Ratio Calculator
Simplify ratios, find equivalent ratios, solve proportions, and convert ratios to fractions, decimals, and percentages.
Enter a Ratio to Simplify
Enter two numbers to find the simplest form of your ratio.
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About This Calculator
Ratios are everywhere in our daily lives, from the ingredients in your morning coffee to the dimensions of your smartphone screen. Whether you're scaling a recipe for a dinner party, calculating aspect ratios for video editing, mixing paint colors, or solving math homework problems, understanding ratios is an essential skill that connects abstract mathematics to real-world applications.
A ratio is simply a way to compare two or more quantities, showing the relative size of one value to another. When you say your coffee is made with a 1:15 ratio of coffee to water, or that your TV has a 16:9 aspect ratio, you're using this fundamental mathematical concept. Ratios appear in architecture (the golden ratio), finance (debt-to-income ratios), cooking (ingredient proportions), science (chemical concentrations), and countless other fields.
Our comprehensive Ratio Calculator helps you work with ratios in multiple ways. Simplify ratios to their lowest terms by finding the greatest common divisor. Find equivalent ratios by scaling up or down while maintaining the same proportion. Solve proportions to find missing values using cross-multiplication. Convert ratios to fractions, decimals, and percentages for different contexts. Each calculation includes step-by-step explanations so you can understand the process, not just get the answer.
Whether you're a student learning proportional reasoning, a chef perfecting recipes, a designer working with aspect ratios, or anyone who needs to work with comparative quantities, this calculator makes ratio calculations fast, accurate, and educational.
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How to Use the Ratio Calculator
- 1**Select your calculation mode**: Choose from Simplify Ratio, Equivalent Ratio, Solve Proportion, or Convert Ratio based on what you need to do.
- 2**Enter your values**: Input the numbers for your ratio. For simplifying, enter the original ratio values. For proportions, enter three known values and select which one is missing.
- 3**Use presets (optional)**: Click on common ratios like 16:9 (widescreen) or 4:3 (standard) to quickly load frequently used values.
- 4**View your results**: The calculator instantly shows the result along with secondary information like the GCD, scale factor, or equivalent representations.
- 5**Review step-by-step solutions**: Expand the solution steps to see exactly how the calculation was performed, perfect for learning or verification.
- 6**Explore visual representations**: See ratio comparisons displayed as proportional bars to better understand the relationship between values.
- 7**Share or print**: Use the share button to copy a link with your calculation, or print your results for reference.
Formula
a:b = a/b (as fraction) = a divided by b (as decimal) = (a/b) x 100% (as percentage)A ratio a:b can be expressed as a fraction (a/b), a decimal (a divided by b), or a percentage (multiply the decimal by 100). To simplify, divide both terms by their GCD. For proportions (a:b = c:d), cross-multiplication gives a x d = b x c, which can be rearranged to solve for any missing term.
Understanding Ratios: The Basics
A ratio compares two or more quantities, showing how many times one value contains another. Ratios can be written in three ways:
Common Notations:
- Colon notation: 3:4 (most common)
- Fraction notation: 3/4
- Word form: "3 to 4"
Key Concepts:
- Terms: The numbers in a ratio are called terms. In 3:4, both 3 and 4 are terms.
- Order matters: 3:4 is different from 4:3. The first term is compared TO the second term.
- Part-to-part vs. Part-to-whole: 3:4 could mean 3 parts of A for every 4 parts of B (part-to-part), or 3 parts out of a total of 4 (part-to-whole).
Examples in Context:
- Recipe: 2 cups flour to 1 cup sugar = 2:1
- Mixture: 3 parts water to 1 part concentrate = 3:1
- Scale: 1 inch represents 100 feet = 1:1200
Simplifying Ratios with GCD
Simplifying a ratio means reducing it to its lowest terms while maintaining the same relationship. This is done by dividing both terms by their Greatest Common Divisor (GCD).
The Process:
- Find the GCD of both numbers using the Euclidean algorithm
- Divide both terms by the GCD
- The result is the simplified ratio
Example: Simplify 48:36
- Find GCD of 48 and 36:
- 48 = 1 x 36 + 12
- 36 = 3 x 12 + 0
- GCD = 12
- Divide both terms: 48/12 : 36/12 = 4:3
- Simplified ratio: 4:3
Why Simplify?
- Easier to understand and compare
- Reveals the fundamental relationship
- Standard form for communication
- Required for many mathematical operations
Proportions and Cross-Multiplication
A proportion states that two ratios are equal. The key to solving proportions is cross-multiplication: if a:b = c:d, then a x d = b x c.
Solving for Unknown Values:
Given: 3:4 = x:12, find x
- Cross multiply: 3 x 12 = 4 x x
- 36 = 4x
- x = 36/4 = 9
- Verification: 3:4 = 9:12 (both equal 0.75)
Real-World Applications:
- Map scales: If 1 inch = 50 miles, how far is 3.5 inches? (1:50 = 3.5:x, x = 175 miles)
- Recipe scaling: If 2 cups feeds 4 people, how much for 10? (2:4 = x:10, x = 5 cups)
- Currency conversion: If $1 = 0.85 euros, how many euros for $50? (1:0.85 = 50:x, x = 42.50 euros)
- Similar triangles: Corresponding sides of similar shapes are proportional
Ratios in Recipes and Cooking
Cooking is one of the most practical applications of ratios. Professional chefs often work with ratios rather than fixed measurements because they scale perfectly.
