2.3: Decimal Comparisons with Rounding
Let’s Think About It
Joey, Chris, and Brad each had jobs over the summer. Now they are back at school and they are comparing how much money they each made. Joey worked at his dad's ice cream shop and made $632.15. Chris babysat his little cousins and made $610.98. Brad mowed lawns in his neighborhood and made $624.50. If the boys first round their earnings to the nearest dollar, who made the most money? How can they order their earnings from greatest to least?
In this concept, you will learn how to compare and order decimals with rounding.
Guidance
To round a number means to approximate a number with a new number that has fewer non-zero digits.
Recall that there are four steps for rounding numbers.
- Identify the place you want to round to and find that digit within your number.
- Look at the digit to the right of the place you want to round to.
- If the digit to the right is between 5 and 9, increase the digit in the place you want to round to by 1. If the digit to the right is between 0 and 4, keep the digit in the place you want to round to the same. NOTE: If the digit in the place you want to round to is a 9 and you need to increase it by 1, change it to zero and increase the digit to its left by 1.
- Replace all digits after the place you are rounding to with zeros. If these digits occur to the right of the decimal point, it is not necessary to write the zeros.
You can compare rounded numbers in the same way that you compare unrounded numbers - from left to right.
Here is an example.
Compare 77.0949 and 77.0961 after rounding to the nearest hundredth.
First, round each number to the nearest hundredth. Remember that the hundredths place is the second digit to the right of the decimal point. Look at each digit in the hundredths place and then the digit to the right of that to decide if you will need to round up the digit in the hundredths place or keep it the same.
Start with 77.0949. 9 is the digit in the hundredths place. The digit to the right of the 9 is the 4 in the thousandths place. Because the digit to the right of the 9 is between 0 and 4, you will keep the 9 the same.
77.0949 rounds to 77.09.
Next, round 77.0961 to the nearest hundredth. Once again, 9 is the digit in the hundredths place. The digit to the right of the 9 is the 6 in the thousandths place. Because the digit to the right of the 9 is between 5 and 9, you will need to round up. The 9 will become a 0 and the digit to the left of the 9 (the 0) will increase by 1 to 1.
77.0961 rounds to 77.10.
Now, you can compare the two rounded numbers. Write the numbers in a place value chart. Make sure to line up the decimal points.
Tens |
Ones |
. |
Tenths |
Hundredths |
7 |
7 |
. |
0 |
9 |
7 |
7 |
. |
1 |
0 |
Compare the two numbers starting with the digits on the left and moving to the right.
- The digits are the same in the tens and ones places.
- The digits are different in the tenths place. 0 is less than 1. This means 77.09 is less than 77.10.
The answer is \begin{align*}77.09 < 77.10\end{align*}.
You can use these same steps when ordering numbers from least to greatest or from greatest to least. First, round the numbers to the desired place. Then, compare and order them.
Here is an example.
Round the following numbers to the nearest tenth. Then, order from greatest to least.
\begin{align*}5.954, 5.599, 5.533, 6.062\end{align*}
First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.
- 5.954 rounds to 6.0
- 5.599 rounds to 5.6
- 5.533 rounds to 5.5
- 6.062 rounds to 6.1
Finally, you can compare the rounded numbers and order them from greatest to least. Write the numbers in a place value chart. Make sure to line up the decimal points.
Ones |
. |
Tenths |
6 |
. |
0 |
5 |
. |
6 |
5 |
. |
5 |
6 |
. |
1 |
Now, compare the four numbers starting with the digits on the left and moving to the right. In the ones place, there are two digits that are 6 and two digits that are 5. The numbers with a 6 in the ones place are bigger than the numbers with a 5 in the ones place.
Next, compare 6.0 and 6.1. 6.1 has a 1 in the tenths place whereas 6.0 only has a 0 in the tenths place. Because 1 is greater than 0, 6.1 is greater than 6.0.
Now compare 5.6 and 5.5. 5.6 has a 6 in the tenths place whereas 5.5 has a 5 in the tenths place. Because 6 is greater than 5, 5.6 is greater than 5.5.
The answer is that once rounded to the nearest tenth and ordered from greatest to least, the numbers are 6.1, 6.0, 5.6, 5.5.
Guided Practice
Sean weighed his textbooks with these results:
- Math - 3.652 kg
- English - 3.596 kg
- History - 3.526 kg
- Science - 3.628 kg
Round each weight to the nearest hundredth and then order his textbooks from greatest to least weight.
First, round each number to the nearest hundredth. Remember that the hundredths place is the second digit to the right of the decimal point. Look at each digit in the hundredths place and then the digit to the right of that to decide if you will need to round up the digit in the hundredths place or keep it the same.
- 3.652 rounds to 3.65
- 3.596 rounds to 3.60
- 3.526 rounds to 3.53
- 3.628 rounds to 3.63
Now, you can compare the rounded numbers and order them from greatest to least. Write the numbers in a place value chart. Make sure to line up the decimal points.
