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

Since we are going to be dealing primarily with rational numbers, let us begin by recalling the definition of a rational number.

### Definition: Rational Number

A rational number is a number that can be expressed in the form , where and are integers and . In other words, it is a number that can be expressed as a simple fraction.

Since we are going to compare fractions, let us recall what a fraction actually is. A fraction compares a part to a whole and describes what we call a proportion. The denominator of the fraction, , is the number of equal shares (or “portions”) the whole is split into, while the numerator, , is the number of these shares that make the part we are considering.

We can represent fractions with a diagram, using a shape (commonly a circle or a rectangle) to represent the whole and shading a region to represent the part we are considering.

In these diagrams, the division of the whole in a number of equal shares is shown. It is then obvious that (read “three-quarters”) is three times greater than . A quarter (of a given whole) can be seen here as a size unit. As a result, ordering fractions of the same denominator is very easy. We simply need to order their numerators, since the numerator gives us the number of shares (of “units”) the part we are considering is made of.

Let us start with an example where we can compare two quantities by writing them as fractions with the same denominator.

### Example 1: Comparing and Ordering Fractions and Percentages

Which of the following is true?

- out of 5
- out of 5
- out of 5

### Answer

We are given two proportions here: one as a percentage, , and the other as 2 out of 5. Both can be written as fractions:

Recall that, to compare fractions, we rewrite them with the same denominator; then, we can order them according to the sizes of their numerators.

As , we can rewrite as .

As , then , which means that .

Therefore, the correct answer is A: out of 5.

Ordering fractions of different denominators is a little bit like ordering distances measured in miles and in kilometres, where we need all the distances to be in the same unit. With fractions, we need either to rewrite them in such a way that they all have the same denominator or to convert them to decimals.

To rewrite fractions with the same denominator, an efficient method is to use the least common multiple of all the denominators as the common denominator. Another way is simply to multiply all the denominators together to get a common denominator; the disadvantage is that we may end up with large numbers for the denominators and perform more multiplications, but this should lead us to the result all the same. Once it is done, we just order the numerators as in the previous example.

Recall that rational numbers can also be negative. To order a set of fractions that includes negative numbers, it is useful to arrange them on a number line, remembering that the farther left a number appears, the smaller it is. The number line below shows the correct ordering for all fractions with a denominator of 5 (i.e., fifths) between and 1.

It is also important to remember that the higher the absolute value of a number, the farther this number is from zero. For instance, we see that is farther left of zero than , which tells us that . Moreover, as both of these fractions have the same denominator, this ordering automatically follows from the fact that .

We will need to bear this pattern in mind in the next example, where we compare two negative rational numbers.

### Example 2: Comparing Negative Rational Numbers

Which of the following is true?

### Answer

In this question, we have two negative fractions with different denominators. Recall that, to compare fractions, we rewrite them with the same denominator; then, we can order them according to the sizes of their numerators.

The least common multiple (LCM) of 5 and 9 is , so it makes sense for us to rewrite both fractions with denominator 45. Thus, we have

Similarly,

Since , then . Hence, , or equivalently, .

We conclude that the correct answer is option C.

Let us now look at another example with fractions that have different denominators.

### Example 3: Ordering Rational Numbers given in the Form of Fractions

Order these values from least to greatest: .

### Answer

Here, we have to arrange four fractions from least to greatest.

Recall that, to compare fractions, we usually rewrite them with the same denominator and then order them according to the sizes of their numerators. However, in this case, we notice that all the numerators are the same: 8. This means that each fraction is 8 times bigger than the same fraction with a 1 in the numerator. So, can we order , and without rewriting them with the same denominator? Let us look at the diagram.

We see that it is possible to compare these fractions without rewriting them with the same denominator. Since they have the same numerator, their size is given by the number of equal shares the whole has been split into. The higher the denominator, the more shares the whole has been split into, which means the smaller the part. Hence, we find

Given that , , and so on, it follows that

Therefore, the final answer is .

In the next example, we are asked to order rational numbers that are expressed in different formats (i.e., fractions, mixed numbers, and decimals). In cases like this, it is simplest to convert the numbers to decimals to enable a comparison.

