Ratio Calculator

How to Use This Calculator

Start by picking your calculation type from the dropdown. You can simplify ratios, convert to percentages, solve proportions, scale values, or find equivalent forms.

Enter your numbers in Part A and Part B - both are required. For three-part ratios, add Part C. The Part D field appears automatically for four-part ratios when you select certain calculation types.

Click Calculate to see your result with complete step-by-step work. Copy any result using the Copy button.

Whole numbers and decimals both work. For proportions, enter three known values to find the fourth.

Understanding Ratios

What Ratios Mean

A ratio compares two or more quantities. You can write them three ways: colon notation 3:2, fraction notation 3/2, or word form "3 to 2". In the ratio 3:2, there are 3 units of the first quantity for every 2 units of the second. Ratios can compare three or more quantities at once, while fractions compare only two.

Colon notation dominates in practice, though fraction form helps with decimal conversions. Both express the same relationship but serve slightly different purposes in mathematics.

Part-to-Part and Part-to-Whole

If a class has 12 boys and 8 girls, their ratio is 12:8 or 3:2 when simplified - part-to-part comparison. Boys to total students gives 12:20 or 3:5 as a part-to-whole comparison. The data stays the same, but which calculation you need depends on what you're comparing.

Simplifying and Finding Equivalents

Simplify ratios by dividing all parts by their greatest common divisor. See the detailed section below for complete examples.

Different ratios can represent identical relationships. The ratios 3:2, 6:4, 9:6, 12:8, and 15:10 all simplify to 3:2 and equal 1.5 as decimals. Like equivalent fractions, you multiply or divide all parts by the same number.

Working with Ratios

Simplifying Ratios

To simplify, find the greatest common divisor of all parts and divide through. The process resembles fraction simplification but handles any number of parts.

When mixing paint with 18 parts blue and 24 parts white, find GCD(18,24) = 6. Dividing gives 3:4. For RGB colors like 12:18:24, the GCD of all three is 6, producing 2:3:4 when simplified.

A ratio like 5:7 can't be simplified further because the GCD equals 1. But 10:14 reduces to 5:7 since both numbers share a factor of 2.

Converting to Percentages

Converting ratios to percentages requires three steps: add all parts for the total, divide each part by that total, then multiply by 100.

Two business partners splitting profits 3:2 would calculate 3+2 = 5 total parts. The first partner receives 3/5 or 60%, while the second gets 2/5 or 40%. An investment portfolio split 2:3:5 among stocks, bonds, and real estate divides into 20%, 30%, and 50% respectively when calculated against the total of 10 parts.

Dividing $1,000 by this method means $600 goes to the first partner and $400 to the second.

Solving Proportions

If 3:4 equals 6:8, you're looking at a proportion. You can write proportions as a:b = c:d or a/b = c/d.

To solve proportions, cross-multiply to get a×d = b×c, then solve for the unknown. Cross-multiplication works because when you multiply both sides of a/b = c/d by bd, you get a×d = b×c.

If 3 cups of flour makes 4 servings, how much flour for 6 servings? Set up 3:4 = x:6. Cross-multiply to get 4x = 18, so x = 4.5 cups. On a map where 2 cm represents 5 miles, 7 cm translates to 17.5 miles using the same method.

Direct and Inverse Proportions

When two quantities increase or decrease together at the same rate, they're in direct proportion. Three apples cost $2, so nine apples cost $6 - triple the quantity, triple the price.

Inverse proportions work differently. If 4 workers complete a job in 6 days, 8 workers finish in 3 days - double the workers, half the time. The formula a×b = c×d captures this relationship: 4×6 = 8×x gives x = 3 days.

Scaling and Combining

Multiply or divide all parts by the same number to create equivalent ratios. The ratio 3:2 multiplied by 5 becomes 15:10. Divide to scale down: 15:10 divided by 5 returns to 3:2. A recipe calling for 3:2 flour to sugar doubles to 6:4 when you need twice the quantity.

When combining ratios with a common term, find the LCM of that term. Mixing solution A:B in 2:3 with solution B:C in 4:5 requires making B the same in both: 2:3 = 8:12 and 4:5 = 12:15, combining to A:B:C = 8:12:15.

Checking Your Work

Convert both sides to decimals to verify. If 3:4 and 6:8 both equal 0.75, the proportion is correct.

Real-World Applications

Recipes and Cooking

Recipes are built on ratios. Pancakes might need 2 parts flour for every 1 part milk. Doubling the recipe means using 4:2, tripling means 6:3 - the relationship stays consistent while quantities scale. Cocktails follow similar logic: a Margarita might use 3:2:1 for tequila, lime juice, and triple sec.

Maps and Geography

Maps use ratios to show real distances. A scale of 1:100,000 means 1 cm on the map equals 100,000 cm in reality - that's 1 km. Measure 5 cm on the map, and the real distance is 5 km. Larger ratios show broader areas with less detail, while smaller ratios provide more detail over smaller areas.

Screens and Displays

Modern widescreen devices use 16:9, meaning 16 units wide for every 9 units tall. Older equipment used 4:3, still found in some tablets and classic films. Ultrawide monitors typically use 21:9, as do certain movies, while theatrical cinema often employs 2.39:1. Instagram and other square platforms use 1:1, and traditional 35mm photography uses 3:2.

Mechanical Systems

Bicycles control speed and effort through gear ratios. A front gear with 50 teeth and rear gear with 10 teeth gives 5:1 - one pedal rotation turns the rear wheel five times. Higher ratios mean more speed but less torque. Lower ratios like 3:2 mean less speed but more torque for climbing hills.

