# Fractions Calculator

Want to add, subtract, multiply, or divide fractions? Cool, try this fractions calculator.

This fractions calculator adds, subtracts, multiplies, and divides fractions and gives the answer in the form of a fraction (including in mixed fraction form if the answer is an improper fraction). It also provides a detailed breakdown of the method used to get the answer.

### Terminology

The number on top of a fraction is called the "numerator" and the number on the bottom is called the "denominator."

### A Useful Concept

Here's a concept that makes working with fractions much easier once you understand it:

At any time you can change the numbers on the numerator and denominator of a fraction by multiplying it by another fraction that you have completely made up, the only rule being that the fraction you have completely made up must have the same number on both the numerator and denominator.

So, for example, you can at any time multiply a fraction by something like 3 / 3, or 5 / 5, or 9845 / 9845.

Why is it okay to do this? Because any number over top of itself (i.e. divided by itself) is equal to 1. So if you multiply a fraction by 3 / 3, or 5 / 5, or 9845 / 9845, then you are actually just multiplying it by 1, and anything multiplied by 1 is equal to itself, so you haven't really changed the value of the fraction. Another way of looking at it is to note that although the numerator and denominator of your fraction might change, their ratio (i.e. their sizes relative to one another) stays the same because both the numerator and denominator are being multiplied by the same amount, meaning the overall value of the fraction hasn't changed.

Consider the following example:

In this example, we multiply 1 / 2, which we all know is "one half," by 2 / 2. The result is 2 / 4, and yet we all know that 2 / 4 is also "one half," which demonstrates that the overall value of the fraction didn't change, it's now just represented by a different set of numbers.

Okay, but why would you want to do this? Well, it's a necessary tool when you are adding or subtracting fractions because you can't add or subtract fractions until they have the same denominators. So, if we are trying to add or subtract two fractions that have different denominators, the approach shown in the example above allows you to change the denominator of one or both of your fractions to whatever you want without changing the overall value of your fraction(s).

Note that when you multiply one or both of your fractions by some number over itself to make the denominators the same, the number you arrive at that's the same for both denominators is called their "common multiple." Any two numbers will always have lots of common multiples. For example, if we suppose the denominators of your fractions are 3 and 9 then, depending on what you choose to multiply your fractions by, you could end up with any of the following common multiples on the denominator of your fractions:

• 9 (because 3 × 3 = 9, and 9 × 1 = 9)
• 18 (because 3 × 6 = 18, and 9 × 2 = 18)
• 27 (because 3 × 9 = 27, and 9 × 3 = 27)
• 36 (because 3 × 12 = 36, and 9 × 4 = 36)
• and so on...

It doesn't matter which common multiple you get for the denominator the of your fractions, they will all work for adding or subtracting. However, it's always a good idea to try to get the smallest common multiple (also called the least common multiple, or LCM) for your denominator because this will often (although not always) eliminate the need to reduce the fraction later on (see the section below called "How to Reduce Fractions"). So, in the example above where you've got 3 and 9 as the denominators of your fractions, you would ideally want to use 9 as your common multiple because it is the "least common multiple," or LCM, for 3 and 9.

### How to Add or Subtract Fractions

These are the general steps to follow when adding or subtracting fractions:

1. If the denominators of your two fractions are already the same then you can ignore this step and go straight to step 2 below, but if they're not you must first make them the same by using the approach discussed in the section called "A Useful Concept" above (i.e. multiply one or both of your fractions by some number over itself to make the denominators the same).

2. Once you have the denominators the same you can add or subtract the numerators to or from one another the same way you would if they were just two normal, non-fraction numbers. Whatever answer you get from adding or subtracting the numerators goes over your common denominator.

