Comparing & Ordering Mixed Numbers

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Read this lesson, and you'll learn how you can easily compare and order your mixed numbers. You'll learn how fractions can actually help you to easily see which number is larger or smaller.

Mixed Numbers

In this lesson, you'll learn what you need to do to compare and order your mixed numbers. And don't worry, it's not that hard once you know what you need to do. First, the definition of a mixed number is any number that has both an integer part and a fraction part. This fraction part is a proper fraction meaning the numerator is always less than the denominator. All of these numbers below are mixed numbers.

mixed numbers

Do you see how all of them begin with an integer, a whole number, and then a fraction where the numerator is always smaller than the denominator? This is what makes a mixed number.

Improper Fractions

Before you even start thinking about comparing and ordering your mixed numbers, you'll need to turn them into fractions called improper fractions. These are fractions where the numerator is larger than the denominator. Look at these for example:

mixed numbers

Do you see how all of these improper fractions have a larger numerator than denominator? For the 7 / 5, see how the 7 is larger than the 5? And for the 5 / 4, the 5 is larger than the 4. So, remember, whenever your numerator is larger than your denominator, you have an improper fraction.

So, how do you go about changing your mixed numbers into improper fractions? It's actually quite easy. You just have to remember in which order and which operation you need to do. Here are the steps:

  • 1. Multiply the integer with the denominator.
  • 2. Add the numerator to your answer from the first step.
  • 3. Write the number you get from step two in the numerator, leaving your original denominator the same.

Let's take a look at these steps in action. To convert 3 1 / 3 into an improper fraction, you'll first multiply your integer, the 3, with the denominator, also a 3. You get 3 * 3 = 9. Then, you'll need to add the numerator, the 1, to this number. You get 9 + 1 = 10. Now, you'll write this in the numerator and leave your denominator the same. So, you get 10 / 3. And look, you have an improper fraction that equals your mixed number.

Now, you try it. Try converting 5 2 / 3 into an improper fraction.

First, you multiply your integer with your denominator. 5 * 3 = 15. Then, you add your numerator to this. 15 + 2 = 17. This now becomes your numerator. Your denominator remains the same: 17 / 3. And, you are done.

Let's convert another. Let's convert 1 1 / 8. 1 * 8 = 8. 8 + 1 = 9. So, the improper fraction is 9 / 8.

You've just converted three mixed numbers into improper fractions. You now have three improper fractions: 10 / 3, 17 / 3, and 9 / 8.

mixed numbers

You'll want to keep track of which improper fraction represents which mixed number so you can choose the right mixed number in your final answer.

Comparing Mixed Numbers

There is just one more step before you can compare and order your mixed numbers. You now need to rewrite your improper fractions so they all have the same common denominator. To do this, you'll use the skills you learned when you learned about finding the least common multiple. This is the step you perform when you have to add or subtract fractions. Your goal is to rewrite the fractions so all your denominators are the same.

Looking at your improper fractions and their denominators, you see that you have a 3 and an 8 for your denominators. The least common multiple of 3 and 8 is 24. So, you'll need to convert all your fractions so they all have a denominator of 24. To do this, you look at what you need to multiply your denominator by to get to your least common multiple, and then you also multiply your numerator by that amount. So, to convert 10 / 3, you need to multiply both the numerator and denominator by 8. You get 80 / 24. Converting 17 / 3, you also need to multiply both the numerator and denominator by 8. You get 136 / 24. Lastly, to convert 9 / 8, you need to multiply both the numerator and denominator by 3. You get 27 / 24.

Now, your improper fractions are 80 / 24, 136 / 24, and 27 / 24.

mixed numbers

Now, you can compare your numbers. To compare your numbers, all you need to look at now is the numerator. That's right, all you need to do is to compare your numerators to see which is larger or smaller. Your numerators are 80, 136, and 27. Isn't trying to compare these numbers easier than trying to compare the original mixed numbers? You can easily see that the 80 is larger than the 27 but smaller than the 136. You can also see that the 136 is the largest of the bunch. And, 27 is the smallest.

Ordering Mixed Numbers

Now that you've compared your numbers, you can now order your mixed numbers. If you want to order them from least to greatest, your 27 numerator is the smallest, so that number comes first. Then comes the 80 numerator. Last, is the 136 numerator. So, ordering from smallest to largest, you get this order for your mixed numbers. You've simply replaced your improper fraction with the mixed number that it represents.

mixed numbers

And, you are done!

Example

Let's try another problem.

Which of these mixed numbers is the largest?

mixed numbers

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