Ratio Calculator
Enter three values in the proportion A:B = C:D and leave one as zero to solve for the missing value. The calculator also simplifies ratios to their lowest terms and shows the scale factor.
What Ratios and Proportions Are
A ratio is just a way of comparing two quantities by showing how many times one contains the other. When you say a recipe calls for a 2:1 ratio of flour to sugar, you mean there's twice as much flour as sugar. The actual amounts don't matter — whether it's 2 cups of flour and 1 cup of sugar or 6 cups and 3 cups, the ratio stays the same.
A proportion takes things one step further. It's a statement that two ratios are equal: A:B = C:D, or equivalently, A/B = C/D. Proportions are how you scale things up or down while keeping the relationship between quantities intact. If you know that 3 workers can paint a fence in 4 hours, and you want to know how long it takes 6 workers (assuming they work at the same rate), you're setting up a proportion.
The most common way to solve a proportion is cross multiplication. If A/B = C/D, then A times D equals B times C. This gives you a simple equation with one unknown. It works because multiplying both sides of the equation by BD clears the fractions, and what's left is the cross products.
Ratios can be written several ways: 3:4, 3/4, or "3 to 4." They all mean the same thing. Some people get confused between ratios and fractions, but there's an important distinction. The fraction 3/4 represents a part of a whole (three out of four equal parts). The ratio 3:4 compares two separate quantities. A class with 3 boys and 4 girls has a ratio of 3:4, but boys aren't 3/4 of the class — they're 3/7 of it.
The Cross Multiplication Method
Cross multiplication is the standard technique for solving proportions, and once you've seen it a couple of times, it becomes second nature. The idea is simple: in any true proportion A/B = C/D, the cross products A*D and B*C are equal. This means you can set them equal and solve for whichever value is missing.
Let's walk through an example. Suppose you know that 3:4 = 6:D, and you want to find D. Cross multiply: 3 * D = 4 * 6, so 3D = 24, and D = 8. You can verify: 3/4 = 0.75, and 6/8 = 0.75. They match.
The method works regardless of which value is unknown. If A is missing from A:5 = 12:20, cross multiply: A * 20 = 5 * 12, so 20A = 60, and A = 3. Check: 3/5 = 0.6, and 12/20 = 0.6.
One thing to watch out for: the proportion must actually be set up correctly. A common mistake is comparing the wrong quantities. If a map says 1 inch represents 50 miles, and two cities are 3.5 inches apart on the map, the proportion is 1/50 = 3.5/D, giving D = 175 miles. If you accidentally write 1/3.5 = 50/D, you'll get the wrong answer.
Simplifying a ratio is related but different. To simplify 12:18, you find the greatest common divisor of 12 and 18, which is 6, and divide both sides by it: 12/6 : 18/6 = 2:3. The simplified ratio 2:3 contains the same information as 12:18 but in its most compact form. Think of it like reducing a fraction.
Ratios in Cooking and Recipes
Cooking is probably where most people encounter ratios in daily life, even if they don't think of it that way. Professional bakers swear by ratios rather than fixed measurements because ratios scale perfectly. Want to double a recipe? Triple it? Make one-third of it? If you understand the ratio, the math is trivial.
The classic vinaigrette is a 3:1 ratio of oil to vinegar. That means 3 tablespoons of oil with 1 tablespoon of vinegar, or 3/4 cup of oil with 1/4 cup of vinegar, or any other combination that keeps the same relationship. The actual amounts depend on how much dressing you need, but the flavor balance stays consistent.
Bread dough follows what bakers call the "baker's percentage," which is really a ratio system. Everything is measured relative to the flour. A basic bread might be 100:65:2:2 for flour:water:salt:yeast. If you're using 500 grams of flour, you'd need 325 grams of water (65% of 500), 10 grams of salt, and 10 grams of yeast. Professional bakeries can scale from a home batch to a commercial batch instantly because the ratios don't change.
Rice is another good example. Most long-grain white rice cooks at a 2:1 ratio of water to rice. Brown rice needs more water, typically 2.5:1. Sushi rice is about 1.2:1. Knowing these ratios means you never have to look up a recipe — you just measure the rice, then measure the right proportion of water.
Cocktails are ratio-based too. A classic margarita is 2:1:1 — two parts tequila, one part lime juice, one part triple sec. Scale it up for a party pitcher by multiplying each part by the same number, and the drink tastes identical to a single serving.
