Video: Comparing and Ordering Rational Numbers

In this video, we will learn how to compare and order rational numbers in different forms to solve real-world problems.

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Video Transcript

In this video, we will learn how to compare and order rational numbers in different forms to solve real-world problems. If this box is for all numbers, we can divide them into two categories: rational and irrational. Rational numbers can be written as a fraction, and irrational numbers cannot be written as a fraction. Integers would be rational values. For example, negative three is an integer. If we were going to write negative three as a fraction, we could write it as negative three over one, which makes it a rational number. Also, decimal values that terminate are rational. 0.25 in fraction form is one-fourth.

Irrational numbers cannot be written as a fraction. An example of this would be 𝜋 or the square root of two. We won’t be dealing with any irrational numbers here. We’ve just said that rational numbers are values that can be expressed as a fraction. So maybe we should remind ourselves what is a fraction.

A fraction compares a part to a whole. In a fraction, the denominator is the value below the fraction bar, and it tells us the number of equal shares in the whole. The numerator is the value on top of the fraction bar, and it tells us the number of shares we are considering.

We commonly represent fractions with a diagram using a shape, either a circle or a rectangle. The number of pieces the shape is divided into represents the denominator. Here, our rectangle is divided into five even pieces, making our denominator five. The piece we’re considering is usually shaded. Since two out of the five pieces are shaded, we would say that this fraction is two-fifths.

Here’s an example of a circular fraction diagram. The denominator is four, which represents the number of equal shares in the circle. And the numerator would be three because three of the pieces are shaded. This is the fraction three-fourths.

But what if this diagram looked like this, favorite color, and then had the labels blue and yellow? Just like before, the blue is three-fourths, which means that the yellow is one-fourth. It’s very obvious here that three-fourths is more than one-fourth. What this means is that ordering fractions with the same denominator is very easy.

To compare five-eighths, one-eighth, and three-eighths, the way we would order these values is to look at the numerator. Five is the largest numerator. Therefore, out of these three fractions, five-eighths is the largest. Three is the next largest of the numerators, which makes three-eighths the second-largest fraction out of this set.

But we need to consider what about when the denominators are not the same. Comparing fractions with different denominators would be a bit like this.

Daniel has nine coins, and Will has three notes. If someone said, “Nine is greater than three; therefore, Daniel has more money,” they’ve made an error in their comparison because of the units. You can’t compare the number of coins to the number of notes. You would either need to convert Daniel’s coins into the number of notes he would have or convert the amount of money Will has into coins.

When we deal with fractions that don’t have the same denominator, we’ll have to do the same thing. We’ll have to convert them to a value that we can compare. Let’s consider an example where we would need to do that.

Compare one-half to two-eighths.

These two values do not have a common denominator, which means that in order to compare them, we’ll need to find a common denominator. Both of these values are even numbers. In fact, two is a factor of eight. If we multiply two by four, we get eight. But it’s very important when we’re working with fractions if we multiply our denominator by a value, we multiply the numerator by that same value. One times four is four. This tells us that four-eighths is an equivalent fraction to one-half.

Since now the two values do have a common denominator, we look to their numerators. Four is greater than two. Four-eighths is larger than two-eighths, which tells us that one-half is greater than two-eighths. This was the method of finding a common denominator.

But we could’ve solved this another way. We could have converted these fractions to decimals. One-half written as a decimal is 0.5. And I see that the numerator and the denominator of two-eighths are divisible by two. So we can say that two-eighths equals one-fourth, and one-fourth written as a decimal is 0.25. If we compare the two decimal values, we’ll see that 0.5 is greater than 0.25, since 0.5 has a five in the tenths place and 0.25 has a two in the tenths place. This confirms what we’ve already said, that one-half is greater than two-eighths.

Now, let’s move on to an example where we’re comparing more than two values.

Arrange one twelfth, one-tenth, one-third, and one twentieth in ascending order.

First, we need to know what ascending order means. That would be from the least to the greatest. And then we notice that we do not have a common denominator. But all four values have a numerator of one. Now, of course, we could find a common denominator or convert all values to decimals. But because all four of the values have the same numerator, we can arrange these four values with a different method.

Imagine we’re going to represent these fractions with a circle diagram. The circle cut into thirds would look like this. Here is that same circle cut into 12 pieces. The larger denominator has smaller portions. One-third is much larger than one twelfth. And so we can say when the numerators are the same, the fraction that has the largest denominator will be the first one when you’re putting it from least to greatest. In this case, we would start with one twentieth as the smallest value. And then we would move up to one twelfth followed by one-tenth and then one-third. We’re able to do this kind of comparison because the numerators were the same.

