Ratios and proportions are related concepts in mathematics and statistics. A ratio compares two quantities and shows their relative sizes. For example, a veterinary office might have 1 dog to 2 cats as patients. This comparison tells us that they see twice as many cats as dogs.

A ratio expresses the relationship between quantities using the form of a:b, where a and b are any two non-zero integers. You state the colon (:) as the word “to.” Statisticians refer to this as a between parts comparison because we’re directly comparing the numbers of each quantity.

You can write ratios using the following standard formats:

- 1 to 2
- 1:2
- 1/2

These forms are equivalent. The key is that you always say 1 to 2 for all these forms. That’s crucial for the fractional format because it distinguishes them from regular fractions. More on that later!

The following illustration shows the concept of a 1 to 3 ratio. There is one blue rectangle for every three red rectangles. It’s equally valid to say that it is 3:1 red to blue rectangles.

Ratios can involve more than two quantities. When you have more than two quantities, always use the colon (:) format. For example, 2:1:4 can indicate two parts vinegar to one part wine to four parts water.

## Simplifying Ratios

Ratios help us express relationships between quantities in a way that makes them easier to understand. They communicate information in a concise and meaningful way. Simplifying ratios is critical to this process.

When working with ratios, it’s important to simplify them down to the smallest whole numbers possible. Find the greatest common factor between the two numbers and divide them accordingly. As you’ll learn in the next section, simplified ratios are closely related to the concept of proportion.

For instance, the veterinary office with 1:2 dogs to cats presumably does not have only one dog and two cats. It is a simplified ratio.

The vets might actually have 36 dogs and 72 cats. 36:72 can be simplified to 1:2 by dividing each number by 36.

For example, consider 15:25. Both 15 and 25 can be divided by 5, the greatest common factor between them. This division simplifies it to 3:5. The simplified form represents the same relationship between the quantities as the original ratio, but it is easier to work with and understand.

## Proportion Definition

A proportion states that two ratios are equal.

In other words, if we have a:b and c:d, they are proportional if a/b = c/d. Proportionality means that the relationship between the quantities in the first ratio is equivalent to the relationship between those in the second.

Express a proportion using the notation—a:b::c:d

The double colon (::) indicates the proportions are equal: a is to b as c is to d.

You can think of proportion as scaling up from the simplest ratio. For example, 1:3 is equivalent to 2:6, as shown below. They have the same relationship and are in proportion.

Whenever you multiply or divide both sides by the same number, they’re proportional.

1 * 2: 3 * 2 = 2:6. Similarly, 3:9 and 4:12 are equivalent by multiplying both sides of 1:3 by three and four, respectively. All the ratios are in proportion.

Alternatively, ratios are proportional if you can simplify them to the same one.

Suppose we have 4:6 and 8:12. Both simplify to 2:3. Hence, they are in proportion.

For using the idea of proportion in real life, think about recipes and chemical mixtures that you might want to scale up to make more. Or scaling different screen sizes while retaining the same 16:9 aspect ratio.

## Converting A Ratio to a Percentage

Ratios are related to but different from your standard fractions and percentages. Remember that most ratios are part-to-part comparisons, while the typical percentage is a part-to-whole comparison.

To convert a ratio to a percentage, you first need to construct the appropriate fraction by doing the following:

- Sum its parts to obtain the fraction’s denominator representing the whole.
- Use each part individually for the numerator of the fraction.

This process leaves you with a part in the numerator and the total in the denominator for part-to-whole comparisons. For a:b in general terms, the numerator is either a or b, and the denominator is a+b, as shown below:

Going back to the 1:2 dog-to-cat ratio, if you want to calculate the percentage of dogs and cats out of the whole population, you’ll need to calculate the whole and then divide by the part.

For the denominator, we need to sum the two parts (1 + 2), then use the numerator 1 for dogs and 2 for cats, as shown below.

When you have a 1:2 dogs to cats, dogs comprise 33% of the whole, while cats constitute 67%.

## Special Ratios

There are several special uses for them.

Pi (π) is the ratio of a circle’s circumference (c) to its diameter (d)—c:d.

Odds in probability theory is the ratio that compares the probability of an event happening to the probability it won’t happen or the number of ways an event occurs vs. the number of ways it doesn’t.

Hazard ratios are an effect size that assesses time to an event.

The golden ratio has defined beauty in arts and architecture since ancient times. When the following mathematical expression is true for a:b, it fits the golden ratio (Wikipedia).

## Comments and Questions