# Fraction Calculator

This fraction calculator can be used to add, subtract, multiply, and divide both fractions and mixed numbers. To calculate mixed numbers, click the "Calculate Mixed Numbers" checkbox. The numerator and denominator inputs accept integers only. When calculating mixed numbers, decimals can be used as long as the numerator and denominator are blank. The decimals will be converted to fractions after calculation.

The answers are fractions in the lowest terms or mixed numbers in reduced form. The process used to find the answer is also provided.

## What is a fraction?

A fraction represents a number of equal parts of a whole, and is made up of a numerator and a denominator:

The numerator is the top number (1) of the fraction while the denominator is the bottom number (8). The line separating the numerator and denominator in a fraction is referred to as a fraction bar, and indicates division. The above fraction is read as "one eighth," meaning that it represents 1 equal part of eight total parts, such as 1 slice of pizza for a pizza that is cut into 8 slices. If you eat 3 slices of pizza, then you've eaten 3 parts of the whole pizza, or using fractions, of the pizza. Fractions are an effective way of representing parts of a whole using numbers, rather than having to describe the concept using words.

Fractions can be categorized in a number of different ways. For example, is both a simple fraction and a proper fraction. It is a proper fraction because the numerator is smaller than the denominator, and is a simple fraction because both numerator and denominator are whole numbers.

What are proper fractions, improper fractions, and mixed numbers?

### Proper fractions

Fractions represent a part of a whole, so it makes sense then that "proper" fractions are fractions where the numerator is smaller than the denominator. The numerator represents the number of equal parts of a whole, while the denominator represents the whole. A proper fraction therefore always has a value smaller than 1. Using the example of a pizza that is cut into 8 slices, slices is considered a proper fraction, but (or 1) is not a proper fraction because the value is equal to 1, and 8 parts out of 8 represents the whole pizza, rather than a part of the whole.

### Improper fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For an improper fraction, the fraction represents more than the whole. Using the example of a pizza that is cut into 8 slices, we can think of an improper fraction as having more than 1 whole pizza; if we have slices of pizza, then we have 1 whole pizza, but if we have slices of pizza, we have 1 whole pizza and 1 part out of 8 of a 2^{nd} pizza. Improper fractions are an effective way to express parts of a whole in a way that tells us the size of the whole while also allowing us to represent values larger than 1.

### Mixed numbers

Mixed numbers (or mixed fractions) are another way to represent improper fractions. Rather than writing an improper fraction where the numerator is larger than the denominator, mixed numbers are written as a combination of a whole number and a proper fraction. For example, 1 whole pizza and 1 additional slice of pizza is represented as as an improper fraction; to write this as a mixed number, count the number of wholes and the number of parts of a whole. In this case, there is 1 whole, or , and 1 part of the whole, or . Combining the two, the improper fraction is written as as a mixed number.

#### Converting mixed numbers to improper fractions

To convert a mixed number to an improper fraction, multiply the denominator by the whole number, then add the numerator. For example:

#### Converting improper fractions to mixed numbers

To convert an improper fraction to a mixed number, divide the numerator by the denominator and write the whole number followed by the remainder over the denominator. For example:

## Fraction addition

To add fractions, the denominators of the fractions must be the same. This is because fractions are used to represent parts of a whole, where the denominator tells us how many parts make up the whole, and the numerator tells us how many parts of the whole we have. It makes sense then that fractions cannot be added if the denominators of the fractions are different, since having 1 part of a whole made up of 8 parts is different from having 1 part of a whole made up of 4 parts.

### When the denominators are the same

If the denominators of the fractions being added are the same, add the numerators and write the result over the shared denominator. For example:

This is true regardless whether the fraction is a proper fraction or an improper fraction. For example:

In this case, one of the fractions being added is an improper fraction, so the result is also an improper fraction. Notice however, that the fractions are still added in the same way. As long as the denominators are the same, fraction addition is relatively simple.

### When the denominators are not the same

When the denominators are not the same, convert the fractions so that the fractions all have the same denominator, then add the numerators and write the result over the shared denominator.

The process of converting a set of fractions to fractions with a shared denominator is referred to as finding a common denominator. There are an infinite number of ways to represent the same value in fraction form. For example:

All of the fractions above represent the same value (half of the whole) but each has a different numerator and denominator. is the most simplified form of the fraction that represents half of a whole. Each subsequent fraction listed above is simply the first fraction, , multiplied by a fractional representation of the number 1. Recall that any value multiplied by 1 is equal to that same value, and that any fraction where the numerator and the denominator have the same value is equal to 1. These facts make it possible to rewrite any fraction into a fraction that has the same value represented by different numerators and denominators; these fractions are referred to as equivalent fractions. For example, noting that fraction multiplication involves multiplying the numerator by the numerator and the denominator by the denominator,

and so on. Thus, one way to find a common denominator between two fractions is to simply multiply their denominators, remembering to also multiply the numerators by the appropriate denominator so that the values of the fractions don't change. For example:

In this case, 32 is a common denominator. We can always find a common denominator by simply multiplying each different denominator, but as we can see from the example above, this does not always result in a fraction in simplest form. For this example, we could have left the alone if we recognized that 8 is a common denominator for the two fractions:

8, in this example, is referred to as the least common denominator. If we find the least common denominator of two or more fractions, we may not have to simplify the result.

## Fraction subtraction

Like fraction addition, subtracting fractions requires that the denominators of the fractions be the same. Refer to the fraction addition section for more detail.

