Ratio Calculator

Simplify a ratio to its simplest form

Enter two numbers to compare, and the calculator will simplify the ratio to its lowest terms by finding the greatest common divisor.

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Other Titles
Understanding Ratios: A Comprehensive Guide
Learn how ratios are used to compare the size of two or more quantities and how to simplify them to their most basic form.

What is a Ratio?

A ratio is a way of comparing two or more quantities of the same kind. It tells us how much of one thing there is compared to another. Ratios are typically written in the format 'A:B' and are read as 'the ratio of A to B'.
For example, if there are 8 girls and 12 boys in a class, the ratio of girls to boys is 8:12. Ratios, like fractions, can be simplified. The ratio 8:12 can be simplified by dividing both numbers by their greatest common divisor, which is 4. So, the simplified ratio is 2:3. This means for every 2 girls, there are 3 boys.
Key points about ratios:

Simplification Example

  • Let's simplify the ratio 18:45.
  • 1. Find the greatest common divisor (GCD) of 18 and 45.
  • - The factors of 18 are 1, 2, 3, 6, 9, 18.
  • - The factors of 45 are 1, 3, 5, 9, 15, 45.
  • - The greatest common divisor is 9.
  • 2. Divide both parts of the ratio by the GCD.
  • - 18 ÷ 9 = 2
  • - 45 ÷ 9 = 5
  • The simplified ratio is 2:5.

Step-by-Step Guide to Using the Ratio Calculator

This calculator simplifies any ratio for you instantly.
How to Use It:

Usage Tips

  • The calculator works with both whole numbers and decimals.
  • If you enter decimals, the calculator will first convert them to a ratio of integers and then simplify.
  • For example, a ratio of 1.5 : 2.5 will be simplified to 3:5.

Real-World Applications of Ratios

Ratios are used everywhere, from the kitchen to complex scientific research.
Cooking and Recipes:
Maps and Scale Models:
Technology and Design:

Practical Examples

  • A graphic designer needs to resize an image that is 1920 pixels wide and 1080 pixels high. The aspect ratio is 1920:1080, which simplifies to 16:9. They must maintain this ratio to avoid stretching the image.
  • A landscaper is mixing concrete with a ratio of 1 part cement, 2 parts sand, and 3 parts gravel (1:2:3). This is an example of a ratio with more than two numbers.

Common Misconceptions and Correct Methods

Misconception 1: Ratios are the same as Fractions
While related, they are not identical. A ratio compares part-to-part (e.g., girls to boys), while a fraction typically compares part-to-whole (e.g., girls to total students). In our example of 8 girls and 12 boys (a 2:3 ratio), the fraction of students who are girls is 8/20 (or 2/5), not 2/3.
Misconception 2: You can add ratios together.
Ratios cannot be added in the same way as regular numbers. If one group has a boy-girl ratio of 1:2 and another group has a ratio of 2:3, you cannot simply add them to get a combined ratio of 3:5. You must add the actual numbers of boys and girls from each group and then form a new ratio.

Key Takeaways

  • A ratio compares two quantities.
  • Order matters in a ratio.
  • Always simplify ratios to their lowest terms.

Mathematical Derivation and Examples

The process of simplifying a ratio A:B relies on finding the Greatest Common Divisor (GCD) of A and B.
The Simplification Process
Let the ratio be A:B. The simplified ratio will be A':B' where:
A' = A / GCD(A, B)
B' = B / GCD(A, B)
The GCD is the largest positive integer that divides both A and B without leaving a remainder. The most common algorithm for finding the GCD is the Euclidean algorithm.

Comprehensive Example

  • Let's simplify the ratio 135:210.
  • 1. Find the GCD of 135 and 210 using the Euclidean algorithm.
  • - 210 = 1 * 135 + 75
  • - 135 = 1 * 75 + 60
  • - 75 = 1 * 60 + 15
  • - 60 = 4 * 15 + 0
  • - The last non-zero remainder is the GCD, which is 15.
  • 2. Divide both parts of the ratio by 15.
  • - 135 ÷ 15 = 9
  • - 210 ÷ 15 = 14
  • The simplified ratio is 9:14.