Simplify Ratio

Calculator and formulas to simplify ratios using the Euclidean algorithm

Ratio Simplifier Calculator

What is calculated?

This function reduces a ratio to the smallest integers. The greatest common divisor (gcd) is computed using the Euclidean algorithm and both parts are divided by it.

Enter ratio

:
Examples: 1920:1080, 50:120, 24:36
Simplified ratio
Original
:
→
Simplified
:
The ratio has been reduced to the smallest integers

Ratio Simplification Info

Simplification process

The simplification is performed in three steps:

  1. Find the greatest common divisor
  2. Divide both numbers by the gcd
  3. Output the simplified form

Why simplify? Simplified ratios are easier to understand and use.

Quick examples
12:16 = 3:4
gcd = 4
50:75 = 2:3
gcd = 25
100:150 = 2:3
gcd = 50
7:14 = 1:2
gcd = 7
Euclidean algorithm

The Euclidean algorithm is the most efficient method to compute the gcd.
Principle: gcd(a,b) = gcd(b, a mod b)

Mathematical formulas for ratio simplification

Simplification formula
\[\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}\] Reduce ratio
Euclidean algorithm
\[\gcd(a,b) = \gcd(b, a \bmod b)\] Recursive definition
gcd properties
\[\gcd(a,b) = \gcd(b,a) = \gcd(|a|,|b|)\] Symmetry and sign
Coprimality
\[\gcd(a,b) = 1 \Leftrightarrow \text{coprime}\] Already fully reduced
Swap
\[a:b = \frac{a}{b}, \quad b:a = \frac{b}{a}\] Inverse ratio
Expand
\[a:b = (a \cdot k):(b \cdot k)\] Expand ratio

Detailed example: Euclidean algorithm

Example: simplify 1920:1080

1Apply Euclidean algorithm

gcd(1920, 1080):
1920 ÷ 1080 = 1 R 840
1080 ÷ 840 = 1 R 240
840 ÷ 240 = 3 R 120
240 ÷ 120 = 2 R 0

gcd = 120

2Divide by gcd

\[\frac{1920 \div 120}{1080 \div 120} = \frac{16}{9}\]

Divide both numbers by 120

1920 ÷ 120 = 16
1080 ÷ 120 = 9

3Simplified ratio

\[1920:1080 = \color{blue}{16:9}\]

Result: The ratio 1920:1080 is 16:9 (widescreen format)

Euclidean algorithm in detail

Algorithm steps
Step 1: Is a > b? If not, swap
\[\text{if } a < b: \gcd(a,b) = \gcd(b,a)\]
Step 2: Division with remainder
\[a = b \cdot q + r, \quad 0 \leq r < b\]
Step 3: Recurse or stop
\[\text{if } r = 0: \gcd(a,b) = b\] \[\text{otherwise: } \gcd(a,b) = \gcd(b,r)\]
Example: gcd(48, 18)
48 = 18 × 2 + 12
gcd(48, 18) = gcd(18, 12)
18 = 12 × 1 + 6
gcd(18, 12) = gcd(12, 6)
12 = 6 × 2 + 0
gcd(12, 6) = 6

Result: 48:18 = 8:3

More simplification examples

Simple examples
6:9 = 2:3 \[\gcd(6,9) = 3\]
20:30 = 2:3 \[\gcd(20,30) = 10\]
15:25 = 3:5 \[\gcd(15,25) = 5\]
Screen resolutions
1920:1080 = 16:9 Full HD widescreen
1024:768 = 4:3 Classic format
1440:900 = 8:5 16:10 format
Special cases
17:23 = 17:23 Coprime (gcd = 1)
100:50 = 2:1 Simple divisor
12:12 = 1:1 Equal ratio

Practical applications

Scale recipes

Example: adjust ingredients

Original: 400g flour, 300g sugar

\[400:300 = 4:3\]

For every 4 parts flour, 3 parts sugar

Mixing colors

Example: color ratio

75ml red + 45ml blue

\[75:45 = 5:3\]

Simpler mixing ratio

Blueprints

Scale 1000:50 = 20:1
Simplified: 1 cm corresponds to 20 cm

Profit sharing

Partner A: €3000, Partner B: €2000
Ratio: 3000:2000 = 3:2

Mathematical foundations

Greatest common divisor

The greatest common divisor (gcd) of two numbers is the largest natural number that divides both numbers without remainder. It is fundamental for reducing fractions and ratios.

Algorithm efficiency

The Euclidean algorithm is very efficient and requires only O(log min(a,b)) steps. It was described by Euclid around 300 BC.

Key properties
  • Uniqueness: The gcd is uniquely determined
  • Commutativity: gcd(a,b) = gcd(b,a)
  • Complete reduction: Result is always coprime
  • Scaling: Reduced ratios remain proportional

More Ratio Functions

Aspect Ratio  •  Compare Ratios  •  Decimal to Ratio  •  Fraction to Ratio  •  Ratio-Calculator  •  Ratio Simplifier  •  Ratio to Decimal  •  Ratio to Fraction  •  Scale Ratio  •