Ratio Simplifier
Simplify any ratio a:b to its lowest terms using the GCD.
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How to use this tool
- Enter a and b in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your simplified ratio and the full breakdown beneath it.
Reduce a ratio a:b to its simplest form by dividing both parts by their GCD.
Formula
Simplified ratio = (a ÷ GCD(a, b)) : (b ÷ GCD(a, b))
GCD is found with the Euclidean algorithm: repeatedly replace (a, b) with (b, a mod b) until b = 0; the remaining a is the GCD.
How it works
This calculator reduces a ratio a:b to its lowest terms by dividing both values by their Greatest Common Divisor (GCD). The GCD is computed with the Euclidean algorithm, which is efficient and exact for integers. Both inputs are converted to their absolute values before processing, so the tool handles negative numbers gracefully.
Worked example
- a = 12, b = 8.
- Euclidean algorithm: GCD(12, 8) → GCD(8, 4) → GCD(4, 0) = 4.
- Divide both sides by GCD: 12 ÷ 4 = 3, 8 ÷ 4 = 2.
- Simplified ratio: 3:2.
12:8 simplifies to 3:2 (GCD = 4).
Common mistakes to avoid
- Dividing only one term by the GCD instead of both, producing an unbalanced ratio.
- Entering decimal values instead of integers; the GCD-based algorithm requires whole numbers, so 1.5:2 should be entered as 3:4.
- Confusing ratio simplification with fraction reduction; 6:9 simplifies to 2:3, but a ratio is not a single number.
Key terms
- Ratio
- A comparison of two quantities expressed as a:b, showing how many times one value contains the other.
- Greatest Common Divisor (GCD)
- The largest positive integer that divides both a and b without a remainder.
- Lowest terms
- A ratio where the two numbers share no common factor other than 1; it cannot be simplified further.
- Euclidean algorithm
- An efficient method for finding the GCD by repeatedly taking the remainder of successive divisions until a remainder of 0 is reached.
Frequently asked questions
- How do you simplify a ratio?
- Find the GCD of a and b, then divide both by it. For example, 12:8 → GCD=4 → 3:2.