2.1: Comparing and Ordering Decimals
Introduction
The 100
Connor loves running track for his middle school. In fact, he is one of the fastest runners and he specializes in short distances like the 100. Connor has set a goal this year to run a personal best in the 100. His fastest race time is 13.91 and his slowest race time is 15.16. Connor has set a goal of running the 100 in 12.51. While this is ambitious, he believes that with the right mind-set and perseverance that he will accomplish his goal.
Jeff is a friend of Connor’s who attends the middle school in the next town over. Jeff is also a runner and he also specializes in the 100. The boys are good friends but they definitely love to compete against each other. Sometimes Jeff is faster and sometimes Connor is faster. Jeff’s best race time in the 100 is 13.42. His slowest race time is 15.32. Jeff’s goal this year is to beat his fastest time and finish the 100 in less than 13.00.
One week before the big race, Connor and Jeff decided to try running practice intervals together. They both ran the 100 on the track; Connor’s time was 13.11 and Jeff’s time was 13.14.
Given these times, who was faster? How does Connor’s practice time compare with his best race time? How does Jeff’s practice time compare with his best race time? Given these numbers, who do you think is on track to accomplish his goal? Give some mathematical reasons why.
Track and field times are calculated in decimals. To answer these questions, you will need to know how to compare and order decimals. Pay attention, you will see this problem again at the end of the lesson.
What You Will Learn
In this lesson you will learn the following skills.
- Compare and order decimals without rounding.
- Round decimals to a given place.
- Compare and order decimals after rounding.
- Describe real world portion or measurement situations by comparing or ordering decimals with and without rounding.
Teaching Time
I. Compare and Order Decimals Without Rounding
Before we move on to comparing and ordering decimals, let’s begin by working with whole numbers and decimals.
Yes! You have been working with whole numbers for a long time. Whole numbers were the first numbers that you used to count and figure out quantities.
Whole numbers are numbers such as 1, 8, 56, and 278—numbers that don’t contain fractional parts.
Not all numbers are whole.
The decimal system lets us represent numbers or parts of numbers that are less than 1. Money is one of the most common places that we see decimals in everyday life. You should be familiar with using decimals in your everyday dealings with money.
Take the amount $41.35.
You know that 0.35 represents part of a dollar.
If you had 100 cents, you would have a dollar.
So 35 cents can be represented as 0.35, or \begin{align*}\frac{35}{100}\end{align*}
Decimals, however, are used to describe a lot more than money.
In a decimal number, like 321.43, the decimal point divides the whole number from its fractional part.
Numbers to the left of the decimal point are whole numbers.
Numbers to the right of the decimal point are the fractional part.
Now that we have reviewed a bit about whole numbers and decimals, we can look at comparing decimals without rounding.
First, let’s think about place value. The value of each number in a decimal, whether it is a whole number to the left of the decimal point, or a fractional part to the right of the decimal point, is defined based on the value of the number. The place that each digit is in has a specific name, and this helps us to figure out which numbers are larger and which are smaller.
Here is a place value chart.
Place | Number |
---|---|
Millions place | 1,000,000.0 |
Hundred-thousands place | 100,000.0 |
Ten-thousands place | 10,000.0 |
Thousands place | 1,000.0 |
Hundreds place | 100.0 |
Tens place | 10.0 |
Ones place | 1.0 |
Decimal point | . |
Tenths place |
0.1 or \begin{align*}\frac{1}{10}\end{align*} |
Hundredths place |
0.01 or \begin{align*}\frac{1}{100}\end{align*} |
Thousandths place |
0.001 or \begin{align*}\frac{1}{1000}\end{align*} |
Ten-thousandths place |
0.0001 or \begin{align*}\frac{1}{10,000}\end{align*} |
Hundred-thousandths place |
0.00001 or \begin{align*}\frac{1}{100,000}\end{align*} |
Millionths place |
0.000001 or \begin{align*}\frac{1}{1,000,000}\end{align*} |
This chart shows us the value of each digit. Now let’s look at an example.
10 Th | Thousand | Hundreds | Tens | Ones | Tenths | Hundredths | Thousandths | Ten-Thousandths | |
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 4 | 5 | 9 | . | 3 | 4 | 7 | 8 |
We read this number as: “Twelve thousand, four hundred fifty-nine AND three-thousand, four hundred and seventy-eight ten-thousandths.”
Alright, you look puzzled. Let’s slow down and look at how we read a decimal so that we can understand its value.
