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2.1: Decimal Comparisons without Rounding

Difficulty Level: At Grade Created by: CK-12
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Ariana and Kaela are waitresses at a local Italian restaurant. After their shift one Saturday night, they decide to compare tips. Ariana adds up all her tips and finds that she has $98.12. Kaela adds up all her tips and finds that she has $98.21. How can the girls figure out who made more in tips during the shift?

In this concept, you will learn to compare and order decimals without rounding.

Comparing Decimals without Rounding

Whole numbers are the set of numbers \begin{align*}\{0, 1, 2, 3, \ldots\}\end{align*}. Whole numbers are numbers that are nonnegative and contain no fractional parts.

Not all numbers are whole. The decimal system lets us represent numbers that are less than 1 and other numbers that exist between whole numbers. In a decimal number, the decimal point divides the whole part of the number from the fractional part of the number. Digits to the left of the decimal point represent whole numbers. Digits to the right of the decimal point represent fractional parts.

Here is an example.

\begin{align*}321.43\end{align*}

This is a decimal number between 321 and 322. The digits to the left of the decimal point are 321, so 321 is the whole number. The digits to the right of the decimal point are 43 so .43 is the fractional part.

The value of each digit in a decimal number depends on its placement within the number. This concept is called place value. The further to the left a digit is, the bigger of a number that digit represents. The further to the right a digit is, the smaller of a number that digit represents.

Here is an example of a place value chart for the number 3,212,459.347801.

Example Place Value Chart
Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones
3 2 6 2 4 5 9
. Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths
. 3 4 7 8 0 1

The table shows that, for example, the third digit of 6 in this number represents 6 ten thousands or 60,000.

You read a decimal number in three parts. First, read the whole number to the left of the decimal point. Then, say “and” for the decimal point. Finally, read the fractional part to the right of the decimal point. To read the fractional part, use the name of the last digit of the number.

Here is an example.

Read the decimal number 23.451.

First, look at the whole number part of the number to the left of the decimal point. This part of the number is read as “twenty-three”.

Then, read the decimal point as “and”.

Finally, read the fractional part. Notice that the last digit is in the thousandths place. Read the digits 451 as if they were a whole number, and then put a “thousandths” at the end. You will say “four hundred fifty-one thousandths”.

The answer is “twenty-three and four hundred fifty-one thousandths”.

You can compare two decimal numbers by working from left to right. Starting with the digit on the left and moving right, whichever number first has a larger digit is the larger number overall.

When you compare decimal numbers it is important to start by lining up the decimal points so that you can make sure you are comparing the digits that have the same place value. One way to do this is by using a place value chart. If there are missing digits, you can fill them in with zeros in order to make both numbers have the same number of digits.

Here is an example.

Compare 8.507 and 8.57.

First, write the numbers in a place value chart. Make sure to line up the decimal points.

Ones . Tenths Hundredths Thousandths
8 . 5 0 7
8 . 5 7 0

Notice that the number 8.57 had one less digit than the number 8.507. You can add a 0 to the end of 8.57 to make 8.570 without changing its value so that the two numbers have the same number of digits.

Now, compare the two numbers starting with the digits on the left and moving to the right.

  • The digits are the same in the ones place.
  • The digits are the same in the tenths place.
  • The digits are different in the hundredths place. 0 is less than 7.

Once you find the place where the digits are different you are done. Because 0 hundredths is less than 7 hundredths, the number 8.507 is less than the number 8.570.

The answer is \begin{align*}8.507 < 8.57\end{align*}.

You can use the same strategy to order decimal numbers from least to greatest.

Here is an example.

Order the following numbers from least to greatest: 94.0299, 94.2019, 94.129, 494.019.

First, write the numbers in a place value chart. Make sure to line up the decimal points.

Hundreds Tens Ones . Tenths Hundredths Thousandths Ten Thousandths
0 9 4 . 0 2 9 9
0 9 4 . 2 0 1 9
0 9 4 . 1 2 9 0
4 9 4 . 0 1 9 0

Notice that you can fill in any blank spots on the chart with zeros in order to make the comparison of the digits easier.

Now, compare the four numbers starting with the digits on the left and moving to the right.

  • In the hundreds place, 494.019 is the only number with a value. Therefore, 494.019 is the greatest number. Because the question asked you to order the numbers from least to greatest, 494.019 will go at the end of your list.

Next, focus on comparing the three numbers you have left: 94.0299, 94.2019, 94.129.

  • All the digits are the same in the tens place and in the ones place.
  • In the tenths place, the digits are different. 0 is less than 1 and 1 is less than 2. This means 94.0299 is less than 94.1290 which is less than 94.2019.

Once you have ordered all four numbers, you can stop comparing digits. Note that you didn't need to look in the hundredths, thousandths, or ten thousandths place!

The answer is that the numbers ordered from least to greatest are 94.0299, 94.129, 94.2019, 494.019.

Examples

Example 1

Earlier, you were given a problem about two waitresses.

After a recent shift they decided to compare the money they made in tips. Ariana made $98.12 and Kaela made $98.21. They want to know who made more money in tips during the shift.

First, write the numbers in a place value chart. Make sure to line up the decimal points.

Tens Ones . Tenths Hundredths
9 8 . 1 2
9 8 . 2 1

Now, compare the two numbers starting with the digits on the left and moving to the right.

