Lesson Objectives

- Demonstrate an understanding of long division
- Demonstrate an understanding of decimals
- Learn how to divide a decimal by a whole number
- Learn how to divide a whole number by a decimal
- Learn how to divide a decimal by a decimal
- Learn how to quickly divide by 10 or a power of 10

## How to Divide Decimals

In our lesson on long division, we learned a step by step process for dividing multi-digit whole numbers. What happens when we want to divide when decimals are involved? We will use our long division process along with one or two extra steps, depending on the situation. When we divide with decimals, we will face three scenarios:

Example 1: Find each quotient

68.4 ÷ 36

Set up the long division normally: Place the decimal point from the dividend directly above into the answer: Divide Normally: Our long division process is the same, with the exception of dealing with the decimal point. We are basically dividing 684 by 36 and moving the decimal point in the answer one place to the left.

68.4 ÷ 36 = 1.9

Example 2: Find each quotient

74.04 ÷ 6

Set up the long division normally: Place the decimal point from the dividend directly above into the answer: Divide Normally: Our long division process is the same, with the exception of dealing with the decimal point. We are basically dividing 7404 by 6 and moving the decimal point in the answer two places to the left.

74.04 ÷ 6 = 12.34

Example 3: Find each quotient

21.5 ÷ 2

We will combine steps 1 and 2, set up the long division and bring the decimal point directly above in the answer: Let's perform our long division up to the remainder part: Normally we would stop here, since there are no more numbers to bring down. This is where adding zeros to the right of the decimal point comes in to play. After the 5 in 21.5, we can add as many zeros as we would like without changing the value of the number. Let's add one zero and make our dividend: 21.50. This will allow us to bring down a zero and continue our division: 21.5 ÷ 2 = 10.75

Example 4: Find each quotient

4.15 ÷ 3

We will combine steps 1 and 2, set up the long division and bring the decimal point directly above in the answer: Let's perform our long division up to the remainder part: After the 5 in 4.15, we can add as many zeros as we would like without changing the value of the number. Let's add one zero and make our dividend: 4.150. This will allow us to bring down a zero and continue our division: We still have a remainder, so let's again add another zero to the end of our dividend: We can see a pattern emerging. Each time we add a zero to the end of our dividend, the result will be the same. We will continue to get a 3 in our answer and a 1 as a remainder forever. We can stop at this point and notate the continuously repeating digit 3 with an overbar:

4.15 ÷ 3 = 1.383

When we divide by a decimal, we move the decimal point in the divisor by enough places to the right such that it becomes a whole number. We then match this movement in our dividend.

In other words, if we have something such as: 38.5 ÷ 2.1, we would move our decimal point in 2.1 one place to the right to form the whole number 21. We then match this movement in the number 38.5. We move our decimal point one place to the right and end up with 385. Our division problem becomes: 385 ÷ 21. Let's take a look at a few examples.

Example 5: Find each quotient

8.37 ÷ 2.7

We will begin by moving our decimal point in our divisor 2.7 one place to the right. This will create the whole number 27. We match this movement in our dividend. We move the decimal point in 8.37 one place to the right. This will create the number: 83.7

We can now perform our long division: 8.37 ÷ 2.7 = 3.1

Example 6: Find each quotient

57.81 ÷ 7.05

We will begin by moving our decimal point in our divisor 7.05 two places to the right. This will create the whole number 705. We match this movement in our dividend. We move the decimal point in 57.81 two places to the right. This will create the number: 5781.0 or 5781

We can now perform our long division: 57.81 ÷ 7.05 = 8.2

32.7534 x 100 = 3275.34

In our example above, we multiplied by 100. 100 is a power of 10 with two zeros, so our decimal point was moved 2 places to the right.

When we divide by powers of 10, we use the same thought process. Instead of moving right, we move left. In other words, when we divide by a power of 10, we can move the decimal point one place left for each zero in the power of 10:

5000 ÷ 10 = 500

Write 5000 as 5000.0

Move the decimal point one place to the left, since 10 has one zero

5000.0 » 500.00

500.00 = 500

We easily find 500 as our answer.

280,000 ÷ 10

Write 280,000 as 280,000.0

Move the decimal point three places to the left, since 10

280,000.0 » 280.0000

280.0000 = 280

We easily find 280 as our answer

Let's look at a few examples.

Example 7: Find each quotient

327.91 ÷ 10

Move the decimal point two places left:

3.2791

327.91 ÷ 10

Example 8: Find each quotient

985.004 ÷ 10

Move the decimal point five places left:

0.00985004

985.004 ÷ 10

- Dividing a decimal by a whole number
- Dividing a whole number by a decimal
- Dividing a decimal by a decimal

### Dividing a Decimal by a Whole Number

- Set up the long division normally
- Place the decimal point from the dividend directly above into the answer
- Divide normally

Example 1: Find each quotient

68.4 ÷ 36

Set up the long division normally: Place the decimal point from the dividend directly above into the answer: Divide Normally: Our long division process is the same, with the exception of dealing with the decimal point. We are basically dividing 684 by 36 and moving the decimal point in the answer one place to the left.