You might want to think about it as the more pieces you cut the whole pizza into, the smaller the pieces get. So if you ate one slice of pizza but there were only three slices, that would be more than if you cut the same pizza into 10 slices and only ate one of the slices. For this question, in ascending order, we get one twentieth, one twelfth, one-tenth, and then one-third.

Here’s another example of ordering rational numbers. This time, none of the values are given in fraction form.

Arrange the following in ascending order: 0.2, negative 0.2, negative 2.3, nine, two.

We know that ascending order means least to greatest. It might be helpful here for us to think of these values in terms of a number line. We know all the positive values will go to the right of zero and all the negative values will go to the left of zero. When we’re dealing with values that are negative, values to the left of zero, the larger they are, the further away from zero they will be. So negative 2.3 will be further away from zero than negative 0.2. So we would call negative 2.3 the least value.

After that would come negative 0.2. From here, we need to compare our positive values. The decimal value is the only value that’s less than one. So it would come next. And we know that two is smaller than nine. So we would write two and then nine. In ascending order, these values are negative 2.3, negative 0.2, 0.2, two, and nine.

Here’s an example that has mixed numbers and decimal values.

Arrange the following set of numbers in ascending order: negative three and three-tenths, negative 3.61, and negative 3.5.

Ascending order is least to greatest, and we have three values. All three of the values are negative. Two of them are decimal numbers, and one is a mixed number. We have two choices. We could convert the mixed number to a decimal, or we could convert both of the decimal numbers into mixed numbers. Since two of the values are already in decimal form, let’s go ahead and convert this mixed number to a decimal number.

We know that the negative three will remain the whole-number portion. But how do we convert three-tenths to a decimal? Since this is a fraction out of 10, that’s a really simple procedure. We put a three in the tenths place so that we have negative 3.3. At this point, we have to be really careful because we’re ordering negative values. And when we order negative values, the least value will be the one that’s the furthest away from zero. And we know that the distance from zero is the absolute value of our number.

That means negative 3.61 is 3.61 units away from zero to the left. And it means that negative 3.5 is 3.5 units away from zero to the left. Negative 3.5 is closer to zero than negative 3.61, and negative 3.3 is the closest to zero of the three values. And that means to arrange them from least to greatest, we would list negative 3.61, negative 3.5, and then negative 3.3. However, for the final answer, it’s best to take negative 3.3 and put it back in the format we were given: negative three and three-tenths.

Here’s an example with both positive and negative values and with mixed numbers and fractions.

Arrange the elements in the set one and two-thirds, negative one-eighth, one and one-ninth, and negative one-half in descending order.

Descending order is from greatest to least. And so we notice about all of these values that they do not have a common denominator. But if we think about them in terms of the number line, we know that negative one-half needs to go to the left of zero and one and two-thirds needs to go to the right of zero. At this point, we’ll need to compare negative one-half and negative one-eighth and one and one-ninth and one and two-thirds.

Starting with the negatives, we have negative one-half and negative one-eighth. If we multiply the numerator and the denominator by four, for negative one-half, it becomes negative four over eight. Now, because these values are negative, we need to be very careful how we compare them. Negative one-eighth will be closer to zero on a number line than negative four-eighths. So they belong on the number line like this. If we’re going in descending order, from greatest to least, negative one-eighth would belong on the list before negative one-half.

But now we need to compare the two positive values. One and one-ninth and one and two-thirds both have the whole-number portion of one. And that means to compare them, we’ll simply need to compare their fractions. These two fractions do not have a common denominator. But if we multiply two-thirds by three in the numerator and the denominator, this mixed number becomes one and six-ninths. One and six-ninths is larger than one and one-ninth. And so we could place them on our number line like this. And we’re ready to write them in descending order.

Descending order is greatest to least. We want to again write it in the set notation, so we’ll open the brackets. The largest of the values is one and two-thirds, then one and one-ninth, then negative one-eighth, and negative one-half. We’ll close the brackets. And we’ve rearranged this set into descending order.

Before we finish the video, let’s consider the key points we need to compare and order rational numbers. Rational numbers are real numbers that can be expressed as simple fractions, meaning the denominator and the numerator are integers. To compare fractions, one method is to rename the fractions with common denominators. Another method is to convert all values to decimals. Now you’re ready to try some on your own.

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