Fractions

What is a Fraction?

A fraction is a number that represents parts of a whole.

Consider, for example, a pizza that is divided into 8 pieces (to the right).

Anatomy of a Fraction

The denominator of a fraction is the number on the bottom. The denominator tells you how many parts the whole is broken into. For the above example, the denominator is 8 because the pizza is divided into 8 slices.

The denominator of a fraction can be any integer except for 0. An integer is just a number that doesn’t have a fractional component. For example, 1 or -82737372 are integers, whereas 2.2 or 492.2 is not.

The numerator of a fraction tells you how many parts of the whole we are considering. In the example above, the numerator for the fraction representing the pizza that George ate is 2 because he ate 2 out of 8 slices.

The numerator of a fraction can be any integer, including 0.

Types of Fractions

Fractions on a Calculator

Computing fractions on a calculator is easy. The line separating the numerator and denominator means “divided by” so you just have to divide the top number by the bottom number.

For a proper fraction, you will get a decimal number less than 1 (ex. 4/5 = 0.8). For an improper fraction, you will get a number that is bigger than 1 (4/2 = 2). If two fractions are equivalent, you will get the same number when you compute each fraction.

Comparing Fractions with the Same Denominator

Let’s consider the following fractions:

Imagine you are really hungry, and your mom has baked a pie that is divided into 4 slices. You would obviously rather eat 4/4 slices (which would be the whole pie), over just 1 of the 4 slices. So, to order these fractions from least to greatest, we have:

From this, we can see that for fractions with the same denominator, the fraction with the bigger numerator will be largest.

Comparing Fractions with the Same Numerator

For all of the fractions below, the numerator is the same. So let’s look at the denominators.

Let’s say you are hungry. Would you rather 1/2 of the pizza? Or 1/4 of the pizza? You’d want 1/2 of course - because 1/2 is more!

So if we wanted to order these fractions from least to greatest based on this logic, it would look like this:

This makes sense because the denominator (the number on the bottom) represents the number of pieces or parts of a whole.

For 1/16, you would have a chocolate bar that is divided into 16 pieces and you could eat one piece. But for 1/3, you would have a chocolate bar divided into 3 pieces and you could eat one piece.

From this, we can see that for fractions with the same numerator, the fraction with the smaller denominator will be greater.

Comparing Fractions

Let’s say we want to compare the following fractions:

We can’t compare them in the same ways as we did above, because they don’t have the same denominators or the same numerators.

In order to compare these fractions, we need to manipulate them so that they have the same denominators.

Sometimes, we get an example where we can’t easily think of a number to multiply one of the denominators by to get it to equal the other. We’ll address this type of problem in the next example.

In these last two examples, we were finding a common denominator. We’ll get more practice with finding the common denominator when we are adding and subtracting fractions.

It is important to remember than whenever we manipulated the fractions (in both videos), we didn’t actually change them. We created equivalent fractions.

Simplifying Fractions

Fractions can get really messy - particularly when you starting operating with fractions (ex. adding fractions). Before we start any operations with fractions, we’re going to practice simplifying fractions.

To simplify fractions, we often try to find the Greatest Common Factor (GCF). What does that mean?

Factors are simply numbers that we can multiply together to get a number. As an example, the factors of 6 are 1, 6, 2, and 3 (because 1 x 6 = 6 and 2 x 3 = 6).

Let’s look at an example to illustrate common factors and the Greatest Common Factor (GCF):

In the video, we see that the process for finding the GCF is as follows:

List the factors of both the numerator and denominator (factors are all the different numbers that can be multiplied together to get a specific number)
Select the biggest factor that is common to both

Now that we know how to find the GCF, we can try simplifying a few fractions:

*In the video above, when listing the factor of 48, I left out 8 and 6. This didn't change anything because the GCF was still 12, but just wanted to correct that.

Converting Between Mixed Numbers and Improper Fractions

Recall that a mixed number is a fraction that is made up of a whole number and a fraction (ex. 2 ¾). When doing any operation with fractions (adding, subtracting, etc.), it is usually better to convert any mixed numbers to improper fractions.

The following videos summarize how we can convert between the two forms.

Adding and Subtracting Fractions

To add and subtract fractions, all fractions must have the same denominator. Once all fractions have the same denominator, we can add or subtract fractions by adding or subtracting the numbers in the numerator. We do not change the denominator when adding and subtracting fractions.

Once the addition or subtraction is complete, we can often simplify the fraction to an equivalent but “nicer” fraction. Check out a few examples of adding and subtracting fractions below:

When adding and subtracting with mixed numbers, you should convert your mixed numbers into improper fractions.

Multiplying Fractions

To multiply fractions, we simply multiply the numbers in the numerator of both fractions together to get the numerator of the new fraction, and multiply the numbers in the denominator of both fractions to get the denominator of the new fraction. Simply put, we multiply “straight across”.

When multiplying mixed numbers, you should convert your mixed number into an improper fraction.

To get some practice multiplying fractions, check out the video below.

Let’s look at how we can multiply fractions to help us word problems:

Dividing Fractions

Dividing fractions is very similar to multiplying fractions. In fact, it’s the same process with 1 extra step at the very beginning. The steps for dividing a fraction are as follows:

Invert or “flip” the second fraction (ex. If the fraction was 3/16, we flip the fraction to become 16/3)
Multiply across as you would when multiplying fractions normally
Simplify the fraction as you would normally

Let’s divide some fractions together:

Check Your Understanding

(1) Simplify the following fractions:

More practice here:

Simplifying Practice 1 - with solutions

Simplifying Practice 2 - with solutions

(2) Convert the following mixed numbers into improper fractions:

More practice here:

Convert Between Mixed Number and Improper Fractions Practice 1 - with solutions

(3) Convert the following improper fractions to mixed numbers:

More practice here:

Convert Between Mixed Number and Improper Fractions Practice 1 - with solutions

(4) Complete the following operations:

More practice here:

Operations with Fractions Practice 1 - with solutions

+ See submodule on Operations to practice fraction operations with negative numbers

(5) For each of the fractions listed below, provide 2 additional equivalent fractions.

(6) Nicole was baking a pie, but made a substitution that resulted in her having to add an extra 2/8 cup of flour. If the original recipe called for 3/4 of a cup of flour, how much flour did she use in total?

(7) Katie bought a long piece of thick ribbon to decorate her Christmas gifts with. She bought 2 ¼ meters of ribbon. She used 1/10 of the ribbon. What length of ribbon did she have left after wrapping her last gift?

(8) Kendra baked a pie to share at a party with her friends. Her boyfriend didn’t know what the pie was for, so he ate ⅛ of the pie. She brought the remaining pie to the party. With 6 people at the party, how much pie did each person get?

Extra practice - word problems:

Word Problems Practice 1 - with solutions

Word Problems Practice 2 - with solutions

General practice:

Fractions Practice - with solutions

*See solutions to provided practice problems on Solutions Page*

Page updated

Google Sites

Report abuse