To read a decimal number, we begin with the whole number.
The decimal point is read as “and” or as “point.”
Then the digits to the right of the decimal point are read by naming the place in the last digit of the number. For example, 23.451 is read as twenty-three and four hundred fifty-one thousandths.
How can we use this information to compare decimals?
First, to compare decimals, we can use our symbols for greater than > and less than <. To order decimals, we will be writing them from least to greatest or from greatest to least. First, let’s just focus on comparing.
We compare and order decimal numbers, just like we compare and order whole numbers—by working from left to right.
Begin by lining up the decimal points. If there are missing places in the decimals, we can figure out which one is smaller or larger by adding zeros on the end of the number for the missing places.
You want each decimal to have the same number of digits so that you can get a clear idea of which is larger and which is smaller.
Example
Compare 8.507 and 8.570. Write >, <, or =.
The first thing to do, is to notice is that each number has one whole number, 8. Because the whole number is the same in both numbers, we need to determine which is greater by looking at the decimal part of the number. These decimals both have three decimal places, so we don’t need to add any zeros.
The clearest way to see the decimal values is by lining up the decimal points in a place-value chart, such as the one below.
Ones | . | Tenths | Hundredths | Thousandths |
---|---|---|---|---|
8 | . | 5 | 0 | 7 |
8 | . | 5 | 7 | 0 |
Use the chart to compare the value of each digit from left to right.
The numbers are identical in the ones place (8) and in the tenths place (5).
In the hundredths place they are different.
0 < 7
The number 8.507 has \begin{align*}\frac{0}{100}\end{align*}
0 hundredths is less than 7 hundredths—we don’t even need to look at the thousandths place!
Our answer is that 8.507 < 8.570.
We can use this skill to help us when writing decimals in order from least to greatest or greatest to least.
Example
Order the following numbers from least to greatest: 94.0299, 94.2019, 94.129, 494.019.
We are ordering four numbers from least to greatest.
Use a place-value chart to see the difference in values.
This time we’ll have to add extra zeros to describe our numbers.
Remember to work from left to right when comparing values.
Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths | Ten-thousandths |
---|---|---|---|---|---|---|---|
9 | 4 | . | 0 | 2 | 9 | 9 | |
9 | 4 | . | 2 | 0 | 1 | 9 | |
9 | 4 | . | 1 | 2 | 9 | 0 | |
4 | 9 | 4 | . | 0 | 1 | 9 | 0 |
With just a glance at the place-value chart, it is easy to see which number is the greatest number.
494.019 is the only number that has a value in the hundreds place.
The question asks us to order the numbers from least to greatest, so put 494.019 at the end of our list.
Now let’s look at the other three numbers.
All of them have the same value in the tens and ones places (94), so we need to look to the tenths place. Don’t be thrown off by the 9's in the thousandths and ten-thousandths place!
We don’t need to look further than the tenth place.
0 < 1 < 2, so 94.0299 < 94.129 < 94.2019
Now we can put our numbers in order from least to greatest.
Our answer is 94.0299, 94.129, 94.2019, 494.019
2A. Lesson Exercises
Practice these skills by comparing and/or ordering these decimals.
- 98.065 ______ 98.08
- 5.237 ______ 5.231
- Write in order from least to greatest: .098, 2.45, 2.099, 2.67
Take a few minutes to check your work with a partner. Did you remember to add zeros when needed?
II. Round Decimals to a Given Place
Some decimals can extend to the millionths place or even farther and be difficult to handle in operations.
Rounding decimals is useful when estimating sums and differences or comparing measurements.
You have already learned how to round whole numbers to a given place value.
Let’s take a minute to review rounding whole numbers.
1,537 rounded to the thousands place is 2,000; rounded to the hundreds place is 1,500; rounded to the tens place is 1,540. We can round to any place that we choose to round the number to.
We can use the same steps when rounding decimals. Here are the steps for rounding numbers.
The Steps for Rounding Numbers
- Identify the place you want to round to, notice that number.
- Look at the digit to the right of that number.
- If the digit to the right is 5 or greater, round up. If the number is less than 5, round down.
Example
Round 406.091 to the nearest whole number
First, underline the number you’re rounding to and bold or circle the number directly to the right.
In the case of our example problem, which asks us to round to the nearest whole number, underline the ones place.
Boldface, or circle, the number directly to the right of the number you underlined. The number in the tenths place is the one you’ll look at when deciding to round up or down.
