Proper and Improper Fractions and Mixed Numbers

Hi, and welcome to this review of fractions and mixed numbers!

Before we dive in, let’s review the basic parts of a fraction. Remember, a fraction simply represents a part of a whole. It has a numerator and a denominator, which tells us what the “part” is and what the “whole” is. Let’s look at the fraction \(\frac{3}{4}\) as an example. We can see the 3 is our numerator and the 4 is our denominator. So the fraction \(\frac{3}{4}\) is really saying 3 parts out of 4 parts total. It can also be helpful to visualize \(\frac{3}{4}\) as simply

\(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\)

It’s very important to remember that a denominator of 4 does not represent the value of 4. A denominator of 4 represents the value of 1 that is divided up into 4 equal parts, or fourths. This type of fraction represents a value less than one whole. \(\frac{3}{4}\) is not quite 1. If we had \(\frac{4}{4}\) that would be equivalent to one, but we only have 3 out of 4 parts.

We see and use fractions that are less than one all the time in our daily lives, whether it’s for things like recipes or keeping track of time. Recipes often call for amounts such as “1/2 tsp salt,” and we often keep track of time in terms of quarter hours, like “a quarter past three” for 3:15. Though we observe this type of fraction very frequently in our daily lives, it is not the only type of fraction.

Consider the following scenario. You are ordering pizza for a big celebration. There will be a lot of hungry guests at this celebration, so you order 3 pizzas. Each pizza is cut into 6 slices. This means that each pizza has 6 equal parts, and as a fraction, 6 would be considered our “whole,” or our denominator.

If your first guest eats 2 slices we would represent this as the fraction \(\frac{2}{6}\). Two parts, out of 6 parts total.

But what if that first guest was really hungry and grabbed 7 slices? Again, each pizza was cut into 6 equal slices, so 6 remains as our “whole,” or denominator. But this time our “part” is 7: \(\frac{7}{6}\). In this scenario, our numerator is larger than our denominator.

Fractions with a numerator larger than their denominator are referred to as improper fractions. Essentially, improper fractions equal a value that is more than one. One whole pizza would be represented by \(\frac{6}{6}\), or “six sixths.” \(\frac{7}{6}\) represents “seven sixths,” which is more than one pizza. This could be visualized as \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=7\). It can also be written in another form called a mixed number. An improper fraction and a mixed number will represent the same amount but simply be written in a different form.

For example, the improper fraction \(\frac{7}{6}\) could also be written as the mixed number \(1\frac{1}{6}\). Mixed numbers and improper fractions show the same amount, but as a mixed number, the “parts” are collected and consolidated into as many groups of 1 whole as possible. For example, \(\frac{4}{4}\) would be grouped together as 1. \(\frac{7}{7}\) would also be grouped together as 1. Any value where the numerator is equivalent to the denominator would be expressed simply as 1.

In our pizza example, the guest took 7 slices from a group of pizzas that were sliced into sixths. We said that this could be expressed as the improper fraction \(\frac{7}{6}\), or visualized as \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\). As a mixed number, we would group 6 of these “sixths” in order to form \(1\frac{6}{6}\), or 1 whole. By grouping \(\frac{6}{6}\) together, we can see that \(\frac{1}{6}\) is left over, on its own. We would write our mixed number as \(1\frac{1}{6}\).

Converting Improper Fractions to Mixed Numbers

Let’s try a few more examples. Let’s write the following improper fractions as mixed numbers: \(\frac{4}{3}\) and \(\frac{3}{2}\).

\(\frac{4}{3}\) can be visualized as \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\). We know that \(\frac{3}{3}\) is equal to 1, so let’s group 3 of these “thirds” together. We are now left with \(1\frac{1}{3}\) as our mixed number.

\(\frac{3}{2}\) can be visualized as \(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\). We then know that \(\frac{2}{2}\) makes one whole. And we’re left with \(\frac{1}{2}\) left over. So \(\frac{3}{2}\) as a mixed number is \(1\frac{1}{2}\).

Let’s try one more example: \(\frac{7}{4}\)

\(\frac{7}{4}\) is the same as \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\). Now we know that \(\frac{4}{4}\) is grouped as one whole. So these \(\frac{4}{4}\) pulled over, to equal 1. And we’re left with \(\frac{3}{4}\). \(\frac{7}{4}\) written as a mixed number is \(1\frac{3}{4}\).

This process will take place in reverse in order to convert a mixed number to an improper fraction.

Converting Mixed Numbers to Improper Fractions

For example, if we started with the mixed number \(1\frac{3}{4}\) and we wanted to convert it to an equivalent improper fraction, we would take a look at the whole number, in this case, it is 1. This whole number is really representing our denominator that’s in the fraction. In this case, it’s 4, so the 1 is equal to \(\frac{4}{4}\). When we combine these four fourths with the \(\frac{3}{4}\), we end up with seven fourths total, or \(\frac{7}{4}\).

That’s all there is to it! I hope that this video was helpful. Thanks for watching, and happy studying!

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