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Math & Statisticsratio

Ratio Simplifier

A ratio reduced to its simplest whole-number form, with the option to scale it or to divide a total in its proportions. Works on any number of parts, since a three-way split is where reducing by eye starts to go wrong.

Also called: ratio calculator, simplify ratio.

Two or more parts, separated by colons or commas.

Simplified ratio
9 : 16

9 : 16, divided through by a common factor of 120. All 2 parts share a factor of 120. With three or more parts the divisor has to be common to every one of them, which is why reducing two at a time goes wrong. Enter a total to split it in these proportions.

Common factor
120
Number of parts
2
As a fraction, for two parts
9/16
Scaled to your first part
Enter a first part to scale to
The total divided
Enter a total to divide
Largest part as a percentage
64
On the parts
All 2 parts share a factor of 120. With three or more parts the divisor has to be common to every one of them, which is why reducing two at a time goes wrong.
On dividing a total
Enter a total to split it in these proportions.
How this is calculated
Method and background

How this is calculated

Every part is divided by the greatest common divisor of all of them, taken pairwise across the list. Reducing two parts at a time and then combining is where errors come from with three or more, because a factor shared by two parts but not the third cannot be cancelled. Dividing a total works from the reduced form: each part over the sum of parts, times the total, which is the same answer as the unreduced form and easier to check by hand.

the greatest common divisor of every part at once, which is what keeps a three part ratio consistent
g
Greatest common divisor of all parts

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a 1080 by 1920 screen

Ratio
1080 : 1920
Scale so the first part becomes
0
Divide this total in the ratio
0

Simplified ratio9 : 16

gcd(1080, 1920) = 120, giving 9 : 16

Open this example

a three way split of a total

Ratio
2, 3, 5
Scale so the first part becomes
0
Divide this total in the ratio
1,00,000

Simplified ratio2 : 3 : 5

boundary: already reduced, and 5 of 10 parts is half the total

Open this example

Method and limits

What it assumes

  • Whole-number parts, since a ratio with decimals is scaled up first.

What it deliberately does not model

  • Parts of zero are kept, since a ratio containing a zero is meaningful and not an error.

Formula version 1.0.0 · definition 1.0.0 · India · Report a problem with this calculator

Frequently asked questions

Can I simplify a three part ratio two at a time?
Not reliably. A factor shared by two parts but not the third cannot be cancelled, so the divisor has to be common to all of them.