Comparing & Rounding Decimals Lesson

In our last lesson, we learned about decimal numbers. We also learned how to convert between decimal fractions and decimals. Now, we will learn about decimal place value, comparing decimals, and how to round decimals.

Decimal Place Value

We will begin by looking at an expanded place value chart: When comparing our whole number place value chart to our decimal place value chart, everything to the left of the decimal point is the same. Our rightmost digit of any whole number is in the ones’ place. As we move left, we multiply the previous place by 10 to get to the next place. From right to left:

• 1’s
• 10’s (1 x 10 = 10)
• 100’s (10 x 10 = 100)
• 1000’s (100 x 10 = 1000)
• 10,000’s (1000 x 10 = 10,000)
• 100,000's (10,000 x 10 = 100,000)

This process continues forever, we can keep multiplying the previous place by 10 to obtain the next place to the left. We can also reverse this process as we move to the right. Each time we move right, we divide the previous place by 10. Let's suppose we work left to right, starting with the 100,000's place:

• 100,000's
• 10,000's (100,000 ÷ 10 = 10,000)
• 1000's (10,000 ÷ 10 = 1000)
• 100's (1000 ÷ 10 = 100)
• 10's (100 ÷ 10 = 10)
• 1's (10 ÷ 10 = 1)

If we expand this process to a decimal number, what do you expect to the right of the ones' place?

• 1's
• 1/10 (1 ÷ 10 = 1/10)
• 1/100 (1/10 ÷ 10 = 1/100)
• 1/1000 (1/100 ÷ 10 = 1/1000)

Again, this process continues forever, we can keep dividing the previous place on the left by 10 to obtain the next place to the right. We sometimes see students expecting to see a oneths' place since we have a ones' place with whole numbers. We will not see a oneths' place because dividing 1 by 10 gives us 1/10. We also notice that each place to the right of the decimal point ends with "ths". To find the place value for a given digit in a decimal number, we can use the decimal place value chart. Let's take a look at a few examples.
Example 1: State the place value of the underlined digit in the number:
32.647 The 4 is in the hundredths' place. The digit 4 is multiplied by the fraction 1/100 to obtain a value of 4/100 or 0.04 in decimal form. $$4 \cdot \frac{1}{100}=\frac{4}{100}=0.04$$ Example 2: State the place value of the underlined digit in the number:
197.583 The 3 is in the thousandths' place. The digit 3 is multiplied by the fraction 1/1000 to obtain a value of 3/1000 or 0.003 in decimal form. $$3 \cdot \frac{1}{1000}=\frac{3}{1000}=0.003$$

How to Compare two or more Decimals

We will now learn how to compare the size of two or more decimals. In order to perform this task effectively, we need to learn about working with zero when decimals are involved. First and foremost consider these two numbers:
26 ? 0026
Which is larger?
The two numbers are the same:
26 = 0026
The two zeros in front of the 2 in 0026 don't add any value whatsoever. What if we looked at:
26 ? 2600
In this case, 2600 is larger:
26 < 2600
The two zeros at the end of the number 2600 do add value here. We would all much rather have $2600 versus only $26.
When working with whole numbers zeros added to the right of the number add value, zeros added to the left don't add value.
13 = 0000000000013
We can place as many zeros in front of 13 as we would like, the value doesn't change.
13 ≠ 130,000,000
Placing zeros after 13 changes the value of the number. We added seven zeros to the end. Now the number is one hundred thirty million, which is much larger than 13.
When we work with decimals, adding zeros will not have the same impact. Any zeros added to the right or end of a decimal number do not change the value:
54 = 54.000000
9.375 = 9.375000000
In other words, to the right of our decimal point and to the right of our final non-zero digit, we can add as many zeros as we would like. This will never change the value of a number. If we add a zero in between our decimal point and the final non-zero digit, the value will change:
9.375 ≠ 9.00000375
5.601 ≠ 5.00601

Comparing Decimals

We will often be asked to compare the size of two decimals. If the decimal number contains a whole number part, we will compare those parts first. If the two whole number parts are identical, then we will consider the following procedure:

• Start at the digit immediately after the decimal point and compare between the numbers, if one digit is larger it belongs to the larger number
• When the digits are identical, we continuously move one digit to the right and compare

Example 3: Replace the ? with the correct inequality symbol "<", ">", or "=".
3.1105 ? 3.1005
We can see the whole number part is the same for each number (3)
The next digit right (tenths' place) for each number is the same (1)
The next digit right (hundredths' place) for each number is different:
3.1105
3.1005
We can see the number on the left (3.1105) has a 1 in the hundredths' place versus a 0 for the number (3.1005). This tells us 3.1105 is larger than 3.1005:
3.1105 > 3.1005
Example 4: Replace the ? with the correct inequality symbol "<", ">", or "=".
6.0989 ? 6.098900
We can see the whole number part is the same for each number (6)
The next four digits right (tenths' place » ten thousandths' place) for each number are the same (0989)
The left number 6.0989 terminates after 9 and the right number has an additional two zeros at the end. Do these zeros add any value? We should remember that zeros to the right of the decimal point and after the last non-zero digit do not add any value. The two numbers are equal in value.
6.0989 = 6.098900

How to Round Decimals

When we round with decimals, we use the same procedure from whole numbers with one small change. We will delete any zeros that do not add value to the number. Let's revisit our rounding whole numbers procedure:

1. Locate the digit in the round-off place (e.g., If Rounding to the nearest ten, we identify the digit in the tens' place)
2. If the digit to the right of the round-off place is:
   • less than 5 (0,1,2,3, or 4) - leave the digit in the round-off place unchanged
   • 5 or larger (5,6,7,8, or 9) - increase the round-off place digit by 1
3. Replace each digit to the right of the round-off place with a zero

We can modify this procedure to work with decimal numbers by stating we will delete any zeros that don't add any value. Let's look at some examples.
Example 5: Round 49.783 to the nearest tenth.
Identify the digit in the tenths' place:
49.783
We can see our 7 in the tenths' place.
We look one digit to the right to make our decision. In this case, we have an 8, this falls in the category of 5 or more so we round up. This means we will increase 7 by 1 (7 + 1 = 8) to 8. Here's the different part. With whole numbers, we replace every digit that follows with a zero. If we try that here:
49.800
These zeros add no value in this case, so we can just modify our procedure and say we will round up and delete all digits that follow:
49.783
49.883
49.8
49.783 rounded to the nearest tenth is 49.8
Example 6: Round 6.91303 to the nearest hundredth.
Identify the digit in the hundredths' place.
6.91303
We can see our 1 in the hundredths' place.
We look one digit to the right to make our decision. In this case, we have a 3, this falls in the category of 4 or less so we round down. This means we will keep our digit 1 the same and delete all digits that follow:
6.91303
6.91
6.91303 rounded to the nearest hundredth is 6.91

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