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# Decimals: Multiplication of Decimals

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply decimals. By the end of the module students should understand the method used for multiplying decimals, be able to multiply decimals, be able to simplify a multiplication of a decimal by a power of 10 and understand how to use the word "of" in multiplication.

## Section Overview

• The Logic Behind the Method
• The Method of Multiplying Decimals
• Calculators
• Multiplying Decimals By Powers of 10
• Multiplication in Terms of “Of”

## The Logic Behind the Method

Consider the product of 3.2 and 1.46. Changing each decimal to a fraction, we have

( 3 . 2 ) ( 1 . 46 ) = 3 2 10 1 46 100 = 32 10 146 100 = 32 146 10 100 = 4672 1000 = 4 672 1000 = four and six hundred seventy-two thousandths = 4 . 672 ( 3 . 2 ) ( 1 . 46 ) = 3 2 10 1 46 100 = 32 10 146 100 = 32 146 10 100 = 4672 1000 = 4 672 1000 = four and six hundred seventy-two thousandths = 4 . 672

Thus, (3.2)(1.46)= 4.672(3.2)(1.46)= 4.672 size 12{ $$3 "." 2$$ $$1 "." "46"$$ =" 4" "." "672"} {}.

Notice that the factor

3.2 has 1 decimal place, 1.46 has 2 decimal places, and the product 4.672 has 3 decimal places. 1+2=3 3.2 has 1 decimal place, 1.46 has 2 decimal places, and the product 4.672 has 3 decimal places. 1+2=3

Using this observation, we can suggest that the sum of the number of decimal places in the factors equals the number of decimal places in the product.

## The Method of Multiplying Decimals

### Method of Multiplying Decimals

To multiply decimals,

1. Multiply the numbers as if they were whole numbers.
2. Find the sum of the number of decimal places in the factors.
3. The number of decimal places in the product is the sum found in step 2.

### Sample Set A

Find the following products.

#### Example 1

6.5 4.36.5 4.3 size 12{6 "." "5 " cdot " 4" "." 3} {}

Thus, 6.54.3=27.956.54.3=27.95 size 12{6 "." 5 cdot 4 "." 3="27" "." "95"} {}.

#### Example 2

23.41.9623.41.96 size 12{"23" "." 4 cdot 1 "." "96"} {}

Thus, 23.41.96=45.86423.41.96=45.864 size 12{"23" "." 4 cdot 1 "." "96"="45" "." "864"} {}.

#### Example 3

Find the product of 0.251 and 0.00113 and round to three decimal places.

Now, rounding to three decimal places, we get

### Practice Set A

Find the following products.

#### Exercise 1

5.3 8.65.3 8.6 size 12{5 "." 3 cdot " 8" "." 6} {}

45.58

#### Exercise 2

2.12 4.92.12 4.9 size 12{2 "." "12" cdot " 4" "." 9} {}

10.388

#### Exercise 3

1.054 0.161.054 0.16 size 12{1 "." "054 " cdot " 0" "." "16"} {}

0.16864

#### Exercise 4

0.00031 0.0020.00031 0.002 size 12{0 "." "00031 " cdot " 0" "." "002"} {}

0.00000062

#### Exercise 5

Find the product of 2.33 and 4.01 and round to one decimal place.

9.3

#### Exercise 6

10 5.39410 5.394 size 12{"10 " cdot " 5" "." "394"} {}

53.94

#### Exercise 7

100 5.394100 5.394 size 12{"100 " cdot " 5" "." "394"} {}

539.4

#### Exercise 8

1000 5.3941000 5.394 size 12{"1000" cdot " 5" "." "394"} {}

5,394

#### Exercise 9

10,000 5.39410,000 5.394 size 12{"10,000 " cdot " 5" "." "394"} {}

59,340

## Calculators

Calculators can be used to find products of decimal numbers. However, a calculator that has only an eight-digit display may not be able to handle numbers or products that result in more than eight digits. But there are plenty of inexpensive ($50 -$75) calculators with more than eight-digit displays.

### Sample Set B

Find the following products, if possible, using a calculator.

#### Example 4

2.58 8.612.58 8.61 size 12{2 "." "58 " cdot " 8" "." "61"} {}

 Display Reads Type 2.58 2.58 Press × 2.58 Type 8.61 8.61 Press = 22.2138

The product is 22.2138.