Because that number is a zero, we don’t round up. The whole number stays the same.
Our answer is 406.
Example
Round 206.9595 to the nearest whole number
First, underline the number you’re rounding to and bold or circle the number directly to the right.
Rounding to the nearest whole number means you should have underlined the digit in the ones place.
The number in the tenths place is the one you look at when deciding to round up or down. This number is a nine, so we round up.
Our answer is 207.
Now let’s use these same steps when rounding decimal places.
Example
Round .07285 to the nearest thousandths place
First, identify that the two is in the thousandths place. We can underline that digit because that is the digit that we are going to be rounding.
Next, look to the number to the right of the two. That is the number we use to determine whether we round up or down.
That number is an 8, so we round up.
Our answer is .073.
Notice that we don’t include the digits past the place where we have rounded. This is because we have rounded that digit.
2B. Lesson Exercises
Practice rounding the following numbers.
- Round 1.23439 to the nearest ten-thousandth.
- Round 3035.67 to the nearest whole number.
- Round 0.98734 to the nearest thousandth.
Take a few minutes to check your work with a friend.
III. Compare and Order Decimals after Rounding
Now you know how to compare and order decimals as well as how to round decimals. Combining the two procedures is fairly straightforward. Once decimals are rounded to a specific place value, they can be compared just as they were compared when they weren’t rounded—from left to right.
Example
Compare 77.0949 and 77.0961 after rounding to the nearest hundredth. Write >, <, or =.
The first thing that we need to do is to round each to the nearest hundredth.
77.0949: 9 is in the hundredths place. The number to the right of the 9 is a 4. So we do not round up.
Our first number is 77.09
77.0961: 9 is in the hundredths place here too. But the number to the right is a 6, so we round up.
Our second number is 77.10
What happened is that we needed to round up. We can’t go from 9 to 10 in one digit, so the place of the number moved over and we rounded 9 hundredths up to 1 tenth.
Now we can compare these numbers.
Our answer is 77.09 < 77.10.
We can use these same steps when ordering numbers from least to greatest or from greatest to least. First, you round the numbers to the desired place then you compare and order them.
Let’s look at an example.
Example
Round the following numbers to the nearest tenth. Then order from greatest to least. 5.954, 5.599, 5.533, 6.062.
First, we round each number to the nearest tenth. Remember that the tenths place is the first place to the right of the decimal point. Here the digit being rounded is in bold print and the number that we use to determine whether we round up or down is in the hundredths place and is underlined.
5.954 rounds up to 6.0
5.599 rounds up to 5.6
5.533 rounds down to 5.5
6.062 rounds up to 6.1
Next, we write them in order from greatest to least.
Our answer is 6.1, 6.0, 5.6, 5.5
2C. Lesson Exercises
Practice rounding and comparing.
- Compare each number after rounding to the nearest hundredth, 4.567 and 4.562
- Compare each number after rounding to the nearest tenth, .234 and .245
- Round each to the nearest tenth and write in order from least to greatest, .0567, .291, .1742
Take a few minutes to check your work with a partner.
Use your notebook to write down the steps for rounding numbers. Then continue with the next section.
IV. Describe Real World Portion or Measurement Situations by Comparing and Ordering Decimals with and without Rounding
There are many real world situations where we use decimals. Decimals are especially useful when working with money, measurement and with portions. Think back to the introductory problem, measuring time is what our two track runners were working on. We’ll go back to that problem in a minute, but first let’s look at some situations where decimals are used in the real world.
You already know that money involves the decimal system to describe fractions of a dollar. Where 100 cents is a dollar, 12 cents is written as .12, \begin{align*}\frac{12}{100}\end{align*}
Example
Compare $1.45 and $1.75
To compare, we see that the dollar amount is the same, so we look at the cents or the decimal part of the dollar. We can compare .45 and .75. 75 is greater than 45. Here is our answer.
$1.45 < $1.75
This may seem obvious, but thinking of decimals in terms of money can often help make sense out of them.
The metric system of measurement is based on a system of 10, so it also uses decimals. The metric system includes units of length (meters), weight (grams), and volume (liter). Look at the metric chart below to get an idea of the base-ten relationship among metric units.
