 # Fraction Operations

## Add, subtract, multiply, and divide fractions

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What are fractions?

You may not realize it, but we use fractions every single day. When you go to the movies, the second you leave the house to meet your friend you are coming into contact with fractions. You'll probably text your friend to let them know when you're half(1/2) -way there. By saying that, your buddy can know precisely what time he or she has to meet you.

Even time gets broken down into fractions. When you pick a 7:15 movie, you might tell your friend that the show starts a quarter past seven. By the time the film is done, it'll be a quarter to ten. As you look at your clock or talk about the time, you inevitably use fractions.

When it comes to fractions, it’s best to think of these numbers as part of a whole set. Any value that is less than the whole becomes a fraction. Believe it or not, there are parts of a whole in everything we encounter. It’s actually far more common to come into contact with parts of a whole, than complete sets of any one thing.

Now that you’re at the movies, your friend wants to get a whole pepperoni pizza. Since you’re expecting two more friends, the pizza should be split up so that everyone gets a fair piece. Without even thinking about it, your instinct is likely to split the savory pie into 4 equal slices. Each slice would represent one-fourth of the whole or 1 over 4 with a dividing line in between.

Even the total cost of the pizza could be broken down into a fraction when the price doesn't fall exactly on the dollar. Say you paid \$8.30 for the pepperoni pizza; your payment can be a fraction written as 8 and 1/3rd of a dollar. The numbers in the fraction can move up or down depending on how many times the set is split up.

As an example, if more of your friends showed up, the pizza could get portioned into 1/8ths or even 1/16ths. But in that case, you would probably just opt for another pizza, increasing your total food expenses to \$16.60 or 16 and 3/5ths of a dollar.

Components of a fraction

Each part of the fraction tells you something about the set. Take a look at the plate of cookies below. Each plate has white chocolate chip and double fudge cookies. Using fractions, can you tell which portions of the cookies have white chocolate chips? The correct answer is 3/10. Three out of the ten cookies have white chocolate chips.

The top number, called the numerator, tells you the number of cookies that have white chocolate chips. The bottom number, called the denominator, determines how many cookies make up the whole set. This can be hard to remember but just think of the ‘down’ number as the ‘down-ominator’, and as a result, the ’denominator’. Take a look at this next example. Here you’ll even notice that some fractions may look different but are really the same. Sometimes you’ll end up in a situation where a fraction can be reduced to smaller numbers. In the instance of these two plates, you could scale down each fraction to 1/2. After all, half of the cookies are white chocolate chip, while the other half is double fudge – regardless of the total number of cookies.

It’s always easier to work with numbers when they are smaller, versus larger. Other times, the fraction might be scaled down by three, five, or ten. In the end, it means the same thing but will be much easier to add, subtract, multiply, or divide.

Adding and subtracting fractions

Like whole numbers and decimal numbers, fractions too can be used in various operations. You can add, subtract, multiply and divide fractions. In this section, we'll review how to add and subtract fractions when they have the same base number, plus what to do when they don’t.

• When the denominator is the same

Adding and subtracting fractions with the same base is very straightforward. All you have to do is find the sum or difference of the numerators (top numbers). In the above example, 4/7th plus 1/7th equals 5/7ths of a pizza. If you were to subtract, the answer would change to 3/7ths of a pizza (4-1 = 3).

• When the fractions have different bases

The course of action changes when you have fractions with different denominators. Take a look at the chocolate bars below. When the bars are different sizes, it can be tricky to figure out how many equal pieces of chocolate you’ll end up with. To add or subtract these fractions you will need to make sure the bases are the same. You can do this by finding a common denominator or using the lowest common denominator.

The simplest solution is to multiply the bases together to find a common denominator. Using the above example, you can get a common denominator of 16, because 2 x 8 equals 16. So instead of a chocolate bar having 2 or 8 squares to make up the whole, they will both have up to 16 squares. You’ll also have to adjust the top numbers the same way you altered the bases, and then find the sum (or difference) of those numbers. In the above example, 8/16ths plus 10/16ths equals 18/16ths, or 1 and 2/16ths of a chocolate bar. The result is a large fraction that requires you to reduce your answer further. Luckily, there is another way to get to the correct answer in fewer steps.

In many instances, two denominators can share a common base that results in fewer partitions. When you identify the smallest number that both fractions multiply into, you can save time from unnecessary steps. Let’s write out the multipliers of each denominator to find the lowest common denominator.

• Multiples of two: 2, 4, 6, 8, 10, 12, 16
• Multiples of eight: 8, 16, 24, 32

What’s the smallest number that they both share? The answer is 8, which means you want both chocolate bars to have 8 squares. When you multiply the top and bottom of ½ by 4, you get 4/8ths. The other chocolate bar is already broken up into 8 squares, so you can leave it as is. Now that the bases match, you can add (or subtract) the fractions. In the above example, 4/8ths plus 5/8ths equals 9/8ths, or 1 and 1/8th of a chocolate bar. In the case of subtraction, the result is -1/8th. That means you would owe your friend 1/8th of your chocolate bar next time around.

In the next section, we’ll review the basics of multiplying and dividing fraction numbers. Don’t forget to bookmark this page and save it to the home screen of your smartphone or tablet. You can return as you need to work out fraction operations in day-to-day life. If you found this page useful, please promote us on social media by using the share features.

Multiplying and dividing fractions

When it comes to multiplication and division of fractions, there are specific rules you must follow. Luckily, you won’t have to take any extra steps to find a common denominator as you did with addition and subtraction.

• Multiplying fractions

Fractions are very straightforward to multiply; First, you have to multiply the top numbers (numerators) in the fraction. Next, you need to multiply the bottom numbers (denominators) of the fractions. To finish, you should simplify your final fraction as required. In the above example, 4/7 multiplied by ½ results in 2/7ths. You’ll probably notice that each time you multiply a fraction, it makes more, smaller pieces - this is perfectly normal.

If any of this sounds confusing, just remember this rhyme: Multiply top by top, bottom by bottom, simplify and you got 'em!

• Dividing fractions

Dividing fractions is also very easy; you just have to follow three simple steps. To start, you have to flip the second fraction (the divisor) by swapping the numerator with the denominator. After you change the divisor to its reciprocal form, you can follow the rules of multiplication: Multiply top by top and bottom by bottom. If needed, simplify the resulting fraction. In the above example, 4/7th divided by ½ equals 1 and 1/7th. As you can see, it is customary to end up with a greater number after division.

If you ever get confused, just remember: To divide, take the second to the flipside. Multiply the tops and bottoms. Then simplify, and you got 'em!

Once you get the hang of the rules, fraction operations are pretty easy to work through. As you practice, your skills will continue to get better. It can be helpful to check your work as you are learning, or even to verify numbers you already confident about. You can do this easily using the calculator on this page. Just plug in your fractions, select the operation, and solve for your answer with one click.