Chemistry and Solutions

Diluting fertilizer concentrate works on ratios - mix 1 part concentrate with 10 parts water. For 55 gallons total, you'd use 5 gallons concentrate and 50 gallons water. Paint thinners work similarly, often at 3:1 paint to thinner.

Financial Analysis

Businesses use debt-to-equity ratios to measure financial risk. A company with $200k debt and $500k equity has a 2:5 ratio - lower ratios indicate less financial risk. Price-to-earnings ratios compare stock price to earnings: stock price $100 with earnings $5 gives 20:1.

Photography

Camera aperture uses ratios like f/2.8, meaning the ratio of focal length to aperture diameter is 2.8:1. Smaller f-numbers mean larger openings and more light entering the camera. Larger f-numbers mean smaller openings and less light. Each stop represents a change in this ratio.

Construction

Standard concrete mixes call for 1 part cement, 2 parts sand, and 3 parts gravel. Roof pitch might be 4:12, meaning 4 inches of rise for every 12 inches of horizontal run. Wheelchair ramps use 1:20 - one unit rise per 20 units horizontal.

Types of Ratios

Part-to-Part Ratios

Part-to-part ratios compare one part directly to another. A class with 15 boys and 10 girls has a boy-to-girl ratio of 3:2. You won't know the total number of students unless you add the parts together.

Part-to-Whole Ratios

Part-to-whole ratios compare one part to the complete total. Boys to total students in that same class is 3:5, while girls to total is 2:5. Together these must equal the whole: 3/5 + 2/5 = 100%.

Rates

Rates compare different units like 60 miles per hour or $3 per 2 pounds. A unit rate reduces the second quantity to 1, so 60 miles : 1 hour equals 60 mph. Rates require units - miles per hour, dollars per pound - while pure ratios don't.

Odds and Probability

If rain has 60% probability, the odds are 3:2 favorable to unfavorable outcomes. Odds express the ratio of favorable to unfavorable outcomes. Probability expresses a fraction. Converting: odds a:b equals probability a/(a+b).

Continued Ratios

Continued ratios compare three or more quantities. An RGB color might be 255:128:64 for red, green, and blue values. A recipe might use 3:2:1:0.5 for flour, sugar, butter, and salt. You must keep all parts since these can't be expressed as a single fraction.

Notable Ratios

Golden Ratio

The golden ratio (φ ≈ 1.618:1) appears in the Parthenon and many Renaissance artworks.

Silver Ratio

The silver ratio (δ ≈ 2.414:1) appears in paper sizing. When you cut A4 paper in half, you get A5 with the same aspect ratio - the silver ratio maintains this property through successive divisions.

Pi

The ratio of a circle's circumference to its diameter stays constant at 3.14159...:1 regardless of circle size. Circumference = π × diameter. It appears in calculations involving circles, spheres, and waves.

Common Aspect Ratios

Screen shapes vary by purpose. Most modern displays use 16:9 widescreen, from TVs to YouTube to computer monitors. Older 4:3 equipment survives in classic films and some tablets. Ultrawide monitors stretch to 21:9, matching certain movie formats. Theatrical cinema goes even wider at 2.39:1. Square 1:1 formats work for Instagram and similar platforms, while traditional photography settled on 3:2.

Common Mistakes

Good Practices

Always use consistent units. Converting 3 meters to 9.84 feet before comparing to 2 feet produces the accurate ratio 9.84:2.

Simplify first. The ratio 3:2 is easier to work with than 12:8, even though both represent the same relationship. Start with the simplified form for easier calculations.

Check your work by converting to decimals. The ratio 3:2 equals 1.5, while 6:8 equals 0.75, revealing an error if you expected those to match.

Don't guess when scaling recipes or other proportional problems. Set up a proportion. If a recipe serves 4 and you need 7 servings, use 4:original = 7:x to calculate the exact amount.

Always label what you're comparing. Without context, "3:2" is ambiguous. "3 boys : 2 girls" clarifies what each number represents.

Frequent Errors

You can't add 2:3 and 1:2 to get 3:5. Ratios need a common reference point before adding.

A ratio of 3:2 means 3 parts A and 2 parts B (total 5 parts). It doesn't mean 3 out of 2, which would be impossible.

An answer of 12:8 is mathematically correct but should be simplified to 3:2. Always reduce to lowest terms.

For proportion a:b = c:d, cross-multiply to get a×d = b×c, not a×c = b×d. Multiply diagonally, not straight across.

You can't divide ratios directly. Simplify each separately or convert to fractions first.

Order matters. Writing 2:3 is not the same as 3:2. The first indicates 2 parts of one quantity to 3 of another. Flipping the numbers reverses the relationship.

A speed comparison like 60:30 means nothing without units. With units - 60 mph : 30 mph = 2:1 - we know the first speed is twice as fast.

A part-to-part ratio like 3:2 measures different things than a part-to-whole ratio like 3:5. Context determines which you need.

Frequently Asked Questions

Can I use decimals in this calculator?

Yes. Decimal inputs are converted to whole numbers internally, then the calculation is performed and simplified.

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

A ratio compares quantities, like 3:2. A proportion is an equation showing two ratios are equal, like 3:2 = 6:4.

How do I know if a ratio is fully simplified?

A ratio is fully simplified when no number divides evenly into all parts. The ratio 3:5 is simplified, but 6:10 reduces to 3:5.

Why does order matter in ratios?

The ratio 3:2 means 3 of the first quantity for every 2 of the second. Reversing to 2:3 changes the relationship. In a 3:2 recipe, you'd use more of the first ingredient; in a 2:3 recipe, more of the second.

Can I use negative numbers in ratios?

No. Ratios compare quantities, which are always positive.

What happens if I enter zero?

Zero in most positions will cause an error because ratios compare positive amounts. For proportions, zero in certain positions makes the equation unsolvable.