratios, all in one place: simplify a ratio to its lowest whole‑number terms, solve a proportion A:B = C:x for a missing value, and scale a ratio up or down by a factor.
Simplifying works exactly like reducing a fraction. To put 18:24 in lowest terms, find the greatest common divisor (GCD) of 18 and 24 and divide both sides by it. Their GCD is 6, so 18:24 = (18÷6):(24÷6) = 3:4. This calculator finds the GCD with the Euclidean algorithm — repeatedly replace the larger number with the remainder of the larger divided by the smaller until one becomes zero — which is exact and instant even for big numbers. It also handles decimals like 1.5:2.25 by first scaling both sides up to whole numbers.
Solving a proportion is the workhorse of everyday math. A proportion says two ratios are equal: A:B = C:x. Cross‑multiply to get A·x = B·C, then x = (B × C) ÷ A. If a map scale is 1:50,000 and a road measures 3 units, then 1:50,000 = 3:x gives x = (50,000 × 3) ÷ 1 = 150,000 real units. The same move scales recipes ("this serves 4, I need 6"), unit prices, mixing concentrations, and similar‑triangle geometry.
Scaling multiplies both terms by the same factor so the ratio stays equivalent. Doubling 2:3 gives 4:6 — a different pair of numbers, but the same relationship (both equal the decimal 0.667). Use it to convert a 1:4 concentrate into a batch of 5:20, or a 16:9 screen ratio into actual pixels.
For every answer the calculator also shows the decimal value (A ÷ B) and the percentage, plus what share of the whole each part represents — so 3:4 tells you A is 75% of B, and that A is 3/7 (≈43%) of the total. Enter your numbers, pick a mode, and you'll see the simplified ratio and the full step‑by‑step working, GCD included.
Calculator
Fill in the fields and click "Calculate" for instant results.
📰 Formula
• Simplify: divide both terms by GCD(A, B), found with the Euclidean algorithm — A:B = (A÷g):(B÷g) • Euclidean GCD: repeatedly set (A, B) ← (B, A mod B) until B = 0; the last A is the GCD • Solve a proportion A:B = C:x: cross‑multiply A·x = B·C, so x = (B × C) ÷ A • Scale a ratio by n: A:B → (A × n):(B × n) — stays equivalent • Decimal value = A ÷ B; Percentage = (A ÷ B) × 100
📰 Formula
• Simplify: divide both terms by GCD(A, B), found with the Euclidean algorithm — A:B = (A÷g):(B÷g) • Euclidean GCD: repeatedly set (A, B) ← (B, A mod B) until B = 0; the last A is the GCD • Solve a proportion A:B = C:x: cross‑multiply A·x = B·C, so x = (B × C) ÷ A • Scale a ratio by n: A:B → (A × n):(B × n) — stays equivalent • Decimal value = A ÷ B; Percentage = (A ÷ B) × 100
🧪 Worked examples
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Example 2
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Example 3
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Example 4
⚠️ Common mistakes
- Dividing only one side by the GCD instead of both terms.
- Cross‑multiplying in the wrong order — x = (B × C) ÷ A, not (A × C) ÷ B.
- Adding to scale a ratio instead of multiplying both terms by the same factor.
- Reading 18:24 as a fraction 18/24 but forgetting it reduces the same way (to 3:4).
- Mixing up the order of the parts so the ratio is reversed (A:B is not the same as B:A).
💡 Tips
- A ratio reduces exactly like a fraction — divide both terms by their greatest common divisor.
- To set up a proportion, keep the same kind of quantity on the same side: parts:whole = parts:whole.
- Scaling never changes the ratio's value — 2:3, 4:6 and 20:30 are all equal to 0.667.
- The decimal value (A ÷ B) is the quickest way to check whether two ratios are equivalent.
- For three‑part ratios like 2:3:5, simplify by the GCD of all three terms at once.
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❓ Frequently asked questions
How do I simplify a ratio?
Find the greatest common divisor (GCD) of both terms and divide each by it. For 18:24 the GCD is 6, so 18:24 simplifies to 3:4. It's the same process as reducing a fraction to lowest terms.
How do I solve a proportion for the missing term?
Write it as A:B = C:x, then cross‑multiply: A × x = B × C, so x = (B × C) ÷ A. For 3:4 = 9:x, x = (4 × 9) ÷ 3 = 12. This is the most common use of ratios in everyday problems.
What is the Euclidean algorithm and why use it?
It's the fastest exact way to find a GCD: repeatedly replace the larger number with the remainder of the larger divided by the smaller until one becomes zero; the last non‑zero value is the GCD. It works instantly even for very large numbers, so this calculator uses it to reduce ratios.
What's the difference between a ratio and a fraction?
A fraction is one number (a part of a whole), while a ratio compares two separate quantities. But they reduce the same way. 3:4 as a fraction can mean 3/4 (A relative to B) or 3/7 (A as a share of the total A+B) — context decides which.
How do I scale a ratio up or down?
Multiply both terms by the same factor. To triple 2:3, compute 2×3 : 3×3 = 6:9. To halve it, multiply by 0.5 to get 1:1.5. The ratio's value never changes — only the size of the numbers does.
How do I turn a ratio into a percentage?
Divide the first term by the second and multiply by 100. For 3:4, that's 3 ÷ 4 × 100 = 75%, meaning A is 75% of B. To find each part's share of the whole, divide by the sum: in 3:4, A is 3/7 ≈ 43% of the total.
Can a ratio have decimals?
Yes. A ratio like 1.5:2.25 is perfectly valid. This calculator scales decimal terms up to whole numbers first, finds the GCD, and returns the simplest whole‑number form — here 1.5:2.25 reduces to 2:3.
How do I use a ratio to scale a recipe?
Set up a proportion. If a recipe's flour‑to‑sugar ratio is 2:1 and you use 3 cups of flour, solve 2:1 = 3:x to find x = (1 × 3) ÷ 2 = 1.5 cups of sugar. Keep the same ingredient on the same side of both ratios.
Does the order of the numbers in a ratio matter?
Yes. A:B is not the same as B:A. A ratio of 3 cats to 4 dogs (3:4) is different from 4 cats to 3 dogs (4:3). Always keep the quantities in the order the problem states them.