Ratio Calculator
Simplify a ratio A:B to lowest terms and find the decimal quotient A / B. Supports whole numbers and decimals, with clear sign handling.
How It's Calculated
Formula
\text{ratio} = \frac{A}{\gcd(A, B)} : \frac{B}{\gcd(A, B)}A ratio compares two quantities, A and B, showing how many times one contains the other. This calculator reduces A:B to its simplest form and also reports the plain decimal quotient A ÷ B. Simplifying works the same way as reducing a fraction: find the greatest common divisor (GCD) of A and B, then divide both terms by it, leaving a ratio with no smaller whole-number equivalent. If either A or B is entered with a decimal part, both terms are first scaled up by a power of 10 — just enough to make both whole numbers — before the same GCD reduction is applied, so a ratio like 2.5:4 still reduces to an exact simplified form (5:8) instead of being left as decimals. Sign is always shown on A: a ratio with exactly one negative term (either A or B) normalizes to a negative A with a positive B, and a ratio with both terms negative cancels out to a positive result — this keeps the sign convention consistent no matter which term was entered as negative. A zero A is valid (it means A has no share of B at all) and always simplifies to 0:1. A zero B is not valid, since both the simplified ratio and the decimal quotient are undefined when the second term is zero.
Worked Examples
Simplify 4:8
- Find the GCD of 4 and 8: GCD(4, 8) = 4
- Divide both terms by the GCD: 4 ÷ 4 = 1, 8 ÷ 4 = 2
- Simplified ratio: 1:2
- Decimal quotient: 4 ÷ 8 = 0.5
Simplify a decimal ratio: 2.5:4
- A has 1 decimal place and B has 0, so scale both terms by 10^1 = 10
- Scaled terms: 2.5 × 10 = 25, 4 × 10 = 40
- Find the GCD of the scaled terms: GCD(25, 40) = 5
- Divide both by the GCD: 25 ÷ 5 = 5, 40 ÷ 5 = 8
- Simplified ratio: 5:8
- Decimal quotient: 2.5 ÷ 4 = 0.625
Frequently Asked Questions
Why is a ratio with a zero second term (B) not allowed?
Both outputs depend on dividing by B: the decimal quotient is literally A ÷ B, and simplifying the ratio also requires dividing by B's greatest common divisor with A. Since division by zero is undefined, any ratio entered with B = 0 is rejected before any calculation runs.
Why does 0:5 simplify to 0:1 instead of staying as 0:5?
A ratio of zero to anything means the first quantity has no share of the second at all, and that's true regardless of what the second number was — 0:5, 0:12, and 0:1 all describe the same relationship. This calculator reduces that relationship to its simplest form, 0:1, the same way it would reduce any other ratio to lowest terms.
Where does the negative sign go when one term is negative?
The sign is always normalized onto A, the first term. So both -2:4 and 2:-4 simplify to -1:2 — the same ratio expressed the same way — rather than leaving a negative sign on whichever term happened to be entered as negative. This mirrors how a simplified fraction always keeps its denominator positive.
What happens if both A and B are negative?
Two negatives describe a positive ratio, so the signs cancel out. For example, -2:-4 simplifies to 1:2, identical to the result for 2:4.
How are decimal inputs like 2.5:4 simplified exactly?
Both terms are scaled up by the smallest power of 10 that turns them into whole numbers — using the number of decimal places actually entered, not floating-point guesswork — and then reduced using the same greatest-common-divisor method used for whole numbers. 2.5:4 scales to 25:40, which reduces to 5:8. This keeps the simplified ratio exact rather than just echoing the original decimals back unsimplified.