2.3: Decimal Comparisons with Rounding
Remember how Connor ran track? Well, his sister Leah volunteers at the local animal shelter. Here is a dilemma that she faced when she arrived on Saturday morning.
The animal shelter assigns sleeping boxes based on the length of kittens. The longest kitten gets the largest sleeping box. Leah was assigned the task of providing each kitten with the correct box. She is puzzled as to how to do this.
Here is the list of 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*} |
Working with decimals is essential for sorting out this problem.
Do you know how to do it?
This Concept is about comparing and ordering decimals with rounding. By the end of it, you will be able to help Leah with her dilemma.
Guidance
You already 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.
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 nine 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.
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
Now it's time for you to try a few on your own.
Example A
Compare each number after rounding to the nearest hundredth, 4.567 and 4.562
Solution: 4.57 > 4.56
Example B
Compare each number after rounding to the nearest tenth, .234 and .245
Solution: .2 = .2
Example C
Round each to the nearest tenth and write in order from least to greatest, .0567, .291, .1742
Solution:: .0567, .1742,.291
Now back to Leah and the kittens. Here is the original problem once again.
Remember how Connor ran track? Well, his sister Leah volunteers at the local animal shelter. Here is a dilemma that she faced when she arrived on Saturday morning.
The animal shelter assigns sleeping boxes based on the length of kittens. The longest kitten gets the largest sleeping box. Leah was assigned the task of providing each kitten with the correct box. She is puzzled as to how to do this.
Here is the list of 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*} |
To solve this problem, let's first write the kittens in order from smallest to largest.
Boxer = 20.31
Ursula = 23.11
Mittens = 23.51
Josh = 25.01
Now the boxes in order from least to greatest.
Box B = 637.56
Box A = 647.06
Box C = 647.46
Box D = 647.9
Next, we put them together.
Boxer = Box B
Ursula = Box A
Mittens = Box C
Josh = Box D
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.
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.
Answer
To complete this problem, we have to round each to the nearest tenth.
Math = 3.7
English = 3.60
History = 3.5
Science = 3.62
Because both the English book and the Science book round to 3.6, we have to add a hundredths digit. Now we can write them from least to greatest.
The answer is History, English, Science, Math.
Video Review
Here is a video for review.
This is a Khan Academy Video on rounding decimals.
Practice
Directions: Round each decimal to the nearest hundredth; then compare. Write <, >, or = for each ___.
1. 1.346 ___ 1.349
2. 0.0589 ___ 0.0559
3. 62.216 ___ 62.301
4. 5.011 ___ 5.001
5. 65.47 ____65.047
6. 12.324 ____ 12.325
7. .00897 _____.00967
8. .0009876 _____.0001020
9. .9806_____.9870
Directions: Round each decimal to the nearest thousandth. Then order from greatest to least.
10. 2.03489, 2.03266, 2.0344, 2.03909
11. 16.0995, 16.0875, 16.0885, 16.089
12. 3.8281, 3.8208, 3.8288, 3.8218
13. .05672, .05972, .05612, .0575
Directions: Complete each word problem.
14. Tamara’s famous holiday punch follows a precise recipe: 0.872 l orange juice; 0.659 l grapefruit juice; 1.95 l club soda; 0.981 1 lemonade; and 0.824 l limeade. Round her ingredient list to the nearest tenth; then order from least to greatest.
15. 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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Image Attributions
Here you'll learn to compare and order decimals with rounding.