  • The digits are the same in the tens place.
  • The digits are the same in the ones place.
  • The digits are different in the tenths place. 2 is greater than 1.

Because 2 tenths is greater than 1 tenth, the number 98.21 is greater than the number 98.12.

The answer is Kaela made more money in tips because \begin{align*}\$98.21 > \$98.12\end{align*}.

Example 2

Compare .00456 and .00458.

First, write the numbers in a place value chart. Make sure to line up the decimal points.

. Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths
. 0 0 4 5 6
. 0 0 4 5 8

Now, compare the two numbers starting with the digits on the left and moving to the right.

  • The digits are the same in the tenths place.
  • The digits are the same in the hundredths place.
  • The digits are the same in the thousandths place.
  • The digits are the same in the ten thousandths place.
  • The digits are different in the hundred thousandths place. 6 is less than 8.

Because 6 hundred thousandths is less than 8 hundred thousandths, the number .00456 is less than the number .00458.

The answer is \begin{align*}.00456 < .00458\end{align*}.

Example 3

Compare the decimals by filling in the blank with an inequality symbol.

\begin{align*}98.065 \ \underline{\;\;\;\;\;\;\;\;} \ 98.08\end{align*}

First, write the numbers in a place value chart. Make sure to line up the decimal points.

Tens Ones . Tenths Hundredths Thousandths
9 8 . 0 6 5
9 8 . 0 8 0

Notice that a 0 was added to the end of 98.08 so that the two numbers would have the same number of digits.

Now, compare the two numbers starting with the digits on the left and moving to the right.

  • The digits are the same in the tens place.
  • The digits are the same in the ones place.
  • The digits are the same in the tenths place.
  • The digits are different in the hundredths place. 6 is less than 8.

Because 6 hundredths is less than 8 hundredths, the number 98.065 is less than the number 98.080.

The answer is \begin{align*}98.065 < 98.08\end{align*}.

Example 4

Compare the decimals by filling in the blank with an inequality symbol.

\begin{align*}5.237 \ \underline{\;\;\;\;\;\;\;} \ 5.231\end{align*}

First, write the numbers in a place value chart. Make sure to line up the decimal points.

Ones . Tenths Hundredths Thousandths
5 . 2 3 7
5 . 2 3 1

Now, compare the two numbers starting with the digits on the left and moving to the right.

  • The digits are the same in the ones place.
  • The digits are the same in the tenths place.
  • The digits are the same in the hundredths place.
  • The digits are different in the thousandths place. 7 is greater than 1.

Because 7 thousandths is greater than 1 thousandth, the number 5.237 is greater than the number 5.231.

The answer is \begin{align*}5.237 > 5.231\end{align*}.

Example 5

Write in order from least to greatest: .098, 2.45, 2.099, 2.67.

First, write the numbers in a place value chart. Make sure to line up the decimal points.

Ones . Tenths Hundredths Thousandths
0 . 0 9 8
2 . 4 5 0
2 . 0 9 9
2 . 6 7 0

Remember that you can fill in any blank spots on the chart with zeros in order to make the comparison of the digits easier.

Now, compare the four numbers starting with the digits on the left and moving to the right.

  • In the ones place, there is one digit of 0 and three digits that are 2. 0 is less than 2. This means that the number 0.098 is the smallest number.

Next, focus on comparing the three numbers you have left: 2.45, 2.099, 2.67.

  • In the tenths place, the digits are different. 0 is less than 4 and 4 is less than 6. This means 2.099 is less than 2.45 which is less than 2.67.

Remember that once you have ordered all four numbers, you can stop comparing digits. You didn't need to look in the hundredths or thousandths place!

The answer is that the numbers ordered from least to greatest are 0.098, 2.099, 2.45, 2.67.

Review

Compare. Write <, >, or = for each blank.

  1. \begin{align*}701.304 \ \underline{\;\;\;\;\;\;} \ 701.33\end{align*} 
  2. \begin{align*}2,012.201 \ \underline{\;\;\;\;\;\;\;} \ 2,012.021\end{align*} 
  3. \begin{align*}6.951 \ \underline{\;\;\;\;\;\;\;} \ 6.9313\end{align*} 
  4. \begin{align*}45.081 \ \underline{\;\;\;\;\;\;\;\;} \ 45.108\end{align*} 
  5. \begin{align*}4.5670 \ \underline{\;\;\;\;\;\;\;} \ 3.452\end{align*} 
  6. \begin{align*}0.0017 \ \underline{\;\;\;\;\;\;\;} \ 0.0019\end{align*} 
  7. \begin{align*}1.0056 \ \underline{\;\;\;\;\;\;\;} \ 1.0560\end{align*} 
  8. \begin{align*}3.4501 \ \underline{\;\;\;\;\;\;\;} \ 3.4510\end{align*} 
  9. \begin{align*}67.001 \ \underline{\;\;\;\;\;\;\;} \ 67.1\end{align*} 

Order the following from least to greatest.

  1. 373.291, 373.192, 373.129, 373.219
  2. 0.4755, 0.4764, 0.4754, 0.4674
  3. 7.16, 7.2, 7.06, 7.21
  4. 25.417, 25.741, 25.074 25.407
  5. 15.001, 15.067, 15.100, 15.0001
  6. 6.780, 6.087, 6.870, 6.008

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.1.

Resources

 

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Vocabulary

Decimal point

A decimal point is a period that separates the complete units from the fractional parts in a decimal number.

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...

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