2.1: Decimal Comparisons without Rounding
Have you ever run track? Take a look at this dilemma.
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 and here were their times. Connor’s time was 13.11 and Jeff’s time was 13.14.
Given these times, who had the faster time? 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 and you will see this problem again at the end of the Concept.
Guidance
Before we move to comparing and ordering decimals, let’s first begin by thinking about 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 like 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 are certainly familiar with 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*} or 35 hundredths of a dollar.
Decimals 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. As you move right from the decimal point, each value is divided by 10.
Numbers to the right of the decimal point are whole numbers.
Numbers to the left 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 and a part of a whole 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.
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 just “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. But first, let’s just focus on comparing.
We compare and order decimal numbers just like whole numbers—by working from left to right.
With decimals, 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 filling in zeros 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.
Compare 8.507 and 8.570. Write >, <, or =.
The first thing to notice is that each number has one whole number, 8. Because the whole number is the same, 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.
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), while the number 8.570 has \begin{align*}\frac{7}{100}\end{align*} (7 hundredths).
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.
Now we can use this information to help us when writing decimals in order from least to greatest or greatest to least.
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 help us see the difference in values.
This time we’ll have to add extra places 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 | ||
4 | 9 | 4 | . | 0 | 1 | 9 |
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 9s 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
Practice these skills by comparing and/or ordering these decimals.
Example A
98.065 ______ 98.08
Solution: <
Example B
5.237 ______ 5.231
Solution: >
Example C
Write in order from least to greatest: .098, 2.45, 2.099, 2.67
Solution: .098, 2.099, 2.45, 2.67
Now back to the 100.
Here is the original problem once again.
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 and here were their times. Connor’s time was 13.11 and Jeff’s time was 13.14.
Given these times, who had the faster time? 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 in this Concept.
- 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.
Guided Practice
Here is one for you to try on your own.
Compare .00456 and .00458
Answer
If you look at these two decimals, we can see that the first few four values are all the same. Therefore, we have to look at the last value to figure out which number is larger and which is smaller.
6 < 8
Therefore, .00456 < .00458.
This is our answer.
Video Review
Here is a video for review.
- This is a James Sousa video on ordering decimals from least to greatest.
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
5. 4.5670 _____ 3.452
6. .0017 ______.0019
7. 1.0056 ______1.0560
8. 3.4501 ______3.4510
9. 67.001 ______67.1
Directions: Order the following from least to greatest.
10. 373.291, 373.192, 373.129, 373.219
11. 0.4755, 0.4764, 0.4754, 0.4674
12. 7.16, 7.2, 7.06, 7.21
13. 25.417, 25.741, 25.074 25.407
14. 15.001, 15.067, 15.100, 15.0001
15. 6.780, 6.087, 6.870, 6.008
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Image Attributions
Here you'll learn to compare and order decimals without rounding.