#### Example 5

0.006 0.00420.006 0.0042 size 12{0 "." "006 " cdot " 0" "." "0042"} {}

 Display Reads Type .006 .006 Press × .006 Type .0042 0.0042 Press = 0.0000252

We know that there will be seven decimal places in the product (since 3 + 4 = 73 + 4 = 7 size 12{"3 "+" 4 "=" 7"} {}). Since the display shows 7 decimal places, we can assume the product is correct. Thus, the product is 0.0000252.

#### Example 6

0.0026 0.119760.0026 0.11976 size 12{0 "." "0026 " cdot " 0" "." "11976"} {}

Since we expect 4 + 5 = 94 + 5 = 9 size 12{"4 "+" 5 "=" 9"} {} decimal places in the product, we know that an eight-digit display calculator will not be able to provide us with the exact value. To obtain the exact value, we must use "hand technology." Suppose, however, that we agree to round off this product to three decimal places. We then need only four decimal places on the display.

 Display Reads Type .0026 .0026 Press × .0026 Type .11976 0.11976 Press = 0.0003114

Rounding 0.0003114 to three decimal places we get 0.000. Thus, 0.0026 0.11976 = 0.0000.0026 0.11976 = 0.000 size 12{0 "." "0026 " cdot " 0" "." "11976 "=" 0" "." "000"} {} to three decimal places.

### Practice Set B

Use a calculator to find each product. If the calculator will not provide the exact product, round the result to four decimal places.

#### Exercise 10

5.126 4.085.126 4.08 size 12{5 "." "126 " cdot " 4" "." "08"} {}

20.91408

#### Exercise 11

0.00165 0.040.00165 0.04 size 12{0 "." "00165 " cdot " 0" "." "04"} {}

0.000066

#### Exercise 12

0.5598 0.42810.5598 0.4281 size 12{0 "." "5598 " cdot " 0" "." "4281"} {}

0.2397

#### Exercise 13

0.000002 0.060.000002 0.06 size 12{0 "." "000002 " cdot " 0" "." "06"} {}

0.0000

## Multiplying Decimals by Powers of 10

There is an interesting feature of multiplying decimals by powers of 10. Consider the following multiplications.

 Multiplication Number of Zeros in the Power of 10 Number of Positions the Decimal Point Has Been Moved to the Right 10 ⋅ 8 . 315274 = 83 . 15274 10 ⋅ 8 . 315274 = 83 . 15274 size 12{"10" cdot 8 "." "315274"="83" "." "15274"} {} 1 1 100 ⋅ 8 . 315274 = 831 . 5274 100 ⋅ 8 . 315274 = 831 . 5274 size 12{"100" cdot 8 "." "315274"="831" "." "5274"} {} 2 2 1, 000 ⋅ 8 . 315274 = 8, 315 . 274 1, 000 ⋅ 8 . 315274 = 8, 315 . 274 size 12{1,"000" cdot 8 "." "315274"=8,"315" "." "274"} {} 3 3 10 , 000 ⋅ 8 . 315274 = 83 , 152 . 74 10 , 000 ⋅ 8 . 315274 = 83 , 152 . 74 size 12{"10","000" cdot 8 "." "315274"="83","152" "." "74"} {} 4 4

### Multiplying a Decimal by a Power of 10

To multiply a decimal by a power of 10, move the decimal place to the right of its current position as many places as there are zeros in the power of 10. Add zeros if necessary.

### Sample Set C

Find the following products.

#### Example 7

10034.87610034.876 size 12{"100" cdot "34" "." "876"} {}. Since there are 2 zeros in 100, Move the decimal point in 34.876 two places to the right.

#### Example 8

1,0004.80581,0004.8058 size 12{1,"000" cdot 4 "." "8058"} {}. Since there are 3 zeros in 1,000, move the decimal point in 4.8058 three places to the right.

#### Example 9

10,00056.8210,00056.82 size 12{"10","000" cdot "56" "." "82"} {}. Since there are 4 zeros in 10,000, move the decimal point in 56.82 four places to the right. We will have to add two zeros in order to obtain the four places.

Since there is no fractional part, we can drop the decimal point.

### Practice Set C

Find the following products.

#### Exercise 14

100 4.27100 4.27 size 12{"100 " cdot " 4" "." "27"} {}

427

#### Exercise 15

10,000 16.5218710,000 16.52187 size 12{"10,000 " cdot " 16" "." "52187"} {}

165,218.7

#### Exercise 16

(10)(0.0188)(10)(0.0188) size 12{ $$"10"$$ $$0 "." "0188"$$ } {}

0.188

#### Exercise 17

(10,000,000,000)(52.7)(10,000,000,000)(52.7) size 12{ $$"10,000,000,000"$$ $$"52" "." 7$$ } {}

527,000,000,000

## Multiplication in Terms of “Of”

Recalling that the word "of" translates to the arithmetic operation of multiplica­tion, let's observe the following multiplications.

