Ratio Calculator

This ratio calculator solves ratio problems of the form A:B = C:D. Depending on the values entered, the calculator either solves for the missing value, determines whether a proportion is true or false, reduces a ratio to lowest terms, or lists some equivalent ratios for a ratio that is input in lowest terms. The aspect ratio function of the calculator lists equivalent ratios for various aspect ratios and also provides an image that demonstrates what a screen would look like at the given aspect ratio. To use the calculator, please provide two, three, or four values and click the "Calculate" button.

What is a ratio?

A ratio is a relation between two quantities that indicates the number of times that one quantity contains another. For example, if there are 5 dogs and 3 cats, the ratio of dogs to cats is 5 to 3. In other words, there are 5 dogs for every 3 cats. Ratios can be expressed in a number of different ways. Using A and B to represent two different quantities:

-  The ratio of A to B
-  A:B

-

A B

Note that the ratio of A to B is not the same as the ratio of B to A. Referencing the example above with 5 dogs and 3 cats, the ratio of dogs to cats is not the same as the ratio of cats to dogs. The ratio of dogs to cats is 5:3, while the ratio of cats to dogs is 3:5. Thus, when writing ratios, it is important to be sure of which ratio we are trying to represent.

A statement of the equality of two ratios is referred to as a proportion. Given that A, B, C, and D are four different quantities, the following is an example of a proportion,

A:B = C:D

and can be read as "A is to B as C is to D." Ratios and proportions have many practical uses. One such example is the use of ratios and proportions in baking. For example, a recipe that makes 30 cookies may call for 2 eggs and 4 cups of flour. If we wanted to make half as many cookies or twice as many cookies, we could use proportions to determine how many eggs and cups of flour are needed in either case. To determine how many eggs and cups of flour we need to make half the number of cookies, divide the cups of flour and number of eggs by 2:

Similarly, the proportion of eggs and flour needed to make double the number of cookies as the original recipe is as follows:

Thus, to make half the number of cookies, or 15 cookies, we would use a ratio of 1 egg : 2 cups of flour; to make double the number of cookies, or 60 cookies, we would use a ratio of 4 eggs : 8 cups of flour. We can similarly find equivalent ratios that allow us to make different numbers of cookies relative to the original recipe. This is because as long as we multiply or divide both components of a ratio by the same value, we will find a new ratio that represents the same relation with a different scale. These are referred to as equivalent ratios.

Although the examples above used ratios that involve only two terms for the sake of simplicity, ratios can have more than 2 terms. For example, given that there are 3 cats, 2 dogs, and 7 bunnies in a room, the ratio of animals in the room can be expressed as 3 cats : 2 dogs : 7 bunnies.

Ratio vs. fraction

Although a fraction is one of the ways in which a ratio can be expressed, ratios and fractions are not the same thing. A ratio is a relation between two or more quantities, while a fraction is a value that represents the number of parts in a whole. For example, the ratio indicates the number of cats to dogs. However, even though it is written in the form of a fraction, it does not make sense to think of the ratio as a fraction. If it were a fraction, we would read the fraction as 3 cats out of a total of 2 dogs, which does not make any sense, because cats are not a part of a whole that is made up of dogs.

Thus, although ratios and fractions share similarities in the way they are expressed, and are also reduced to lowest terms in the same manner, it is important to realize that ratios written in fraction form are not the same as fractions. To further illustrate this point, unlike fractions, which can be added and subtracted, it is not possible to add or subtract ratios. For example, 3 cats : 5 dogs - 2 cats : 7 dogs is not a subtraction problem that can be solved. Notice also that the quantities in a ratio can have different units (the units can also be the same), such as cats and dogs. This is not the case for fractions. A fraction represents a part of a whole, so the numerator and denominator of a fraction share the same units.

How to calculate a ratio

Calculating a ratio has a few different meanings. For this calculator, calculating a ratio can mean solving a proportion. Given the proportion

A:B = C:D

It is possible to calculate the ratio of C:D given A:B, or vice versa. It is also possible to calculate any of the other values given 3 of the values. It not possible however, to calculate the ratios if we only have one value on each side, or only one value. For example, if we only know A and C, there is no way to find either B or D. If we only know A, there is no way to find B, C, or D. The same is true for other similar combinations of values.

Calculating one value given 3:

The process for solving for missing values is the same, and can be done by setting up the ratios as equivalent fractions. For example, given 3 values, A, B, and C, we can solve D as follows:

Cross multiplying yields:

3D = 20

This same method can be used to find any of the values, given the other 3.

Calculating equivalent ratios:

If we only know either A and B or C and D, there are an infinite number of possible solutions, since the problem involves finding equivalent ratios. For example, given A = 1 and B = 2,

and there are an infinite number of solutions because we can multiply or divide 1 and 2 by the same factor to find any number of solutions for , such as , and so on.

Calculating the lowest terms of a ratio:

Calculating the lowest terms of a ratio involves the same process as simplifying a fraction. There are a few ways to reduce a ratio to lowest terms, the most straightforward of which is to divide all the terms in a ratio by a shared factor until there are no more shared factors. This can also be accomplished more efficiently by dividing each term by the greatest common divisor (GCD). This of course requires knowledge of how to find a GCD. Refer to the GCD calculator for more information. The GCD method of reducing to lowest terms is what this calculator uses, but the GCD will not be discussed in detail on this page.

As an example of reducing a ratio to lowest terms, consider the ratio .

12 and 19 have no shared factors, so is in lowest terms. In this example, the GCD of 48 and 76 is 4. Given that we know the GCD, we could have skipped one step in the above problem by immediately dividing both 48 and 76 by 4:

Verifying a proportion:

This calculator can also be used to verify a proportion. If 4 values are entered into the calculator, it will determine whether the proportion is true or false. To verify a proportion by hand, compare the two ratios to determine whether or not one ratio can be converted into the other by multiplying or dividing the terms by the same factor. Using the same example above, where we found that is in lowest terms, we know that we can either divide the ratio by a factor of 4, or multiply by a factor of 4 to produce the other ratio. Thus, the two are equivalent ratios, and the proportion is therefore true.

Another way to verify a proportion is to reduce both ratios to lowest terms. If the ratios in lowest terms are equal, then the proportion is valid. For example, consider the proportion

Both of the ratios above reduce to , so the proportion is true. If two ratios reduced to lowest terms are not equal, then the proportion is false. Similarly, if two ratios are given, and it is not possible to multiply or divide one ratio by some factor to convert it into the other ratio, then the proportion is false.

Aspect ratio

An aspect ratio is an everyday application of ratios that typically relates the width and height of a screen such as a TV, monitor, or phone screen. Aspect ratios are typically expressed in the form A:B, where A represents the quantity of width, and B represents the quantity of height. It is important to note that A and B in an aspect ratio are not actual width or height measurements. Rather, they indicate the ratio of width to height of the object being measured.

For example, given an aspect ratio of 16:9, if we knew the width of a given screen were 12 inches, we could find the height of the screen using the aspect ratio:

16D = 108

D = 6.75

Thus, the height of the screen is 6.75 inches. Given 2 values, A:B or C:D, this calculator provides equivalent aspect ratios as well as the factor that relates the corresponding equivalent ratio to the aspect ratio of choice.