Lesson Objectives

- Demonstrate an understanding of decimals
- Demonstrate an understanding of multiplication with whole numbers
- Learn how to quickly multiply a decimal by 10 or a power of 10
- Learn how to multiply two or more decimals

## How to Multiply Decimals

We previously learned how to multiply multi-digit whole numbers.
We will build on this process and show how to find the product of two or more decimals. When we multiply two or more decimals together:

Example 1: Find each product:

3.25 x 7.17

Ignore the decimal points in each factor and find the product:

325 x 717 = 233,025

Place a visible decimal point after the final digit of the number:

233,025.

Count the total number of decimal places between all factors:

3.25 » (2 decimal places)

7.17 » (2 decimal places)

Between the two factors, we have a total of four decimal places.

We will move our decimal point in the number (233,025.) four places to the left:

23.3025

3.25 x 7.17 = 23.3025

Example 2: Find each product:

4.5 x 1.02 x 0.009

Ignore the decimal points in each factor and find the product:

45 x 102 x 9 = 41,310

Place a visible decimal point after the final digit of the number:

41,310.

Count the total number of decimal places between all factors:

4.5 » (1 decimal place)

1.02 » (2 decimal places)

0.009 » (3 decimal places)

Between the three factors, we have a total of six decimal places.

We will move our decimal point in the number (41,310.) six places to the left:

0.041310

4.5 x 1.02 x 0.009 = 0.04131

5 x 1 = 5

5 x 10 = 50

5 x 100 = 500

5 x 1000 = 5000

At this point, we know the trick for multiplying when we have powers of ten or trailing zeros. We learned that we could multiply the non-zero numbers and attach the total number of trailing zeros between all factors to the end of the number. When we multiply by 10 or a power of 10, we can just move the decimal point one place to the right for each 0 in the power of 10. We previously learned that to the right of a decimal point, we can add zeros after the final non-zero digit and not change the value of a number. Let's look at an example. Suppose we start with the number 3:

We can write 3 as 3.0:

3.0

If we multiply by 10, we have one zero in the power of 10, we can just move the decimal point one place to the right to obtain our answer.

3.0 x 10 = 30.0

Suppose we tried the same technique with 10,000. This power of 10 has four zeros:

3 x 10,000

3.0 x 10,000

We move our decimal point four places to the right:

3.0 x 10,000 = 30,000.0

Example 3: Find each product:

17.215 x 1000

We are multiplying by 1000, this power of 10 has three zeros:

17.215 x 1000

We move our decimal point three places to the right:

17,215.0

Example 4: Find each product:

1.907516 x 10,000

We are multiplying by 10,000, this power of 10 has four zeros:

1.907516 x 10,000

We move our decimal point four places to the right:

19,075.16

When we see a power of 10 in exponent form, we can use the same process. Recall that a power of 10 such as 10

10

10

When we see a decimal multiplied by a power of 10 in exponent form, we move our decimal point to the right by the same number of digits as the exponent on 10. Let's look at a few examples.

Example 5: Find each product

3.99156 x 10

Our exponent on 10 is a 4.

We move our decimal point in the number 3.99156 four places to the right:

3.99156 » 39,915.6

3.99156 x 10

Example 6: Find each product

0.228195001 x 10

Our exponent on 10 is an 8.

We move our decimal point in the number 0.228195001 eight places to the right:

0.228195001 » 22,819,500.1

0.228195001 x 10

- Ignore all decimal points in each factor (numbers being multiplied) » pretend the numbers are whole numbers
- Find the product of the whole numbers and place a visible decimal point after the final digit (rightmost) of the answer
- Count the total number of decimal places (places to the right of the decimal point) between all factors
- Move the decimal point in the answer to the left by the number of decimal places counted in the last step

Example 1: Find each product:

3.25 x 7.17

Ignore the decimal points in each factor and find the product:

325 x 717 = 233,025

Place a visible decimal point after the final digit of the number:

233,025.

Count the total number of decimal places between all factors:

3.25 » (2 decimal places)

7.17 » (2 decimal places)

Between the two factors, we have a total of four decimal places.

We will move our decimal point in the number (233,025.) four places to the left:

23.3025

3.25 x 7.17 = 23.3025

Example 2: Find each product:

4.5 x 1.02 x 0.009

Ignore the decimal points in each factor and find the product:

45 x 102 x 9 = 41,310

Place a visible decimal point after the final digit of the number:

41,310.

Count the total number of decimal places between all factors:

4.5 » (1 decimal place)

1.02 » (2 decimal places)

0.009 » (3 decimal places)

Between the three factors, we have a total of six decimal places.

We will move our decimal point in the number (41,310.) six places to the left:

0.041310

4.5 x 1.02 x 0.009 = 0.04131

### Multiplying by 10 or a Power of 10

When we multiply by 10 or a positive multiple of 10 (10, 100, 1000,...) we can use a shortcut. Observe the following pattern:5 x 1 = 5

5 x 10 = 50

5 x 100 = 500

5 x 1000 = 5000

At this point, we know the trick for multiplying when we have powers of ten or trailing zeros. We learned that we could multiply the non-zero numbers and attach the total number of trailing zeros between all factors to the end of the number. When we multiply by 10 or a power of 10, we can just move the decimal point one place to the right for each 0 in the power of 10. We previously learned that to the right of a decimal point, we can add zeros after the final non-zero digit and not change the value of a number. Let's look at an example. Suppose we start with the number 3:

We can write 3 as 3.0:

3.0

If we multiply by 10, we have one zero in the power of 10, we can just move the decimal point one place to the right to obtain our answer.

3.0 x 10 = 30.0

Suppose we tried the same technique with 10,000. This power of 10 has four zeros:

3 x 10,000

3.0 x 10,000

We move our decimal point four places to the right:

3.0 x 10,000 = 30,000.0

Example 3: Find each product:

17.215 x 1000

We are multiplying by 1000, this power of 10 has three zeros:

17.215 x 1000

We move our decimal point three places to the right:

17,215.0

Example 4: Find each product:

1.907516 x 10,000

We are multiplying by 10,000, this power of 10 has four zeros:

1.907516 x 10,000

We move our decimal point four places to the right:

19,075.16

When we see a power of 10 in exponent form, we can use the same process. Recall that a power of 10 such as 10

^{3}can be evaluated quickly by writing a 1 followed by the exponent number of zeros:10

^{3}= 100010

^{7}= 10,000,000When we see a decimal multiplied by a power of 10 in exponent form, we move our decimal point to the right by the same number of digits as the exponent on 10. Let's look at a few examples.

Example 5: Find each product

3.99156 x 10

^{4}Our exponent on 10 is a 4.

We move our decimal point in the number 3.99156 four places to the right:

3.99156 » 39,915.6

3.99156 x 10

^{4}= 39,915.6Example 6: Find each product

0.228195001 x 10

^{8}Our exponent on 10 is an 8.

We move our decimal point in the number 0.228195001 eight places to the right:

0.228195001 » 22,819,500.1

0.228195001 x 10

^{8}= 22,819,500.1
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