### Sample Set D

#### Example 12

Find 4.1 of 3.8.

Translating "of" to "×", we get

4.1 × 3.8 ̲ 328 123   ̲ 15.58 4.1 × 3.8 ̲ 328 123   ̲ 15.58

Thus, 4.1 of 3.8 is 15.58.

#### Example 13

Find 0.95 of the sum of 2.6 and 0.8.

We first find the sum of 2.6 and 0.8.

2.6 + 0.8 ̲ 3.4 2.6 + 0.8 ̲ 3.4

Now find 0.95 of 3.4

3.4 × 0.95 ̲ 170 306   ̲ 3.230 3.4 × 0.95 ̲ 170 306   ̲ 3.230

Thus, 0.95 of (2.6 + 0.8)(2.6 + 0.8) size 12{ $$2 "." "6 "+" 0" "." 8$$ } {} is 3.230.

### Practice Set D

Find 2.8 of 6.4.

17.92

Find 0.1 of 1.3.

0.13

#### Exercise 20

Find 1.01 of 3.6.

3.636

#### Exercise 21

Find 0.004 of 0.0009.

0.0000036

Find 0.83 of 12.

9.96

#### Exercise 23

Find 1.1 of the sum of 8.6 and 4.2.

14.08

## Exercises

For the following 30 problems, find each product and check each result with a calculator.

### Exercise 24

3.49.23.49.2 size 12{3 "." 4 cdot 9 "." 2} {}

31.28

### Exercise 25

4.56.14.56.1 size 12{4 "." 5 cdot 6 "." 1} {}

### Exercise 26

8.05.98.05.9 size 12{8 "." 0 cdot 5 "." 9} {}

47.20

### Exercise 27

6.176.17 size 12{6 "." 1 cdot 7} {}

### Exercise 28

(0.1)(1.52)(0.1)(1.52) size 12{ $$0 "." 1$$ $$1 "." "52"$$ } {}

0.152

### Exercise 29

(1.99)(0.05)(1.99)(0.05) size 12{ $$1 "." "99"$$ $$0 "." "05"$$ } {}

### Exercise 30

(12.52)(0.37)(12.52)(0.37) size 12{ $$"12" "." "52"$$ $$0 "." "37"$$ } {}

4.6324

### Exercise 31

(5.116)(1.21)(5.116)(1.21) size 12{ $$5 "." "116"$$ $$1 "." "21"$$ } {}

### Exercise 32

(31.82)(0.1)(31.82)(0.1) size 12{ $$"31" "." "82"$$ $$0 "." 1$$ } {}

3.182

### Exercise 33

(16.527)(9.16)(16.527)(9.16) size 12{ $$"16" "." "527"$$ $$9 "." "16"$$ } {}

### Exercise 34

0.00210.0130.00210.013 size 12{0 "." "0021" cdot 0 "." "013"} {}

0.0000273

### Exercise 35

1.00371.000371.00371.00037 size 12{1 "." "0037" cdot 1 "." "00037"} {}

### Exercise 36

(1.6)(1.6)(1.6)(1.6) size 12{ $$1 "." 6$$ $$1 "." 6$$ } {}

2.56

### Exercise 37

(4.2)(4.2)(4.2)(4.2) size 12{ $$4 "." 2$$ $$4 "." 2$$ } {}

### Exercise 38

0.90.90.90.9 size 12{0 "." 9 cdot 0 "." 9} {}

0.81

### Exercise 39

1.111.111.111.11 size 12{1 "." "11" cdot 1 "." "11"} {}

### Exercise 40

6.8154.36.8154.3 size 12{6 "." "815" cdot 4 "." 3} {}

29.3045

### Exercise 41

9.01681.29.01681.2 size 12{9 "." "0168" cdot 1 "." 2} {}

### Exercise 42

(3.5162)(0.0000003)(3.5162)(0.0000003) size 12{ $$3 "." "5162"$$ $$0 "." "0000003"$$ } {}

