Ratio Calculator
Simplify a ratio with two or more terms to its lowest whole-number form by dividing every term by the greatest common factor they share.
Calculator
- Two or more terms
- GCF reduction
- Decimal ratio for two terms
- Coprime proof
Result & calculation receipt
- Method
- Divide every term by the greatest common factor of all terms
- Formula
a:b:… = (a ÷ g):(b ÷ g):…, g = gcd(a, b, …)- Substitution
- 16:9 = 16:9
- Steps
- gcd(16, 9) = 1
- Divide each term by 1: 16:9
- Verification
- gcd(16, 9) = 1 = 1, so the ratio is in simplest form.
- Exact result
- 16:9
- Decimal result
- 1.7777777778
- Engine
- math-solvers v1.4.0
How this calculator works
The calculator finds the greatest common factor of all the terms with the Euclidean algorithm and divides each term by it. The result is the smallest set of whole numbers in the same proportion, and the calculator confirms it cannot reduce further by checking that the simplified terms share no common factor.
Worked example
20:30 has a greatest common factor of 10, so dividing both terms gives 2:3. A three-term ratio like 10:20:30 shares a factor of 10 across all terms and reduces to 1:2:3. For a two-term ratio the calculator also reports the equivalent decimal, so 16:9 is shown as about 1.778.
Key insight
Ratios and fractions are the same idea written differently — 20:30 is 20/30 is 2/3 — so simplifying a ratio is exactly simplifying a fraction. Reducing to lowest terms makes proportions easy to compare, which is why screen aspect ratios are quoted as 16:9, not 1920:1080.
Frequently asked questions
How do I simplify a ratio?
Divide every term by the greatest common factor of all the terms. For 20:30 the GCF is 10, giving 2:3. The result keeps the same proportion in the smallest whole numbers.
Can I simplify a ratio with three or more terms?
Yes. The calculator finds the greatest common factor across all terms and divides each by it, so 10:20:30 becomes 1:2:3.
What is the relationship between ratios and fractions?
A two-term ratio a:b is the fraction a/b. Simplifying the ratio is identical to reducing the fraction, and the calculator shows the decimal value as well.
How do I know a ratio is fully simplified?
When the terms share no common factor greater than 1 — they are collectively coprime. The calculator verifies this for the reduced ratio.
Why must the terms be whole numbers?
Simplifying by a greatest common factor is defined for integers. Scale a ratio with decimals up to whole numbers first (for example, multiply 1.5:2 by 2 to get 3:4).
Methodology & verification
The ratio is simplified by dividing every term by the greatest common factor of all terms, computed with the Euclidean algorithm. Terms are validated as non-negative integers. For a two-term ratio the equivalent decimal is also reported. The receipt confirms the simplified terms are collectively coprime (overall GCF = 1), proving the ratio is in lowest terms.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.