Ratios in Scale Drawings and Maps
Scale drawings and maps rely entirely on proportions to represent large objects or distances in a compact format. When an architect draws a floor plan at 1:50 scale, every centimeter on the drawing represents 50 centimeters (half a meter) in real life. A room that's 4 meters wide appears as 8 centimeters on the plan.
Map scales work the same way. A scale of 1:100,000 means 1 centimeter on the map equals 100,000 centimeters (1 kilometer) in reality. If two towns are 7.3 cm apart on the map, they're 73 kilometers apart on the ground. GPS has made paper maps less common, but the concept of proportional representation still underlies every digital map's zoom levels.
Model builders use ratios constantly. A 1:72 scale model airplane means every dimension is 1/72 of the real thing. If the actual aircraft has a wingspan of 36 meters, the model's wingspan is 36/72 = 0.5 meters, or 50 centimeters. Different modeling communities have standardized on specific scales: 1:87 for HO-scale model trains, 1:24 for large die-cast cars, 1:48 for military aircraft models.
Engineering drawings typically note the scale in their title blocks. Structural plans might be 1:100 or 1:200 for overview drawings, then 1:10 or 1:5 for detail sections showing how specific connections work. Every dimension on the drawing is a proportion of the real-world measurement, and construction workers use scale rulers that automatically convert drawn lengths to actual lengths.
Photography uses ratios for aspect ratios — the width-to-height relationship of an image. A 3:2 ratio is standard for full-frame cameras, 4:3 for most smartphones, and 16:9 for widescreen video. When you crop a photo from 3:2 to 16:9, you're changing the proportion, which is why some of the image gets cut off at the top and bottom.
Cross Multiplication
A/B = C/D → A × D = B × C
A proportion states that two ratios are equal. Cross multiplication converts this into a simple equation: A times D equals B times C. When one value is unknown, you solve for it by rearranging. For example, if A/B = C/D and D is unknown, then D = (B × C) / A. The simplified ratio is found by dividing both parts by their greatest common divisor (GCD). The scale factor is the multiplier that converts one ratio to its equivalent.
Where:
- A = First value of the first ratio
- B = Second value of the first ratio
- C = First value of the second ratio
- D = Second value of the second ratio
Example Calculations
Solving for the Fourth Value
Finding D when A:B = C:D, given 3:4 = 6:?
Using cross multiplication: 3/4 = 6/D, so 3D = 24, giving D = 8. The scale factor from 3:4 to 6:8 is 2, meaning both values were multiplied by 2. The ratio 3:4 is already in simplest form since the GCD of 3 and 4 is 1.
Solving for the First Value
Finding A when ?:5 = 12:20
Cross multiplying: A/5 = 12/20, so 20A = 60, giving A = 3. Checking: 3/5 = 0.6 and 12/20 = 0.6 confirms the proportion is correct. The scale factor from 3:5 to 12:20 is 4.
Frequently Asked Questions
The calculator needs exactly three known values to solve for the fourth. If two or more values are zero or missing, there isn't enough information to determine a unique solution. Enter three non-zero values and set the unknown to zero to solve the proportion.
Yes. While ratios are often written with whole numbers for simplicity, there's nothing wrong with ratios like 2.5:3.7 or 1:1.618 (the golden ratio). The calculator handles decimals in all four positions. If you want a whole-number equivalent, you can multiply both parts by the same number to eliminate decimals. For instance, 2.5:3.5 is the same as 5:7.
A ratio compares two quantities of the same kind, like 3 apples to 5 apples. A rate compares two quantities of different kinds, like 60 miles per 1 hour. Rates include units (miles per hour, dollars per pound), while pure ratios are unitless. Proportions work with both: if gas costs $3.50 per gallon and you need 12 gallons, the proportion $3.50/1 = x/12 gives x = $42.
To simplify a ratio, divide both numbers by their greatest common divisor (GCD). For 24:36, the GCD is 12, so the simplified ratio is 2:3. For ratios with decimals, first multiply both numbers by a power of 10 to make them whole numbers, then find the GCD. For example, 1.5:2.5 becomes 15:25 (multiply by 10), and the GCD of 15 and 25 is 5, giving a simplified ratio of 3:5.
Mathematically, yes. A ratio like -3:4 is valid and represents an inverse relationship. In practical applications, negative ratios are less common since most real-world quantities (lengths, counts, prices) are positive. However, they appear in contexts like financial gain-to-loss ratios or direction changes. This calculator handles negative values in proportions.