Ratio

In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.

A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a:b", or by giving just the value of their quotient ⁠_a_/_b_⁠. Equal quotients correspond to equal ratios. A statement expressing the equality of two ratios is called a proportion.

Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers.

A more specific definition adopted in physical sciences (especially in metrology) for ratio is the dimensionless quotient between two physical quantities measured with the same unit. A quotient of two quantities that are measured with different units may be called a rate.

Notation and terminology

A ratio shows how many times one number fits into another. For example, if you have 8 oranges and 6 lemons, the ratio of oranges to lemons is 8 to 6, which can also be written as 8:6 or as the fraction 8 divided by 6.

When we write a ratio like 8:6, the numbers 8 and 6 are called the terms of the ratio. We can also say that 8 is to 6 as 4 is to 3, which is a way of showing that the ratios are equal. Ratios can also have more than two terms, like thickness:width:length = 2:4:10 for a piece of wood. This means the thickness, width, and length all relate to each other in the same way.

History and etymology

The word "ratio" comes from ancient Greek, where it was called logos. Early translators turned this into the Latin word ratio, meaning "reason." Later thinkers used the word proportio to talk about ratios.

Euclid, a famous ancient mathematician, wrote about ratios in his work The Elements. He defined ratios as ways to compare two amounts of the same kind, like two lengths or two areas. He also explained how to tell when two ratios are equal. These ideas helped shape how we understand ratios today.

Number of terms and use of fractions

Ratios can be written as fractions. For example, a ratio of 2:3 means there are 2 parts of one thing for every 3 parts of another. This can be written as the fraction 2/3.

If we have 2 oranges and 3 apples, the ratio of oranges to apples is 2:3. We can also say this as a fraction: there are 2/3 as many oranges as there are apples. Ratios and fractions help us compare amounts clearly, and it’s important to know what we are comparing.

Proportions and percentage ratios

When we change all numbers in a ratio by the same amount, the ratio stays the same. For example, the ratio 3:2 is also 12:8. We can make ratios easier by using the smallest numbers possible or by showing them as parts out of 100, called percentages.

Imagine a mix with four things—let’s call them A, B, C, and D—in the ratio 5:9:4:2. This means for every 5 parts of A, there are 9 parts of B, 4 parts of C, and 2 parts of D. All together, there are 20 parts (5 + 9 + 4 + 2). So, A is 5 out of 20, B is 9 out of 20, C is 4 out of 20, and D is 2 out of 20. If we turn these into percentages, A is 25%, B is 45%, C is 20%, and D is 10%.

A proportion compares a part to the whole. For example, if a fruit basket has 2 apples and 3 oranges, the whole basket has 5 fruits. The apples are 2 out of 5, or 40% of the basket, and the oranges are 3 out of 5, or 60%.

Ratios with two numbers can be shown as fractions. Old televisions had a ratio of 4:3, meaning the width is 4/3 of the height. Newer widescreen TVs use a 16:9 ratio. Showing ratios as decimals makes it easier to compare them. For example, 1.33, 1.78, and 2.35 clearly show which TV format is wider. This works best when we always compare the same thing, like width to height.

Main article: aspect ratio

Reduction

Ratios can be made simpler, just like fractions. For example, the ratio 40:60 can be reduced to 2:3 by dividing both numbers by 20. This means 40 is to 60 as 2 is to 3.

A ratio is in its simplest form when the numbers can't be reduced any further. Sometimes, ratios are written with one number as 1, like 1:1.25 for the ratio 4:5. This helps compare different ratios more easily.

Irrational ratios

Ratios can also be made between things that don’t fit into simple fractions. One famous example is the ratio of the diagonal to the side of a square, which is the square root of 2. Another is the ratio of a circle’s circumference to its diameter, called π, which is also special.

There is also the golden ratio, which appears in art and nature. It is a special ratio between two lengths where one part is to the whole as the whole is to the larger part. Even though we can use whole numbers to get close, the true golden ratio needs at least one number that isn’t a whole number to be exact. It shows up, for example, when you look at the ratios of numbers in the Fibonacci sequence.

Odds

Main article: Odds

Odds are a way to show the chance of something happening, using a ratio. For example, if the odds are "7 to 3 against," this means there are seven chances the event will not happen for every three chances it will happen. This means there is a 30% chance of success. In ten tries, you would expect to win three times and not win seven times.

Units

Ratios can sometimes not have units. This happens when they compare amounts that are the same kind, even if they start with different units. For example, the ratio of one minute to 40 seconds can be changed by turning one minute into 60 seconds. Then the ratio becomes 60 seconds to 40 seconds, which simplifies to 3:2 once the units are the same and can be left out.

However, some ratios do have units. These are called rates. In chemistry, for example, mass concentration ratios are often shown as weight to volume fractions. A concentration of 3% w/v means there are 3 grams of a substance in every 100 milliliters of solution. This cannot be turned into a ratio without units, unlike ratios that compare weight to weight or volume to volume.

Triangular coordinates

The locations of points related to a triangle with points A, B, and C can be shown using special ratios called triangular coordinates.

In barycentric coordinates, a point is found by thinking of a weightless sheet shaped like the triangle. If we place weights at points A, B, and C, the ratios of these weights tell us where the point is. For example, the ratio of weights at A to B might be α : β, and at B to C might be β : γ.

In trilinear coordinates, we look at how far a point is from the sides of the triangle. The distances from the point to each side are in certain ratios, helping us find the point’s location. Because these coordinates use ratios, they work for triangles of any size.

vertices barycentric coordinates trilinear coordinates perpendicular