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What is an improper fraction?

Fractions get separated into two categories. The first is the category of proper fractions. These fractions have numerator (the top number) smaller than the denominator (the bottom number). The positive ones will be between $0$ and $1$ on the number line. The negative ones will be between $-1$ and $0$ on the number line.

The second category is the improper fractions. These fractions have numerator bigger than the denominator. The positive ones will be to the right of $1$ on the number line. The negative ones will be to the left of $-1$ on the number line.

Of course, it is easier to visualize what the positive fractions represent.

The positive fractions in the second category will always be greater than $1$. Think of a fraction as a container with some stuff in it. The denominator tells you how much one container can hold. However, the numerator tells you how much is actually in the container.

A fraction is improper when there's too much stuff for the container. In other words, you need more than one container to hold all of it. In this picture, one container is being overfilled. Yikes!

The "stuff" can be whatever you want. In this picture we're using rocks.

Proper vs. Improper Fractions

Before we start, let's quickly review the difference between proper and improper fractions.

If absolute values aren't your cup of tea, there's another way you can identify an improper fraction. We can ignore any negative signs and check to see if the numerator is bigger than the denominator. If it is, the fraction is improper. Otherwise, the fraction is proper.

The fraction $\frac{5}{4}$ has a bigger numerator than denominator. On the other hand, $\frac{4}{5}$ is the other way around. This means $\frac54$ will be to the right of $1$ on the number line. Also, $\frac45$ will be to the left of $1$ on the number line.

Now, let's use those definitions to answer a few practice questions.

Converting improper fractions to mixed numbers

Sometimes, you will be asked to take an improper fraction and rewrite it as a mixed number.

Since it is just a re-written form of an improper fraction, mixed numbers also have an absolute value greater than $1$.

To write improper fractions as mixed numbers, follow these steps:

• Determine how many times the numerator can be evenly divided by the denominator. This number will be the integer in your mixed number.

• Write the remainder over the original denominator. This will be the fraction portion of your mixed number.

Let's try it out:

This information tells us that the integer in our mixed number is $3$ and the numerator of the fraction portion is $2$ (the remainder). The denominator remains the same as it did in the improper fraction ($3$). We can write our mixed number like this:

To check your work, multiply the integer by the denominator and add the numerator to that product. This should give you the original numerator of your improper fraction.

(Video) Mixed Numbers

by Math Antics

Here’s a video by Math Antics that covers improper fractions. It also covers some other topics. These include mixed numbers, converting fractions to mixed numbers, and converting mixed numbers to fractions. All of the topics are important. However in this explanation we'll focus on the first one.

$\large{\text{Summary}}$

Let's talk about the two main types of fractions. We'll summarize what the video went over.

A proper fraction is such that the top number is smaller than the bottom number.

An improper fraction is a fraction where that the top number is larger than the bottom number. In other words, the numerator is greater than or equal to the denominator. We can convert these to mixed numbers. These numbers are also called mixed fractions. Moreover, the other ones are made by combining a whole number and a proper fraction. But the whole number is hidden inside the fraction. It's what makes the numerator bigger than the denominator. $$1+\frac12=\frac32$$

On the number line, proper fractions are between $0$ and $1$. The other fractions are above $1$.

Finally, mixed numbers are equivalent to improper fractions. That is, every mixed number can be written with a numerator larger than the denominator. The reverse is also true.

So why use one over the other? If you are comparing two numbers, mixed numbers are easier to handle. You compare the whole number parts, and then compare the fractions. However, other operations are done more easily with improper fractions. If you need to add, subtract, multiply, or divide, don't convert to a mixed number.