### When the denominators are the same

When the denominators of the fractions are the same, subtract the numerators of the fractions, then write the result over the shared denominator and simplify:

### When the denominators are not the same

When the denominators of the fractions are not the same, convert the fractions to equivalent fractions with a common denominator, then subtract their numerators and write the result over the common denominator. There are a number of different ways to find a common denominator. The most straightforward way to find a common denominator is to multiply the denominators of the fractions involved, but this method often results in a fraction that is not in simplest form. For example, suppose that we want to subtract the following:

The numerators cannot be subtracted yet since the denominators of the fractions are not the same. Multiplying 4×8=32 provides us with a common denominator for the fractions. To convert the fractions to equivalent fractions with this common denominator, multiply the numerator and denominator of each fraction by the denominator of the other fraction:

Now that the fractions have the same denominator, the numerators can be subtracted and the fractions simplified as follows:

It is also possible to find the least common denominator, but methods for doing so will not be discussed here. In this case, we could have noticed that 8 is the least common denominator between the two fractions, and could have instead subtracted the fractions by converting into an equivalent fraction with a denominator of 8:

## Fraction multiplication

Fraction multiplication is much the same as regular multiplication; simply find the product of the numerators of the fractions being multiplied and write the result over the product of the denominators. The number of fractions being multiplied doesn't change the process. For example:

To multiply mixed numbers, first convert them to improper fractions, then multiply:

When multiplying two mixed numbers, be careful not to multiply the whole number portions and fraction portions separately. It is necessary to convert the mixed numbers to improper fractions first.

## Fraction division

Dividing fractions is similar to multiplying fractions with the exception that we multiply the fraction being divided by what is referred to as the reciprocal of the other fraction. The reciprocal of a fraction is simply that fraction with the positions of the numerator and denominator switched. For example, the reciprocal of is . Thus, to divide two fractions, just find the reciprocal of the divisor, then multiply the numerator by the numerator and the denominator by the denominator. For example:

## Fraction simplification

Fraction simplification involves finding the form of a fraction where all shared factors of the numerator and denominator have been factored out such that the only common factor between the numerator and denominator is 1; this final form of a fraction is referred to as its simplest form. This process is also sometimes referred to as reducing the fraction.

One way to simplify fractions is to divide the numerator and denominator by shared prime factors (or their multiples), proceeding from the first prime number, 2, through each subsequent prime number until the fraction can no longer be reduced. This can be tedious depending on the fraction, but is a sure and straightforward way to reduce the fraction. For example, given that we want to simplify , we know that 78 and 156 share a factor of 2 because they are both even, so we can start by dividing both by 2:

The next prime number is 3, and in this case both 39 and 78 share this factor, so divide both by 3:

At this point, the next prime number that 13 and 26 share is 13, so divide them both by 13:

Since the only shared factor at this point is 1, the fraction cannot be reduced further, and is therefore in simplest form. Even though the numerator and denominator in this example are relatively small, we see that it can quickly get tedious to reduce fractions using this process.

Another more efficient way to simplify fractions involves finding the greatest common factor (GCF), though the process for finding the GCF won't be discussed here in detail. Briefly, we may have recognized that 78 is the GCF of 78 and 156, and we could immediately have divided both by 78 to arrive at the simplest form. Thus, if we can identify the GCF of the numerator and denominator, simplifying fractions just involves dividing both numerator and denominator by their GCF.

## Converting between fractions and decimals

Fractions and decimals are two different ways to express the same value. Depending on the situation, we may prefer to express a value in either fraction or decimal form, so it is useful to be able to convert between the two.

### Converting decimals to fractions

Converting from a decimal to a fraction is relatively simple as long as we understand what decimals and place values are. Briefly, each position in a decimal corresponds to a power of 10. 0.5 for example, is read as "five tenths," because the 5 is in the 10ths place, which corresponds to 10^{-1}. 0.05 is read as "five hundredths," and corresponds to 10^{-2}. 0.005 is read as "five thousandths," and corresponds to 10^{-3}, and so on.

To convert from a decimal to a fraction, shift the decimal point to the right, keeping track of how many times the point is shifted, until it is placed after the last non-zero value (furthest to the right). The number should then be a whole number, and this number is the numerator of the fraction. The denominator of the fraction is 10 raised to the power of however many times we shifted the decimal point. For 0.5, we would have to shift the decimal place one time to get a whole number of 5. Thus, the numerator is 5, and the denominator is 10^{1} since we shifted the decimal point once. 0.5 in fraction form is therefore:

If the decimal also has a whole number portion, such as 1.5, the process is the same, except that the value before the decimal point will be the whole number portion of the mixed number. Thus, 1.5 in mixed number form is:

### Converting fractions to decimals

Converting fractions to decimals is generally more difficult than converting decimals to fractions. In cases where we can convert the fraction into an equivalent fraction with a denominator that is a power of 10, we can use the reverse of the logic used to convert decimals to fractions (see above). For example:

In many cases however, this is not possible, and converting from fractions to decimals is a tedious process that requires long division. Even then, many fractions may not have a terminating decimal. For example, in decimal form is 0.3333..., where the "..." indicates that the 3 repeats indefinitely. Other values, such as π, simply never end and have no repeating values. Thus, if we have to resort to long division, the decimal form of the fraction will often be an estimate.

To convert a fraction using long division, divide the numerator by the denominator, keeping the decimal point in the same position in the quotient as it is in the numerator. That's all there is to it, but ideally this should be done using a computer or a calculator, since converting fractions in this way is very tedious.