Let’s take a look at ordering rational numbers. But before we do that, I wanna pause quickly and remind us what rational numbers are. All of the numbers found on this chart are rational numbers. Rational numbers are positive and negative numbers that can be written as a fraction, so any number that can be written in a fraction form is considered a rational number. And without further ado, we’re gonna talk about ordering rational numbers. Today we’re gonna focus on numbers that are already written in fraction form.
Here’s an example. Order these values from least to greatest. We have sixth-eighths, two-eighths, five-eighths, three-eighths. What you should have noticed immediately when you saw these values is that they have the same denominator. And that is good news when it comes to ordering values from least to greatest. Since these fractions have the same denominator, ordering them from least to greatest is as simple as comparing their numerators. Let’s remember that we’re working from least to greatest. So we find the smallest numerator, and that’s gonna be our least value here, the two-eighths. From there, we’ll have three-eighths as the next largest value and then five-eighths and, finally, the greatest value in this set, six-eighths. Here’s a picture to help you visualize what’s happening here. Imagine if four people ordered a personal pizza and each personal pizza was cut into eight slices. Since each person’s pizza is cut into eight slices and the slices are the same size, to find out who ate the most, we only need to ask who had the most slices of pizza. Two, three, five, and then six slices.
Here’s our next example. Order these values from greatest to least: one-twelfth, one-tenth, one-third, and one-twentieth. Something slightly different is happening here than in the last example. This time, each of our values has the same numerator instead of the same denominator. So what does that mean for us? How can we compare these values? Well we could draw a picture. Well we could draw a picture. If we drew a picture, we would need to divide the first rectangle into twelve pieces, the second one into ten pieces, the third one into three pieces, and the fourth one into twenty pieces. It might look something like this. What we notice here is the more pieces we divide our whole into, the smaller the pieces get. If we tried to divide one chocolate bar into twenty pieces, the pieces would be really small. However, if we divided that same chocolate bar in only three pieces, the pieces are much bigger. What I’m trying to get you to see here is when the numerators are the same, fractions with smaller denominators represent more of the whole. Here we’re trying to move from greatest to least. One-third is the greatest value here. Sharing one chocolate bar with three people is going to work out better for you than sharing one chocolate bar with twenty people. So after one-third comes one-tenth and then one-twelfth and, finally, the least value out of these four is one-twentieth.
Here’s our next example. Order these values from least to greatest: eight-halves, eight-fourths, eight-sixths, eight-fifths. So first you noticed that they have the same numerator. This means that the fractions divided into more pieces, or the ones with the larger denominator, will have a smaller value. You also remember that this is true as long as the numerators are the same. Moving forward, you check that we’re moving from least to greatest. Since we’re looking for the fraction that represents the least value, we’re going to try and find the highest denominator. Here, that’s eight-sixths followed by eight-fifths and then eight-fourths and, finally, eight-halves representing the greatest of these four values.
Here’s our last example. Order these values from least to greatest: seven-tenths, six-fourths, four-fifths, and five-halves. Immediately, we noticed that this example is different from all the ones we’ve seen. This time there’re no common numerators. There’re also no common denominators, and this is a problem. We might say this is a problem of trying to compare apples and oranges. It’s impossible to compare these in their current state; and therefore to order them, we need to find some kind of format for these four values that makes it easier for us to compare and order them. And you have a few options. You could try and find the least common denominator. You could also draw a picture. You could convert the fractions to decimals.
I’m gonna choose to solve this one by converting these fractions to decimals. Seven-tenths is one of those fractions that’s a power of ten, so it converts to zero decimal seven. To convert six-fourths to decimals, you can divide six by four. That process would look something like this, leaving you with the answer of one point five. To convert four-fifths to decimal format, I first convert this to an equivalent fraction with a base of ten. Eight-tenths is written as zero decimal eight in decimal format. I’m gonna follow that same procedure for five-halves. When I do that, I get twenty-five tenths, which in decimal format is two and five tenths. Okay, we’re no longer trying to compare apples to oranges. Now we have a format that we can work with. We’re trying to order these from least to greatest. Seven-tenths is the smallest value. After that is eight-tenths or four-fifths. Six-fourths comes next. And the greatest of these four values is five-halves.
Now you’re ready to try some on your own.