Ordering Decimal Numbers from Least to Greatest

Decimals are the numbers with fractional part and whole number part. The whole number part is written after the decimal point. The decimal point separates both the fractional part and the whole number part. Some examples of the decimal numbers are 0.34, 9.86, 3.254, etc.

Ordering decimals means arranging the decimal number in ascending or descending order by following certain rules whereas comparing decimal numbers means finding the differences between numbers to decide if it is greater than, smaller than, or equal to another number.

In this article, we will discuss the rules on comparing and ordering decimals. These rules will also help us to understand the ways to arrange and compare two or more decimals. Also, we will look at the order of operations with decimals.

Comparing and Order Decimals 

Let us understand comparing and ordering decimals through two different cases.

Case 1: If the numbers given before the decimal point are not equal, then they can be easily compared. For example, to compare 9.45, 12.06, 8.75, we can easily find 12 is the greatest of the numbers that are given and accordingly, we can arrange the decimals in ascending or descending order.

Case 2: If all the numbers given before the decimal point have the same digits, then we look at the digit just after the decimal point to determine which number has the greatest digit. That number will be considered as the greatest number in the given set. 

For example, if we want to compare 9.45, 9.36, 9.57, we can check the digit given just after the decimal point and here we see that 9.57 is the greatest among all the given decimal number and accordingly we can arrange the decimal numbers in ascending or descending order.

Ordering Decimals Rules

The following rules help you to arrange least to greatest decimals.

1. Construct a table by putting the decimal point in the same place for each number.

1. List down each number.

2. Fill the empty columns with the trailing zeroes.

3. Compare the numbers using the first column on the left

4. If the digits are equal, then move to the other columns until you find the appropriate number.

If you want to arrange decimals in ascending order, then always select the smallest decimal number first.

If you want to arrange decimals in descending order, then always select the greatest decimal number first.

Let us now understand how to arrange least to greatest decimal through solved examples.

Example

Arrange the following decimals in ascending order.

1.406 , 1.46 , 0.7

Solution:

1. Construct a table with a decimal point in the same position for each number. Also, list down each number in the table at the appropriate place.

The table will look like this

2. Now, fill the empty columns with zeros

3. Now, compare the decimals using the first column (ones).

Here, we can see two of them are ‘1’, and another is ‘0’. Ascending order needs the smallest number first. Here ‘0’ is the smallest number.

So, the smallest decimal number is 0.7

Now, we can remove the smallest decimal number i.e. 0.7 from the list.

4. Now compare the tenth place:

Here, we can see two of the numbers with a similar value of ‘4’ in the tenth place, and so move along to the hundredth place to avoid the tie breaker.

5. Now compare the hundredth place:

Here, we can see one of the numbers has ‘6’ in the hundredth place whereas the other has ‘0’. So, 0 is the appropriate number (remember we are finding the smallest number each time).

So, the next smallest decimal number after 0.7  is 1.406

Now, we can remove 1.406  from the list.

At last, only one number is left. It must be the largest.

So, the decimals in ascending order is  is 0.7, 1.406, 1.46

Order of Operations With Decimal Number

To proceed with the order of operation with a decimal number, we follow the same order of operation rules that we use with the integers. We can use the acronym PEDMAS to remember that order of operation with decimal number is calculated in the following order: 

• Parenthesis

• Exponents

• Division

• Multiplication

• Addition

• Subtraction

Comparing and Ordering Decimals Examples With Solution

1. Compare 71.31 and 71.37

Solution: 

Step 1: Arrange the given number vertically in order to put the decimal points exactly one above the other.

71 . 31

71. 37

Step 2: As both the given numbers have the same decimal numbers, so we don't need to add the digit 0 to the right side of the given numbers.

71 . 31

71. 37

Step 3: As the whole number part of both the numbers are the same. So, we will compare the digits given in tenth place to find the greater decimal number.

Here, we can see the digits in the tenth place i.e. 3 are the same in both the numbers. Next, compare the digit given in the hundredth place. We can see that the digit in the hundredth place in the 71.31 is 1 whereas the digit in the hundredth place in 7.37 is 7. As 7 > 1. 

Therefore, 71.37 > 71.31

2. Calculate 58.7 - 8.8 ÷ 2.2 - 2.5

Solution:

Let us first check, if there are any parenthesis. As there is not any parenthesis in the given expression 58.7 - 8.8 ÷  2.2 - 2.5, we will move to the next category in the order of operation with decimals which is the exponent. As there is no exponent, we will move to the next category in the order of operation with decimals which is multiplication or division.

Now, we will divide 8.8 by 2.2. Also, to make the divisor a whole number, we will multiply both dividend and divisor by 10 as shown below.

8.8 ÷  2.2 = \[\frac{8.8}{2.2}\]

\[\frac{8.8\times10}{2.2\times10}\]

\[\frac{88}{22}\]

= 4

Therefore we get, 58.7 - 8.8 ÷  2.2 - 2.5 = 58.7 - 3 - 2.5

Now, we are left with the calculation of two subtractions, so keeping in mind the order of operation, we will perform the calculation from left to right. We can subtract using the column method to simplify the calculation. Here, we can also add the digit 0 to the right of 3 so that both the numbers have the same decimal place.

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We can replace this answer in our calculation to give

58.7 - 3 - 2.5 = 55.7 - 3.25

So, our final calculation is 55.7 - 3.25. This calculation can also be calculated using the column method as shown below. 

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