Metric Units of Length
\begin{align*}& \text{millimeter} \ (mm) && .1 \ cm && .001 \ m && .000001 \ km\\
& \text{centimeter} \ (cm) && 10 \ mm && .01 \ m && .00001 \ km \\
& \text{meter} \ (m) && 1000 \ mm && 100 \ cm && .001 \ km\\
& \text{kilometer} \ (km) && 1,000,000 \ mm && 100,000 \ cm && 1000 \ m\end{align*}
Metric Units of Mass
\begin{align*}& \text{milligram} \ (mg) && .1 \ cg && .001 \ g && .000001 \ kg \\
& \text{centigram} \ (cg) && 10 \ mg && .01 \ g && .00001 \ kg\\
& \text{gram} \ (g) && 1000 \ mg && 100 \ cg && .001 \ kg\\
& \text{kilogram} \ (kg) && 1,000,000 \ mg && 100,000 \ cg && 1000 \ g\end{align*}
Metric Units of Volume
\begin{align*}& \text{milliliter} \ (ml) && .1 \ cl && .001 \ l && .000001 \ kl\\
& \text{centiliter} \ (cl) && 10 \ ml && .01 \ l && .00001 \ kl \\
& \text{liter} \ (l) && 1000 \ ml && 100 \ cl && .001 \ kl \\
& \text{kiloliter} \ (kl) && 1,000,000 \ ml && 100,000 \ cl && 1000 \ l\end{align*}
Example
Mr. Wang is pricing batteries for his car. He finds the following prices: $100.01, $110.10, $101.10, $100.11. Order the battery prices from least to greatest.
The problem asks us to order to prices from least to greatest. Don’t be confused by all the ones and zeros in the numbers. By now, you may not need to use a place-value chart, but it still helps to line up the decimal points and compare values from left to right.
$100.01
$110.10
$101.11
$100.10
We can see that the values with $100 are going to be the least, so we look at the decimal places, .01 < .10, so $100.01 < $100.10. We can put these two numbers as our least and next to least values.
Now let’s compare the other two numbers. We don’t even need to look at the decimals because we can see that $101 < $110.
Our answer is $100.01, $100.10, $101.11, $110.10
Sometimes, it helps to use rounding when problem solving. Let’s look at an example where rounding is very helpful.
Example
Four families all live at different distances from downtown. The Murrays live 1.75 kilometers from downtown; the Smiths live 1.89 kilometers from downtown; the Giles live 1.67 kilometers from downtown; and the Hofflers live 1.51 kilometers from downtown. Round each distance to the nearest tenth, and then order the distances from greatest to least.
The problem asks us to compare the numbers after we have rounded them to the nearest tenth. The tenth place is directly to the right of the decimal point. We’ll need to look to the hundredth place (second to the right of the decimal place) to determine whether to round up or down. Let’s begin by rounding the four numbers separately. The tenth place is underlined; the hundredth place is bolded.
Murrays: 1.75 \begin{align*}\rightarrow\end{align*}
Smiths: 1.89 \begin{align*}\rightarrow\end{align*}
Giles: 1.67 \begin{align*}\rightarrow\end{align*}
Hofflers: 1.51 \begin{align*}\rightarrow\end{align*}
Next, the problem asks us to order the numbers from greatest to least. Be sure to double-check which direction the problem asks for (greatest to least or least to greatest) before you order. Once we have the rounded numbers it is pretty simple to order them from greatest to least.
Our answer is 1.9 km, 1.8 km, 1.7 km, 1.5 km.
Real Life Example Completed
The 100
Here is the original problem once again. Reread this problem and then underline any important information before beginning to solve it.
Connor loves running track for his middle school. In fact, he is one of the fastest runners and he specializes in short distances like the 100. Connor has set a goal this year to run a personal best in the 100. His fastest race time is 13.91 and his slowest race time is 15.16. Connor has set a goal of running the 100 in 12.51. While this is ambitious, he believes that with the right mind-set and perseverance that he will accomplish his goal.
Jeff is a friend of Connor’s who attends the middle school in the next town over. Jeff is also a runner and he also specializes in the 100. The boys are good friends but they definitely love to compete against each other. Sometimes Jeff is faster and sometimes Connor is faster. Jeff’s best race time in the 100 is 13.42. His slowest race time is 15.32. Jeff’s goal this year is to beat his fastest time and finish the 100 in less than 13.00.
One week before the big race, Connor and Jeff decided to try running practice intervals together. They both ran the 100 on the track: Connor’s time was 13.11 and Jeff’s time was 13.14.