68.4 ÷ 36 = 1.9

Example 2: Find each quotient

74.04 ÷ 6

Set up the long division normally: Place the decimal point from the dividend directly above into the answer: Divide Normally: Our long division process is the same, with the exception of dealing with the decimal point. We are basically dividing 7404 by 6 and moving the decimal point in the answer two places to the left.

74.04 ÷ 6 = 12.34

### Decimals and Remainders in Long Division

When we studied long division with whole numbers, we generally stopped dividing when we had a remainder. If we saw a problem such as: 13 ÷ 2, we would say the result is 6 R1. When we start dividing with decimals, we can continue the process and get an answer of 6.5. This is the same result we will see with a calculator. We achieve this result by placing enough zeros after the decimal point and continuing the division. Recall that 13, 13.0, 13.00, and 13.000 are all the same value. We continue to add zeros until we have no remainder. In some cases, we will stop when we see a repeating digit or series of digits. When this occurs, we can write the digit or digits that repeat under an overbar (vinculum). Let's take a look at a few examples:Example 3: Find each quotient

21.5 ÷ 2

We will combine steps 1 and 2, set up the long division and bring the decimal point directly above in the answer: Let's perform our long division up to the remainder part: Normally we would stop here, since there are no more numbers to bring down. This is where adding zeros to the right of the decimal point comes in to play. After the 5 in 21.5, we can add as many zeros as we would like without changing the value of the number. Let's add one zero and make our dividend: 21.50. This will allow us to bring down a zero and continue our division: 21.5 ÷ 2 = 10.75

Example 4: Find each quotient

4.15 ÷ 3

We will combine steps 1 and 2, set up the long division and bring the decimal point directly above in the answer: Let's perform our long division up to the remainder part: After the 5 in 4.15, we can add as many zeros as we would like without changing the value of the number. Let's add one zero and make our dividend: 4.150. This will allow us to bring down a zero and continue our division: We still have a remainder, so let's again add another zero to the end of our dividend: We can see a pattern emerging. Each time we add a zero to the end of our dividend, the result will be the same. We will continue to get a 3 in our answer and a 1 as a remainder forever. We can stop at this point and notate the continuously repeating digit 3 with an overbar:

4.15 ÷ 3 = 1.383

### Dividing by a Decimal

In many cases, we will need to divide a whole number by a decimal or a decimal by a decimal. In this situation, we only need to make one adjustment:When we divide by a decimal, we move the decimal point in the divisor by enough places to the right such that it becomes a whole number. We then match this movement in our dividend.

In other words, if we have something such as: 38.5 ÷ 2.1, we would move our decimal point in 2.1 one place to the right to form the whole number 21. We then match this movement in the number 38.5. We move our decimal point one place to the right and end up with 385. Our division problem becomes: 385 ÷ 21. Let's take a look at a few examples.

Example 5: Find each quotient

8.37 ÷ 2.7

We will begin by moving our decimal point in our divisor 2.7 one place to the right. This will create the whole number 27. We match this movement in our dividend. We move the decimal point in 8.37 one place to the right. This will create the number: 83.7

We can now perform our long division: 8.37 ÷ 2.7 = 3.1

Example 6: Find each quotient

57.81 ÷ 7.05

We will begin by moving our decimal point in our divisor 7.05 two places to the right. This will create the whole number 705. We match this movement in our dividend. We move the decimal point in 57.81 two places to the right. This will create the number: 5781.0 or 5781

We can now perform our long division: 57.81 ÷ 7.05 = 8.2

### Dividing by Powers of 10

In our lesson on multiplying decimals, we learned a shortcut for multiplying by positive multiples of 10 (powers of 10). We learned that we can quickly multiply by a power of 10 by moving the decimal point one place to the right for each zero in the power of ten.32.7534 x 100 = 3275.34

In our example above, we multiplied by 100. 100 is a power of 10 with two zeros, so our decimal point was moved 2 places to the right.

When we divide by powers of 10, we use the same thought process. Instead of moving right, we move left. In other words, when we divide by a power of 10, we can move the decimal point one place left for each zero in the power of 10:

5000 ÷ 10 = 500

Write 5000 as 5000.0

Move the decimal point one place to the left, since 10 has one zero

5000.0 » 500.00

500.00 = 500

We easily find 500 as our answer.

280,000 ÷ 10

^{3}= 280Write 280,000 as 280,000.0

Move the decimal point three places to the left, since 10

^{3}is a 1 followed by three zeros: 1000280,000.0 » 280.0000

280.0000 = 280

We easily find 280 as our answer

Let's look at a few examples.

Example 7: Find each quotient

327.91 ÷ 10

^{2}Move the decimal point two places left:

3.2791

327.91 ÷ 10

^{2}= 3.2791Example 8: Find each quotient

985.004 ÷ 10

^{5}Move the decimal point five places left:

0.00985004

985.004 ÷ 10

^{5}= 0.00985004 Ready for more?

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