Ratio Calculator - Simplify Ratios and Solve Proportions Online

What is a Ratio?

A ratio is a mathematical comparison between two or more quantities, expressing how many times one value contains or is contained within another. Ratios are fundamental in mathematics, science, engineering, and everyday life.

a : b or a/b

Ratios can be written in three different ways:

Colon notation: 3:4 (read as "three to four")
Fraction notation: 3/4 (read as "three fourths")
Word notation: "3 to 4"

Key Properties of Ratios

Order matters: 3:4 is different from 4:3
No units: Ratios are dimensionless quantities
Can be simplified: 6:8 = 3:4 (both divided by 2)
Can be scaled: 3:4 = 6:8 = 9:12

Types of Ratio Calculations

1. Simplifying Ratios

To simplify a ratio, divide both terms by their Greatest Common Divisor (GCD). This gives you the ratio in its lowest terms.

Example: Simplify 12:18

GCD of 12 and 18 is 6

12 ÷ 6 = 2, 18 ÷ 6 = 3

Simplified ratio: 2:3

2. Solving Proportions

A proportion is an equation stating that two ratios are equal. You can solve for a missing value using cross multiplication.

a : b = c : d

Example: Solve 3:4 = 6:?

Cross multiply: 3 × ? = 4 × 6

3 × ? = 24

? = 24 ÷ 3 = 8

Answer: 3:4 = 6:8

3. Scaling Ratios

To scale a ratio, multiply both terms by the same number (scale factor).

Example: Scale 2:3 by factor of 5

2 × 5 = 10, 3 × 5 = 15

Scaled ratio: 10:15

4. Comparing Ratios

To compare two ratios, convert them to decimal form and compare the values.

Example: Compare 3:4 and 5:7

3:4 = 0.75, 5:7 ≈ 0.714

Since 0.75 > 0.714, 3:4 is greater than 5:7

Real-World Applications

Cooking and Recipes

Ratios are essential in cooking for scaling recipes up or down. For example, if a recipe calls for 2 cups of flour to 1 cup of sugar (2:1 ratio), you can easily scale it for different serving sizes.

Architecture and Engineering

The golden ratio (approximately 1.618:1) is used in architecture and design for aesthetically pleasing proportions. Scale models use ratios to represent real-world objects.

Finance and Business

Financial ratios like debt-to-equity ratio, price-to-earnings ratio, and current ratio help analyze company performance and make investment decisions.

Science and Medicine

Ratios are used in chemistry for stoichiometry, in medicine for drug dosages, and in physics for various calculations involving proportions.

Common Ratio Examples

Aspect Ratios

16:9 (HD TV)
4:3 (Traditional TV)
21:9 (Ultrawide)
1:1 (Square)

Mathematical Constants

π ≈ 3.14159:1
Golden Ratio ≈ 1.618:1
e ≈ 2.718:1
√2 ≈ 1.414:1

Tips for Using the Ratio Calculator

Enter whole numbers, decimals, or fractions as decimals
For proportions, leave exactly one field empty to solve for it
Use positive numbers for most calculations
Check your results by verifying the calculations manually
Use the step-by-step solutions to understand the process
Copy results for use in other applications

Mathematical Formulas

Greatest Common Divisor (GCD)

Use Euclidean algorithm to find GCD of two numbers

Cross Multiplication

If a:b = c:d, then a × d = b × c

Ratio to Decimal

a:b = a ÷ b

Ratio to Percentage

a:b = (a ÷ b) × 100%

Frequently Asked Questions

What's the difference between a ratio and a fraction?

While ratios and fractions use similar notation, they serve different purposes. A ratio compares two quantities and shows their relationship, while a fraction represents a part of a whole. For example, 3:4 as a ratio means "3 parts to 4 parts," while 3/4 as a fraction means "three-fourths of a whole."

Can ratios have more than two terms?

Yes, ratios can have multiple terms. For example, 2:3:5 represents a ratio of three quantities. This is common in recipes (flour:sugar:butter) or mixing solutions (water:alcohol:acid). However, our calculator focuses on two-term ratios for simplicity.

How do I convert a ratio to a percentage?

To convert a ratio a:b to a percentage, divide a by b and multiply by 100. For example, 3:4 = (3÷4) × 100% = 75%. This tells you that the first quantity is 75% of the second quantity.

What if one of the ratio values is zero?

If the second value is zero, the ratio is undefined (division by zero). If the first value is zero, the ratio is 0:1, meaning the first quantity is zero. Our calculator handles these cases appropriately and shows clear error messages.

How accurate are the calculator results?

The calculator provides results accurate to 6 decimal places for decimal values and 2 decimal places for percentages. For most practical purposes, this level of precision is sufficient. The step-by-step solutions show the exact calculations performed.

Can I use negative numbers in ratios?

While mathematically possible, negative numbers in ratios are rarely used in practical applications. Our calculator accepts negative numbers but focuses on positive ratios, which are more common in real-world scenarios like cooking, engineering, and business.

What's the difference between ratio and proportion?

A ratio is a comparison between two quantities (a:b), while a proportion is an equation stating that two ratios are equal (a:b = c:d). Proportions are used to solve for unknown values when you know three of the four terms.

How do I scale a recipe using ratios?

First, identify the ratio of ingredients in the original recipe. Then, determine your scale factor (e.g., if doubling, use factor 2). Multiply each ingredient by the scale factor. For example, if the ratio is 2:1 (flour:sugar) and you want to triple it, use 6:3.