0.00000105486

### Exercise 43

(0.000001)(0.01)(0.000001)(0.01) size 12{ $$0 "." "000001"$$ $$0 "." "01"$$ } {}

### Exercise 44

(10)(4.96)(10)(4.96) size 12{ $$"10"$$ $$4 "." "96"$$ } {}

49.6

### Exercise 45

(10)(36.17)(10)(36.17) size 12{ $$"10"$$ $$"36" "." "17"$$ } {}

### Exercise 46

10421.884210421.8842 size 12{"10" cdot "421" "." "8842"} {}

4,218.842

### Exercise 47

108.0107108.0107 size 12{"10" cdot 8 "." "0107"} {}

### Exercise 48

1000.196211000.19621 size 12{"100" cdot 0 "." "19621"} {}

19.621

### Exercise 49

1000.7791000.779 size 12{"100" cdot 0 "." "779"} {}

### Exercise 50

10003.59616810003.596168 size 12{"1000" cdot 3 "." "596168"} {}

3,596.168

### Exercise 51

100042.7125571100042.7125571 size 12{"1000" cdot "42" "." "7125571"} {}

### Exercise 52

100025.01100025.01 size 12{"1000" cdot "25" "." "01"} {}

25,010

### Exercise 53

100,0009.923100,0009.923 size 12{"100","000" cdot 9 "." "923"} {}

### Exercise 54

(4.6)(6.17)(4.6)(6.17) size 12{ $$4 "." 6$$ $$6 "." "17"$$ } {}

 Actual product Tenths Hundreds Thousandths

#### Solution

 Actual product Tenths Hundreds Thousandths 28.382 28.4 28.38 28.382

### Exercise 55

(8.09)(7.1)(8.09)(7.1) size 12{ $$8 "." "09"$$ $$7 "." 1$$ } {}

 Actual product Tenths Hundreds Thousandths

### Exercise 56

(11.1106)(12.08)(11.1106)(12.08) size 12{ $$"11" "." "1106"$$ $$"12" "." "08"$$ } {}

 Actual product Tenths Hundreds Thousandths

#### Solution

 Actual product Tenths Hundreds Thousandths 134.216048 134.2 134.22 134.216

### Exercise 57

0.00831.0909010.00831.090901 size 12{0 "." "0083" cdot 1 "." "090901"} {}

 Actual product Tenths Hundreds Thousandths

### Exercise 58

726.518726.518 size 12{7 cdot "26" "." "518"} {}

 Actual product Tenths Hundreds Thousandths

#### Solution

 Actual product Tenths Hundreds Thousandths 185.626 185.6 185.63 185.626

For the following 15 problems, perform the indicated operations

Find 5.2 of 3.7.

### Exercise 60

Find 12.03 of 10.1

121.503

Find 16 of 1.04

Find 12 of 0.1

1.2

### Exercise 63

Find 0.09 of 0.003

### Exercise 64

Find 1.02 of 0.9801

0.999702

### Exercise 65

Find 0.01 of the sum of 3.6 and 12.18

### Exercise 66

Find 0.2 of the sum of 0.194 and 1.07

0.2528

### Exercise 67

Find the difference of 6.1 of 2.7 and 2.7 of 4.03

### Exercise 68

Find the difference of 0.071 of 42 and 0.003 of 9.2

2.9544

### Calculator Problems

For the following 10 problems, use a calculator to determine each product. If the calculator will not provide the exact product, round the results to five decimal places.

### Exercise 74

0.019 0.321 0.0190.321

0.006099

### Exercise 75

0.261 1.96 0.2611.96

### Exercise 76

4.826 4.827 4.8264.827

23.295102

9.46 2 9.46 2

0.012 2 0.012 2

0.000144

### Exercise 79

0.00037 0.0065 0.000370.0065

### Exercise 80

0.002 0.0009 0.0020.0009

0.0000018

### Exercise 81

0.1286 0.7699 0.12860.7699

### Exercise 82

0.01 0.00000471 0.010.00000471

0.0000000471

### Exercise 83

0.00198709 0.03 0.001987090.03

### Exercises for Review

#### Exercise 84

((Reference)) Find the value, if it exists, of 0 ÷ 150 ÷ 15 size 12{"0 " div " 15"} {}.

0

#### Exercise 85

((Reference)) Find the greatest common factor of 210, 231, and 357.

#### Exercise 86

((Reference)) Reduce 2802,1562802,156 size 12{ { {"280"} over {2,"156"} } } {} to lowest terms.

10771077

#### Exercise 87

((Reference)) Write "fourteen and one hundred twenty-one ten-thousandths, using digits."

#### Exercise 88

((Reference)) Subtract 6.882 from 8.661 and round the result to two decimal places.

1.78

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