Given these times, who was faster? How does Connor’s practice time compare with his best race time? How does Jeff’s practice time compare with his best race time? Given these numbers, who do you think is on track to accomplish his goal? Give some mathematical reasons why.
To answer the first question, we have to compare the two times that the boys ran during their practice.
13.11 and 13.14
Notice that the thirteen is the same, so we need to look at the decimal. 11 is less than 14, so Connor ran the faster time.
Next, we compare Connor’s practice time and his best race time.
His practice time was 13.11
His fastest race time was 13.91
Connor is definitely getting faster, because his practice time was faster than his best race time.
Now we can compare Jeff’s times.
His practice time was 13.14.
His fastest race time was 13.42.
His practice time is faster than his race time. Jeff is also getting faster.
The last question involves your thinking. You need to decide whether or not you think both boys will meet their time goals for the race. Discuss this with a partner. Be sure to justify your thinking by using mathematical data.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Decimal System
- a system of measuring parts of a whole by using a decimal point.
- Decimal point
- the point that divides a whole number from its parts.
- Decimals
- a part of a whole located to the right of the decimal point.
- Whole Numbers
- counting numbers that do not include fractions or decimals. A whole number is found to the left of the decimal point.
Technology Integration
James Sousa, Example of Ordering Decimals from Least to Greatest
James Sousa, Another Example of Ordering Decimals from Least to Greatest
James Sousa, Rounding Decimals
Other Videos:
http://www.schooltube.com/video/ef265b169dea4d4aa3c5/Reading-and-Writing-Decimals – This is a great basic video that goes through place value as a means for reading and writing decimals.
Time to Practice
Directions: Compare. Write <, >, or = for each ___.
1. 701.304 ___ 701.33
2. 2,012.201 ___ 2,012.021
3. 6.951 ___ 6.9313
4. 45.081 ___ 45.108
Directions: Order the following from least to greatest.
5. 373.291, 373.192, 373.129, 373.219
6. 0.4755, 0.4764, 0.4754, 0.4674
7. 7.16, 7.2, 7.06, 7.21
8. 25.417, 25.741, 25.074 25.407
Directions: Round the following to the nearest whole number.
9. 621.891
10. 1,318.0999
11. 17.275
12. 49.64
Directions: Round the following to the designated place.
13. 32.295 to the nearest hundredth
14. 0.1062461 to the nearest ten-thousandth
15. 2.4004728 to the nearest hundred-thousandth
16. 4,062.03 to the nearest tenth
Directions: Round each decimal to the nearest hundredth; then compare. Write <, >, or = for each ___.
17. 1.346 ___ 1.349
18. 0.0589 ___ 0.0559
19. 62.216 ___ 62.301
20. 5.011 ___ 5.001
Directions: Round each decimal to the nearest thousandth. Then order from greatest to least.
21. 2.03489, 2.03266, 2.0344, 2.03909
22. 16.0995, 16.0875, 16.0885, 16.089
23. 3.8281, 3.8208, 3.8288, 3.8218
24. .05672, .05972, .05612, .0575
25. A pet store assigns sleeping boxes based on the length of kittens. The longest kitten gets the largest sleeping box. Match the following kitten lengths with the area of the sleeping boxes.
Kittens: | Sleeping Boxes: |
---|---|
Ursula—23.11 cm |
A—\begin{align*}647.06 \ cm^2\end{align*} |
Mittens—23.51 cm |
B—\begin{align*}637.56 \ cm^2\end{align*} |
Josh—25.01 cm |
C—\begin{align*}647.46 \ cm^2\end{align*} |
Boxer—20.31 cm |
D—\begin{align*}647.9 \ cm^2\end{align*} |
26. Tamara’s famous holiday punch follows a precise recipe: 0.872 liters of orange juice; 0.659 liter of grapefruit juice; 1.95 liters of club soda; 0.981 liters of lemonade; and 0.824 liter of limeade. Round her ingredient list to the nearest tenth; then order from least to greatest.
27. Sean weighed his textbooks with these results: Math—3.652 kg; English—3.596 kg; History—3.526 kg; Science—3.628 kilograms. Order his textbooks from greatest to least weight.
28. Mrs. King is pricing cabins at the state park for a weekend getaway with the family. A 2-person cabin is $53.90 for the weekend; a 3-person cabin is $67.53 for the weekend; a 4-person cabin is $89.72 for the weekend. Round each price to the nearest whole number; then estimate the cheapest combination of cabins if there are 6 people in the King family.
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