Classic Cooking Ratios:
- Bread: 5:3 flour to water (by weight)
- Vinaigrette: 3:1 oil to vinegar
- Rice: 2:1 water to rice (by volume)
- Roux: 1:1 butter to flour (by weight)
- Custard: 2:1 cream to eggs (by volume)
Scaling Recipes: Original recipe serves 4, you need to serve 10:
- Scale factor = 10/4 = 2.5
- Multiply all ingredients by 2.5
- 2 cups flour becomes 5 cups
- 1 cup sugar becomes 2.5 cups
Tips for Scaling:
- Scale by weight for accuracy, not volume
- Some ingredients (spices, leaveners) may not scale linearly
- Cooking times may need adjustment for larger quantities
Aspect Ratios in Media and Design
Aspect ratio describes the proportional relationship between width and height of an image or screen. Understanding aspect ratios is crucial for photography, videography, and design.
Common Aspect Ratios:
| Ratio | Decimal | Common Use |
|---|---|---|
| 1:1 | 1.00 | Instagram square, album covers |
| 4:3 | 1.33 | Classic TV, iPad, old photos |
| 3:2 | 1.50 | 35mm film, DSLR photos |
| 16:9 | 1.78 | HDTV, YouTube, most monitors |
| 21:9 | 2.33 | Ultrawide monitors, cinema |
| 2.39:1 | 2.39 | Anamorphic widescreen films |
Calculating Dimensions: For a 16:9 display that's 32 inches wide:
- 16:9 = 32:x
- x = (9 x 32) / 16 = 18 inches tall
Resizing Images: To maintain aspect ratio when resizing:
- New height = (new width x original height) / original width
- Always lock aspect ratio in editing software to prevent distortion
The Golden Ratio in Nature and Design
The golden ratio (approximately 1.618:1, represented by the Greek letter phi) appears throughout nature and has been used in art and architecture for millennia.
Mathematical Definition:
- If a/b = (a+b)/a, then a/b = phi
- phi = (1 + sqrt(5)) / 2 = 1.6180339887...
- phi has the unique property: phi^2 = phi + 1
In Nature:
- Spiral patterns in sunflowers and pinecones
- Shell spirals (nautilus)
- Leaf arrangements on stems
- DNA molecule dimensions
- Galaxy spiral arms
In Art and Architecture:
- Parthenon dimensions
- Renaissance paintings (da Vinci, Michelangelo)
- Modern logos (Apple, Twitter)
- Typography and page layouts
- Photography composition (similar to rule of thirds)
Golden Rectangle: A rectangle with sides in the golden ratio. When you cut a square from it, the remaining rectangle is also golden.
Ratios in Maps and Scale Drawings
Maps and scale drawings use ratios to represent real-world distances in a manageable size. Understanding map scales is essential for navigation, architecture, and engineering.
Types of Scale:
- Representative fraction: 1:50,000 (1 unit on map = 50,000 units in reality)
- Verbal scale: "1 inch = 1 mile"
- Graphic scale: A bar showing distances
Common Map Scales:
| Scale | 1 cm represents | Best for |
|---|---|---|
| 1:1,000 | 10 meters | Floor plans |
| 1:10,000 | 100 meters | City maps |
| 1:50,000 | 500 meters | Hiking maps |
| 1:250,000 | 2.5 km | Regional maps |
| 1:1,000,000 | 10 km | Country maps |
Calculating Real Distances: Map scale 1:25,000, measured distance 4 cm:
- Real distance = 4 x 25,000 = 100,000 cm = 1 km
For Architects:
- 1/4" = 1' means 1:48 scale
- 1/8" = 1' means 1:96 scale
Pro Tips
- 💡Always keep ratio terms in order. The ratio of length to width (3:2) is different from width to length (2:3). Label your ratios when working with multiple quantities.
- 💡To quickly simplify a ratio, look for obvious common factors first. If both numbers are even, divide by 2. If both end in 0 or 5, divide by 5. Then use GCD for the final simplification.
- 💡When scaling recipes, convert everything to the same unit (preferably weight) before applying the ratio. This prevents errors from different measurement systems.
- 💡For aspect ratios, remember that 16:9 = 1.78 and 4:3 = 1.33. If you need to quickly assess an aspect ratio, divide width by height.
- 💡Use cross-multiplication to verify your proportion solutions. If a:b = c:d, then a x d should equal b x c. If they don't match, recheck your calculation.
- 💡When converting ratios to percentages, decide first whether it's a part-to-part or part-to-whole ratio. The interpretation differs significantly.
- 💡For map scales, remember that larger scale numbers mean smaller map detail. 1:10,000 shows more detail than 1:100,000 for the same physical map size.
- 💡The golden ratio is approximately 1.618:1. For quick golden rectangle approximations, multiply the smaller side by 1.618 to get the larger side.
- 💡When working with photo aspect ratios, remember that 3:2 (DSLR standard) is slightly wider than 4:3 (phone standard). Cropping between them loses about 11% of the image.
- 💡To find equivalent ratios quickly, think of them like fractions. 2:3 = 4:6 = 6:9 = 8:12. Each pair is the original multiplied by 2, 3, 4, etc.
- 💡When simplifying ratios with very large numbers, factor each number into primes first. This reveals the GCD directly without iterative division.
Frequently Asked Questions
While ratios and fractions look similar and can often be used interchangeably, they serve different purposes. A fraction represents a part of a whole (3/4 of a pizza), while a ratio compares quantities (3 boys to 4 girls, written as 3:4). A ratio can compare parts of different wholes or the same whole. In math, the ratio 3:4 equals the fraction 3/4, but conceptually, the ratio compares two separate quantities while the fraction shows one quantity relative to a total.