How to Compare and Order Rational Numbers?

Comparing numbers is just a process of identifying greater and smaller numbers. The order of the numbers includes comparing them and arranging them in ascending or descending order.

Related Topics

• How to Order Integers and Numbers

Step by step guide to comparing and ordering rational numbers

Comparing and ordering numbers is a concept in mathematics in which we compare numbers according to their values and identify smaller and bigger numbers and then arrange them in ascending or descending order.

What is comparing numbers?

Comparing numbers is a way of comparing two or more numbers and identifying whether one number is equal, lesser, or greater than the other numbers. We can compare numbers using different methods such as on a number line, by counting, or by counting the number of digits, using place values of the numbers, etc. In our daily lives, comparing numbers is a common practice because we compare numbers with similar properties to determine whether one number is equal to, smaller than, or greater than the other numbers.

What is the ordering number?

Ordering numbers is a way of arranging them in order – either from small to big or big to small. When we arrange the numbers in ascending order, we arrange them from small to big, and when we arrange the numbers from big to small, they are called descending.

Symbols for comparing numbers

To compare numbers, we use special symbols to identify greater, smaller, or equal numbers. There are three such symbols. The table below shows the meaning of each symbol used to compare numbers.

The less than and greater than symbols look like the alphabet \(V\) placed horizontally. An easy way to remember symbols is to always have the open side of the symbol facing the greater number and the pointed end points toward the smaller number.  So, if the greater number comes first, then it is greater than symbol \(>\), and if the smaller number comes first, then it is a less than symbol \(<\).

Comparing rational numbers

When comparing rational numbers, we consider the \(LCM\) denominator of rational numbers. We convert rational numbers into similar fractions and then compare rational numbers. There are some points to remember before learning how to compare rational numbers:

All negative rational numbers are less than \(0\). All negative rational numbers are less than \(0\). And All positive rational numbers are greater than all negative rational numbers.

Now let’s compare two rational numbers to understand the process. Compare \(\frac{2}{3}\) and \(\frac{6}{7}\). First, we find the \(LCM\) of the denominators of the two given rational numbers. \(LCM (3, 7) = 21\). Now, convert the rational numbers into like rational numbers.

\(\frac{2}{3} = \frac{(2 × 7)}{(3 × 7)} = \frac{14}{21}\)

\(\frac{6}{7} = \frac{(6 × 3)}{ (7 × 3)} = \frac{18}{21}\)

Now, compare the rational numbers by comparing the numerators of the two like fractions. Since, \(18 > 14\), so \(\frac{18}{21} > \frac{14}{21}\). Therefore, we have \(\frac{6}{7} > \frac{2}{3}\).

Ordering rational numbers

To order rational numbers use these steps:

• Convert all numbers to a common format.
• Put them required ordered.
• Rewrite in the original format.

Comparing and Ordering Rational Numbers – Example 1:

Order the following rational numbers from greatest to least: \(2^2,\:40\%,\:4.53,\:\frac{19}{4},\frac{25}{6}\).

Solution:

First, convert all rational numbers into decimals:

\(2^2=2×2=4\)

\(40\%=40\div 100=0.40\)

\(4.53= 4.53\)

\(\frac{19}{4}=19\div 4=4.75\)

\(\frac{25}{6}=25\div 6=4.16\)

Next, compare each place value digit, working from left to right, and order the decimals from greatest to least:

\(4.75,\:4.53,\:4.16,\:4,\:0.4\)

Finally, replace each decimal with the corresponding rational number from the original list:

\(\frac{19}{4},\:4.53,\:\frac{25}{6},\:2^2,\:40\%\)

Exercises for Comparing and Ordering Rational Numbers

Order the following rational numbers from least to greatest.

• \(\color{blue}{0.780,\:72.5\%,\:\frac{4}{5},\:\sqrt{4},\:\frac{3}{4}}\)
• \(\color{blue}{135\%,\:13.5,\:8\frac{3}{5},\:3.9\times 10^2}\)
• \(\color{blue}{0.16,\:16.7\%,\:\frac{2}{5}, \:3.2\times 10^{-3}}\)

Compare the following rational numbers using \(>, <, or =\).

• \(\color{blue}{10\%\:▢\:\frac{1}{8}}\)
• \(\color{blue}{\frac{2}{5}\:▢\:\frac{3}{4}}\)
• \(\color{blue}{\frac{5}{-8}\:▢-\frac{17}{23}}\)

• \(\color{blue}{72.5\%,\:\frac{3}{4},\:0.780,\:\frac{4}{5},\:\sqrt{4}}\)
• \(\color{blue}{135\%,\:\:8\frac{3}{5},\:13.5,\:3.9\times \:10^2}\)
• \(\color{blue}{\:3.2\times 10^{-3},\:0.16,\:16.7\%,\:\frac{2}{5}}\)

• \(\color{blue}{10\%\:<\:\frac{1}{8}}\)
• \(\color{blue}{\frac{2}{5}\:<\:\frac{3}{4}}\)
• \(\color{blue}{\frac{5}{-8}\:>-\frac{17}{23}}\)