Here’s an example fraction addition problem solved according to the steps described above:

### How to Multiply Fractions

Multiplying fractions is pretty easy. You just multiply the two numerators together, and you multiply the two denominators together. Here’s an example fraction multiplicaton problem:

### How to Divide Fractions

Dividing fractions is also pretty easy. You just invert (i.e. flip upside down) the fraction that you are dividing by, change the division symbol to a multiplication symbol, and multiply. Here’s an example fraction division problem:

### How to Reduce Fractions

Sometimes, when you've finished adding, subtracting, multiplying, or dividing your fractions, you'll end up with an answer like 2 / 4, or 6 / 18, or 21 / 28, etc. These are examples of fractions that can be reduced, which just means that it's possible to show them with smaller numerators and denominators than what you're currently seeing, without changing their overall value. For example:

• 2 / 4 can be reduced to 1 / 2
• 6 / 18 can be reduced to 1 / 3
• 21 / 28 can be reduced to 3 / 4

You should always reduce a fraction so that it's got the lowest possible numerator and denominator, it's just considered good form.

You'll know you can reduce a fraction if both the numerator and denominator numbers can be evenly divided by the same whole number (other than 1). There's often more than one whole number by which you can evenly divide both the numerator and denominator, and if this is the case then you want to find the largest possible one (it's called the greatest common divisor, or GCD) because it will reduce the fraction to the greatest extent possible. When you find such a number then it can be factored out (i.e. removed) from both the numerator and denominator of your fraction, after which it cancels itself out, becomes 1, and can disappear from the equation. Here's an example:

### How To Deal With Negative Signs

Sometimes you'll have a negative sign on both the numerator and denominator of a fraction. When this happens they can be removed by factoring -1 out of both the numerator and denominator. Then the -1's can cancel each other out, become 1, and can disappear from the equation. Here's an example:

If you've got a negative sign on the denominator of a fraction you can move it to the numerator, or vice versa, by multiplying the fraction by -1 / -1. Here's an example:

### Proper, Improper, and Mixed Fractions

#### Proper Fractions

A proper fraction is any fraction in which the numerator is smaller than the denominator. For example:

• 3 / 5 is a proper fraction
• 9 / 15 is a proper fraction
• 873 / 9832 is a proper fraction
• 999 / 1000 is a proper fraction

When you have a proper fraction for your answer there's no need to show it any other way, proper fraction form is the best you can do.

#### Improper Fractions

An improper fraction is any fraction in which the numerator is bigger than the denominator. For example:

• 7 / 5 is an improper fraction
• 19 / 15 is an improper fraction
• 9995 / 9832 is an improper fraction
• 1001 / 1000 is an improper fractions

When you have an improper fraction for your answer it can always be converted to a mixed fraction (although it's not always required, it's simply a matter of what your audience prefers).

#### Mixed Fractions

A mixed fraction is just another way of displaying your answer if your answer is an improper fraction. Some people like seeing improper fractions converted to mixed fractions because they find mixed fractions easier to understand.

A mixed fraction is made up of a whole number (like 1, 2, 8, etc.) combined with a proper fraction (like 1 / 2, 4 / 5, 1 / 999, etc.). Here are some examples of improper fractions and their equivalent mixed fractions:

To convert an improper fraction to a mixed fraction do the following:

1. Divide the numerator of your improper fraction by the denominator. The whole number portion of the result will be the whole number portion of your mixed fraction. So, if your improper fraction is 14 / 5, then 14 ÷ 5 = 2.8 and so 2 will be the whole number portion of your mixed fraction.

2. Multiply the whole number portion of your mixed fraction by the denominator of your improper fraction and subtract what you get from the numerator of your improper fraction, the result will be the numerator of the proper fraction portion of your mixed fraction. So, 14 - (2 × 5) = 4, which means that 4 is the numerator of the proper fraction portion of your mixed fraction.

3. Keep the denominator of the proper fraction portion of your mixed fraction the same as the denominator of the improper fraction that you started with.

Note that you don't need to worry about whether your fractions are proper, improper, or mixed until you've arrived at the end of your calculation and have an answer in the form of a single fraction. Only then should you think about what type it is (proper or improper) and, if it's improper, whether or not you want to also show it as a mixed fraction.