### Example 4: Ordering Rational Numbers given in the Form of Fractions, Decimals, and Mixed Numbers

Order the numbers from least to greatest.

### Answer

Here, we have two positive and two negative numbers. We know that the negative numbers will come before the positive ones when ordered from least to greatest, but we still need to find the precise ordering.

Since we have a combination of decimals, mixed numbers and fractions, our strategy will be to convert all the numbers to decimals. Note that 0.83 and are already in decimal form.

To convert the fraction , we divide 3 by 4 to get the decimal 0.75.

To convert the mixed number , we divide 7 by 8 to get the decimal 0.875, which means that the whole mixed number becomes .

Finally, we can arrange all four numbers on a number line as follows:

Recall that the farther left a number appears, the smaller it is.

Thus, ordering the original numbers from least to greatest, we get , , , and 0.83.

Being able to compare rational numbers that are expressed in different forms is a very useful skill, as it can help us to solve real-world problems. Here is an example of this type.

### Example 5: Solving a Real-World Problem by Comparing Rational Numbers

Fady has been offered a discount at the local diner since he always has his breakfast there on work days. He can either have or off his next meal. Which offer should Fady take?

### Answer

Fady has two different discount options, one given as a fraction and the other as a percentage. The greater the discount, the lower the cost, so Fady should choose the larger of the two given numbers.

Note that we already have the fraction , and we can easily convert the percentage of to the equivalent fraction .

Now, recall that, to compare fractions, we rewrite them with the same denominator; then, we can order them according to the sizes of their numerators.

Looking at the denominators, we see that , so we can rewrite as .

As , then , which means that .

Therefore, Fady should take the offer of off his next meal.

Rational numbers have a further interesting property, which we can outline with the following brief discussion.

Suppose that we were asked to find a rational number between and .

The least common multiple (LCM) of 5 and 3 is 15, so and .

Therefore, we can see that is a rational number between the two fractions because

It may seem at first as if there is no easy way to obtain a rational number between and since these two fractions have the same denominator and there is no integer between 3 and 4. However, we can get around this by multiplying the numerators and denominators by 2 to get the equivalent fractions and . Then, we can see that is a rational number between the two fractions because

It turns out that this pattern carries on forever, which is a consequence of the following fact.

### Fact: Density of Rational Numbers

Between any two different rational numbers there exists an infinite number of rational numbers. Therefore, we say that the rational numbers are dense.

The above information will prove useful when tackling our final example.

### Example 6: Identifying Which Rational Number Completes an Inequality

Which of the following expressions completes ?

### Answer

Here, we have two proper fractions with the same denominator and numerators that differ by 1. We must find which one of the four given fractions lies between them.

Notice that we can immediately rule out options A and C, as in both cases the numerator is larger than the denominator. This means that they are improper fractions and hence have values greater than 1, so neither can possibly lie between the proper fractions and . Both of the remaining two answer options, B and D, are proper fractions with a denominator of 30.

Now, recall that, to compare fractions, we rewrite them with the same denominator. Once this is done, we can order them according to the sizes of their numerators.

To obtain a common denominator, we could multiply 45 by 30 to get 1 350. However, we note that, in this case, a much smaller LCM. is 90, so we will use that instead.

Since , we can rewrite as and as .

Similarly, since , in B we can rewrite as and in D we can rewrite as .

Ordering the fractions according to the sizes of their numerators, we get

We conclude that is the missing fraction, so D is the correct answer.

Let us finish by recapping some key concepts from this explainer.

### Key Points

- To compare fractions, we rewrite them with the same denominator. Once this is done, we can order them according to the sizes of their numerators.
- To order a set of fractions that includes negative numbers, it is useful to arrange them on a number line, remembering that the farther left a number appears, the smaller it is. It is also important to remember that the higher the absolute value of a number, the farther this number is from zero.
- To compare a mixture of fractions, mixed numbers, and decimals, it is usually simplest to convert all numbers to decimals and then represent them on a number line.
- Between any two different rational numbers there exists an infinite number of rational numbers; we call this property the density of the rational numbers. This means that, whenever we are given two rational numbers, we can always find a further one that lies between them.