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Ratio Simplifier: a : b in Simplest Form

By Hesaplayıcı

The ratio simplifier reduces the ratio a : b to its simplest whole-number form. Enter the two terms. The calculator divides both by their GCD and shows the simplified ratio, its value and the 1 : n form. For example, 12 : 18 becomes 2 : 3. Decimal terms are made whole first: 4.8 : 11.2 = 3 : 7.

Worked example: Simplify the ratio 12 : 18

Value of the ratio (a / b)

0.67

First term, simplified
2
Second term, simplified
3
GCD
6
Whole-number multiplier
1
n in 1 : n (b / a)
1.5
Step by step

  1. GCD

    GCD = GCD(a, b)

    GCD = GCD(12, 18) = 6

  2. First term, simplified

    simplified a = a / GCD

    simplified a = 12 / 6 = 2

  3. Second term, simplified

    simplified b = b / GCD

    simplified b = 18 / 6 = 3

math.simplify-ratio

How to use

  1. Enter a (first term), the first number of the ratio.
  2. Enter b (second term), the second number. Both terms must be greater than zero.
  3. Press Calculate. You get the simplified ratio and the steps.

Formula

  • a : b = (a × multiplier) : (b × multiplier), when the terms have decimals
  • GCD = GCD(a, b)
  • simplified a = a / GCD, simplified b = b / GCD

The multiplier is the smallest power of ten that makes both terms whole. The GCD is the largest whole number that divides both terms; the calculator finds it with Euclid’s algorithm. The value of the ratio is a / b, and n in the 1 : n form is b / a.

Worked example

The worked example on this page simplifies the ratio 12 : 18. The steps give the GCD first and then the two terms of the simplified ratio: 12 : 18 = 2 : 3.

Limits

The calculator simplifies ratios of two terms; ratios of three or more terms and solving a proportion are not included (the proportion calculator does that). Terms must be greater than zero with at most 6 decimal places. Each term can be at most 10⁹. The value of the ratio and the 1 : n form are not rounded.

Frequently Asked Questions

How do you simplify a ratio?

Divide both terms by their greatest common divisor (GCD). In 12 : 18 the GCD is 6, so the simplest form is 2 : 3.

How do you simplify a ratio with decimals?

Multiply both terms by the same power of ten to make them whole, then divide by the GCD. For example, 4.8 : 11.2 = 48 : 112 = 3 : 7.

Is a ratio the same as a fraction?

The ratio a : b simplifies like the fraction a / b and has the same value. A ratio compares two quantities; a fraction is a part of one whole.

What does the 1 : n form mean?

The first term is made 1: the ratio 2 : 3 becomes 1 : 1.5. Map scales and mixing ratios often use this form.

Calculation rules

  • A ratio is simplified by dividing both terms by their greatest common divisor: in 12 : 18 the GCD is 6, so the simplest form is 2 : 3.
  • Decimal terms are first multiplied by the same power of ten to make them whole: 4.8 : 11.2 = 48 : 112 = 3 : 7. Terms can have at most 6 decimal places.
  • Both terms must be greater than zero. The value of the ratio and the 1 : n form are not rounded; the page may shorten numbers.
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