Explainer: Comparing and Ordering Rational Numbers

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

Recall that a rational number is a real number that can be expressed as a simple fraction (i.e., whose denominator and numerator are integers).

Here, 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 34 (read “three-quarters”) is three times greater than 14. 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.

Example 1: Comparing and Ordering Fractions, Decimals, and Percentages

Which of the following is true?

  1. 65%>2 out of 5
  2. 65%=2 out of 5
  3. 65%<2 out of 5

Answer

We are given two proportions here, one as a percentage, 65%, and the other as 2 out of 5. Both can be written as fractions: 65100 and 25. To compare them, we need to rewrite them with the same denominator. As 100=20×5, we can rewrite 25 as 2×205×20=40100. As 40<65, 40100<65100, which means that 25<65100.

Ordering fractions of different denominators is a little bit like ordering distances measured in miles and in kilometers, 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 rename fractions with the same denominator, an efficient method is to use the least common multiple of all the denominators as 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 numerators 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.

Example 2: Comparing and Ordering Fractions, Decimals, and Percentages

Arrange the elements in the set 0.5,25%,0.3,45 in descending order.

Answer

We are asked to order decimals, a percentage, and a negative quotient in descending order. Remember that, in this case, the percentage is to be understood as this percentage of 1. We can then express 25% of 1 as a decimal: this is 0.25. Then, it is easy to order the number. The negative quotient is the only negative number, so it is the smallest and will come at last as we need to order the numbers in descending order. The three other numbers have 0 as unit. Looking at the first decimal number, we see that 0.5>0.3>0.25.

The answer is, therefore, 0.5,0.3,25%,45.

In the previous example, we have seen how a fraction can be used to describe a number by considering this fraction of 1. It is the meaning of the fraction in a mixed number (it is a fraction of 1).

Let us look at another example with fractions with unlike denominators.

Example 3: Arranging Fractions in Ascending Order

Arrange 82, 84, 86, 85 in ascending order.

Answer

Here, we have to order fractions in ascending order. We notice that all the denominators are different. However, we notice also that all the numerators are the same: 8. So each fraction is 8 times bigger than the unitary fraction. Can we then order 12, 14, 16, and 15 without renaming them with the same denominator? Let us look at the diagram.

We see that it is possible to compare these fractions without renaming 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 16<15<14<12.

It follows (given that 86=8×16,85=8×15, etc.) that 86<85<84<82.

The final answer is 86, 85, 84, 82.

Note that when we are asked to order rational numbers expressed in different ways (as fractions or decimals), the whole in the given fractions is implied to be one. Let us look at an example.

Example 4: Comparing Decimals and Percentages

Arrange the following in ascending order: 31.43%, 3.143, 0.03143, 35.43%.

Answer

We are asked to arrange, in ascending order, different rational numbers written either as a decimal (3.143 and 0.03143) or as a percentage (31.43% and 35.43%). Since we are asked to order them, it means that the percentages are to be understood as a percentage of the number 1. In other words, 31.43% is seen here as the rational number 31.43100 and 35.43% as 35.43100.

Let us express these fractions as decimals so that we can compare them with the decimals given in the question. We find that 31.43100=0.3143 and 35.43100=0.3543.

We can now order the four decimals in ascending order, that is, from least to greatest: 0.03143<0.3143<0.3543<3.143. The final answer is 0.03143, 31.43%, 35.43%, 3.143.

When we compare fractions where the whole is known, two situations may arise. Consider, for instance, the following statement: In school A, 35 of the 345 students eat lunch provided by the school, whereas in school B, 52% of the 410 students do. Two questions could be asked here: In which school is the proportion of students who eat school lunch greater? And which school serves more lunches every day? In the first question, we are comparing proportions, that is, how a part compares to a whole, no matter what the value of the whole is. In the second question, by contrast, we are interested in a real size, here the number of students eating school lunch, and not how this number compares to the total student population.

Let us look at an example to check our understanding.

Example 5: Comparing Percentages of a Given Number

Arrange the following in ascending order: 20% of 70, 75% of 70, 310% of 70.

Answer

We are asked here to arrange percentages of a given whole in ascending order, that is, from least to greatest. Since the whole is given, we can work out the real numbers represented by these percentages.

We find that 20% of 70 is 20100×70=14. Similarly, 75% of 70 is 52.5, and 310% of 70 is 0.3% of 70, that is, 0.21.

We can arrange the resulting numbers in ascending order: 0.21 < 14 < 52.5.

The answer is, therefore, 310% of 70, 20% of 70, 75% of 70.

We notice that since the whole is the same in the three figures, it was not necessary to work out the actual number given by the percentages to be able to order the percentages given. We could have simply ordered the percentages: 310%<20%<70%. However, when the whole is given like here, and the wholes are different, the question may be to order the real numbers.

So, be careful in this case not to forget to work out the actual numbers.

Rational numbers can also be negative. To order negative numbers, it is useful to arrange them on a number line, remembering that the farther a number is located on the left, the smaller it is.

Let us see, using an example, how this is done.

Example 6: Ordering Negative Rational Numbers

Daniel and William were asked to order a series of numbers from least to greatest. Daniel wrote 7, 12, 35, 34, 37, 7 and William wrote 7, 35, 12, 37, 34, 7. Who is correct?

Answer

Let us put all the numbers given on a number line.

It is important to remember that the higher the absolute value of a number, the farther this number is from zero. For negative numbers, this means, for instance, that 35<12, whereas 35>12.

As we see in the diagram, the order of the numbers, from least to greatest, is 7,35,12,37,34,7.

Hence, William is correct.

Key Points

  1. A rational number is a real number that can be expressed as a simple fraction (i.e., whose denominator and numerator are integers).
  2. 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.
  3. To compare fractions, a method consists in renaming the fractions with the same denominator.
  4. Another method to compare fractions is to convert these fractions to decimals. By doing so, we actually find the part when the whole is 1. So, 34 can be interpreted as 34=0.751. It is how the fraction in a mixed number is understood (it is a fraction of 1).
  5. A fraction can be used to represent any rational number, which is the result of dividing the numerator by the denominator. For instance, 34=3÷4=0.75, and 124=3.
  6. A fraction can be used to represent both a part-to-whole ratio and a rational number; the methods to compare fractions are valid whatever the fractions represent.

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