**Comparing and Ordering Rational Numbers :**

In this section, we will learn how to compare and order rational numbers.

Let us see the steps involved in comparing two or more rational numbers.

Whenever we want to compare two or more rational numbers in the form a/b, first we have to check the denominators.

**Case 1 :**

If the denominators are same then we can decide that the fraction which is having the greater numerator is greater.

**Case 2 :**

**If the denominators are different then we have to convert each fraction into equivalent fraction with the common denominator.**

**To make the denominator same, we have to apply the concept least common multiple. **

**Example 1 :**

Find which of the given two fractions is greater

6/19 and 16/19

**Solution :**

Since the denominators of those fractions are same, then we can compare the numerators and decide which fraction is greater.

So, 16/19 is greater than 6/19.

**Example 2 :**

Find which fraction is greater 1/2, 3/5

**Solution :**

**Because the denominators are different, we have to convert each fraction into equivalent fraction with the common denominator.**

**To make the denominator same, we have to apply the concept LCM.**

Now we can compare the numerators and decide which fraction is greater.

By comparing the numerators, 6 is greater than 5. So 6/10 is greater than 5/10

So, 3/5 is greater than 1/2

**Example 3 :**

Compare 1.031 and 0.256

**Solution : **

To compare the given decimal numbers, we have to write them in a table as given below.

In the above table, compare the numbers in the first column (units).

We have "1" in 1.0231 and "0" in 0.256.

Because 1 is greater than 0, we have

1.031 is greater than 0.256

**Example 4 :**

Compare 0.231 and 0.228

**Solution : **

To compare the given decimal numbers, we have to write them in a table as given below.

In the above table, compare the numbers in the first column (units).

In the first column, we have have "0" in both the decimal numbers 0.231 and 0.228.

So, we have to compare the numbers in the third column (Tenths).

In the third column (Tenths), we have the same number "2" in both the decimal numbers.

So, we have to compare the numbers in fourth column (Hundredths).

In the fourth column (Hundredths), we have "3" in 0.231 and "2" in 0.228.

Since 3 is greater than 2, we have

0.231 is greater than 0.228

**Question 5 :**

Order the fractions from least to greatest 3/4 , 2/5 ,1/8

**Solution :**

Since the denominators are different then we have to Convert these fractions to equivalent fractions with a common denominator in order to compare them

LCM of (4,5 and 8) = 40

Now we have to make each fraction with the common denominator 40.

Multiply both numerator and denominator of the first fraction by **10**

Multiply both numerator and denominator of the second fraction by **8**

Multiply both numerator and denominator of the third fraction by **5**

Now we can compare the fraction and order the fraction from least to greatest

So, the answer is (1/8) < (2/5) < (3/4)

**Question 6 :**

Order the decimal numbers 1.023, 1.101 and 0.985 from least to greatest.

**Solution :**

To order the given decimal numbers, first we have to write them in a table as given below.

To get the least one , compare the numbers in the first column (units).

We get "0" in 0.985. This is the least among the given three decimal numbers.

To get the next least number, let us compare 1.023 and 1.101.

In both the decimals, we find "1" in the first column (units).

Now, we have to compare the third column (Tenths). There we get "0" in 1.023 and "1" in 1.101.

Since "0" is less than "1", the next least decimal is 1.023.

Therefore, the order of the given decimals from least to greatest is

0.985, 1.023, 1.101

**Question 7 :**

Five friends completed a triathlon that included a 3-mile run, a 12-mile bike ride, and a 1/2 -mile swim. To compare their running times they created a table that shows the difference between each person’s time and the average time, with negative numbers representing times less than the average.

Use a number line to order the numbers from greatest to least.

**Solution :**

**Step 1 : **

Write the fractions as equivalent decimals.

1/2 = 0.5 and -1 1/4 = -1.25

**Step 2 : **

Use the number line to write the decimals in order.

From the above number line, we have the numbers in order from greatest to least are

1.95, 1.4, 1/2, -1 1/4, -2.0

After having gone through the stuff given above, we hope that the students would have understood how to compare and order rational numbers.

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