Ones |
. |
Tenths |
Hundredths |
3 |
. |
6 |
5 |
3 |
. |
6 |
0 |
3 |
. |
5 |
3 |
3 |
. |
6 |
3 |
Now, compare the four numbers starting with the digits on the left and moving to the right. In the ones place, all the digits are the same. In the tenths place there is one 5 and three 6s. The number with the 5 is the smallest number. This means 3.53 will go at the end of your list of numbers ordered from greatest to least.
Next, focus on the three numbers that have a 6 in the tenths place: 3.65, 3.60, 3.63. In the hundredths place the digits are different. 5 is the biggest, then 3, then 0. This means 3.65 is greater than 3.63 which is greater than 3.60.
The answer is that ordered from greatest to least, the numbers are 3.65, 3.63, 3.60, 3.53.
Examples
Example 1
Compare each number after rounding to the nearest hundredth: 4.567 and 4.562.
First, round each number to the nearest hundredth. Remember that the hundredths place is the second digit to the right of the decimal point. Look at each digit in the hundredths place and then the digit to the right of that to decide if you will need to round up the digit in the hundredths place or keep it the same.
- 4.567 rounds to 4.57
- 4.562 rounds to 4.56
Now, you can compare the two rounded numbers. Write the numbers in a place value chart. Make sure to line up the decimal points.
Ones |
. |
Tenths |
Hundredths |
4 |
. |
5 |
7 |
4 |
. |
5 |
6 |
Compare the two numbers starting with the digits on the left and moving to the right.
- The digits are the same in the ones and tenths places.
- The digits are different in the hundredths place. 7 is greater than 6. This means 4.57 is greater than 4.56.
The answer is \begin{align*} 4.57 > 4.56\end{align*}.
Example 2
Compare each number after rounding to the nearest tenth: 0.234 and 0.245.
First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.
- 0.234 rounds to 0.2
- 0.245 rounds to 0.2
Now, you can compare the two rounded numbers. Without even having to make a place value chart you can see that when rounded, the numbers are equal.
The answer is \begin{align*}0.2 = 0.2\end{align*}.
Example 3
Round each number to the nearest tenth and then write in order from least to greatest: 0.0567, 0.291, 0.1742.
First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.
- 0.0567 rounds to 0.1
- 0.291 rounds to 0.3
- 0.1742 rounds to 0.2
Now, you can compare the rounded numbers and order them from least to greatest. Write the numbers in a place value chart. Make sure to line up the decimal points.
Ones |
. |
Tenths |
0 |
. |
1 |
0 |
. |
3 |
0 |
. |
2 |
Now, compare the three numbers starting with the digits on the left and moving to the right. In the ones place, all the digits are the same. In the tenths place, all the digits are different. The smallest number is 1, the next smallest number is 2, and the largest number is 3. This means 0.1 is less than 0.2 which is less than 0.3.
The answer is that rounded to the nearest tenths and ordered from least to greatest, the numbers are 0.1, 0.2, 0.3.
Follow Up
Remember Joey, Chris, and Brad and their summer jobs? Joey made $632.15, Chris made $610.98, and Brad made $624.50. The boys want to round their earnings to the nearest dollar and then figure out who made the most money. They also want to order their rounded earnings from greatest to least.
First, round each number to the nearest dollar. Rounding to the nearest dollar is like rounding to the ones place which is the first digit to the left of the decimal point. Look at each digit in the ones place and then the digit to the right of that to decide if you will need to round up the digit in the ones place or keep it the same.
- 632.15 rounds to 632
- 610.98 rounds to 611
- 624.50 rounds to 625
Now, you can compare the rounded numbers and order them from greatest to least. Start with the digits on the left and move to the right. In the hundreds place, all the digits are the same. In the tens place, all the digits are different. The largest number is 3, the next largest number is 2, and the smallest number is 1. This means 632 is greater than 625 which is greater than 611.
The answer is that Joey made the most, making about $632. Ordered from greatest to least, the boys made $632, $625, $611.
Explore More
Round each decimal to the nearest hundredth; then compare. Write <, >, or = for each \begin{align*}\underline{\; \; \; \; \; \; \;}\end{align*}.
1. \begin{align*}1.346 \underline{\; \; \; \; \; \; \;} 1.349\end{align*}
2. \begin{align*} 0.0589 \underline{\; \; \; \; \; \; \;} 0.0559\end{align*}
3. \begin{align*} 62.216 \underline{\; \; \; \; \; \; \;} 62.301\end{align*}
4. \begin{align*}5.011 \underline{\; \; \; \; \; \; \;} 5.001 \end{align*}
5. \begin{align*} 65.47 \underline{\; \; \; \; \; \; \;} 65.047\end{align*}
6. \begin{align*}12.324 \underline{\; \; \; \; \; \; \;} 12.325\end{align*}
7. \begin{align*} .00897 \underline{\; \; \; \; \; \; \;} .00967\end{align*}
8. \begin{align*}.0009876 \underline{\; \; \; \; \; \; \;} .0001020\end{align*}
9. \begin{align*}.9806 \underline{\; \; \; \; \; \; \;} .9870\end{align*}
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
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
In this concept, you will learn how to compare and order decimals with rounding.