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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.

Vertical multiplication. 1.46 times 3.2. The first round of multiplication yields a first partial product of 292. The second round yields a second partial product of 438, aligned in the tens column. Take note that 2 decimal places in the first factor and 1 decimal place in the second factor sums to a total of three decimal places in the product. The final product is 4.672.

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} {}

Vertical multiplication. 6.5 times 4.3. The first round of multiplication yields a first partial product of 195. The second round yields a second partial product of 260, aligned in the tens column. Take note that 1 decimal place in the first factor and 1 decimal place in the second factor sums to a total of two decimal places in the product. The final product is 27.95.

Thus, 6.5⋅4.3=27.956.5⋅4.3=27.95 size 12{6 "." 5 cdot 4 "." 3="27" "." "95"} {}.

Example 2

23.4⋅1.9623.4⋅1.96 size 12{"23" "." 4 cdot 1 "." "96"} {}

Vertical multiplication. 23.4 times 1.96. The first round of multiplication yields a first partial product of 1404. The second round yields a second partial product of 2106, aligned in the tens column. The third round yields a third partial product of 234, aligned in the hundred column. Take note that 1 decimal place in the first factor and 2 decimal places in the second factor sums to a total of three decimal places in the product. The final product is 45.864.

Thus, 23.4⋅1.96=45.86423.4⋅1.96=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.

Vertical multiplication. 0.251 times 0.00113. The first round of multiplication yields a first partial product of 753. The second round yields a second partial product of 251, aligned in the tens column. The third round yields a third partial product of 251, aligned in the hundred column. Take note that 3 decimal places in the first factor and 5 decimal places in the second factor sums to a total of eight decimal places in the product. The final product is 0.00028363.

Now, rounding to three decimal places, we get

0.251 times 0.00113 = 0.000, if the product is rounded to three decimal places.

Practice Set A

Find the following products.

Exercise 1

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

Solution

45.58

Exercise 2

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

Solution

10.388

Exercise 3

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

Solution

0.16864

Exercise 4

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

Solution

0.00000062

Exercise 5

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

Solution

9.3

Exercise 6

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

Solution

53.94

Exercise 7

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

Solution

539.4

Exercise 8

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

Solution

5,394

Exercise 9

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

Solution

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"} {}

Table 1
    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"} {}

Table 2
    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.

Table 3
    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"} {}

Solution

20.91408

Exercise 11

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

Solution

0.000066

Exercise 12

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

Solution

0.2397

Exercise 13

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

Solution

0.0000

Multiplying Decimals by Powers of 10

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

Table 4
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

100⋅34.876100⋅34.876 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.

100 times 34.876 equals 3487.6. An arrows shows how the decimal in 34.876 is moved two digits to the right to make 3,487.6

Example 8

1,000⋅4.80581,000⋅4.8058 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.

1,000 times 4.8058 equals 4805.8. An arrows shows how the decimal in 4.8058 is moved three digits to the right to make 4,805.8

Example 9

10,000⋅56.8210,000⋅56.82 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.

10,000 times 56.82 equals 568200. An arrows shows how the decimal in 56.82 is moved four digits to the right to make 568,200.
Since there is no fractional part, we can drop the decimal point.

Example 10

1,000,000 times 2.57 equals 2570000. An arrows shows how the decimal in 2.57 is moved six digits to the right to make 2,570,000.

Example 11

1,000 times 0.0000029 equals 0.0029. An arrows shows how the decimal in 0.0000029 is moved six digits to the right to make 0.0029.

Practice Set C

Find the following products.

Exercise 14

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

Solution

427

Exercise 15

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

Solution

165,218.7

Exercise 16

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

Solution

0.188

Exercise 17

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

Solution

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

Exercise 18

Find 2.8 of 6.4.

Solution

17.92

Exercise 19

Find 0.1 of 1.3.

Solution

0.13

Exercise 20

Find 1.01 of 3.6.

Solution

3.636

Exercise 21

Find 0.004 of 0.0009.

Solution

0.0000036

Exercise 22

Find 0.83 of 12.

Solution

9.96

Exercise 23

Find 1.1 of the sum of 8.6 and 4.2.

Solution

14.08

Exercises

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

Exercise 24

3.4⋅9.23.4⋅9.2 size 12{3 "." 4 cdot 9 "." 2} {}

Solution

31.28

Exercise 25

4.5⋅6.14.5⋅6.1 size 12{4 "." 5 cdot 6 "." 1} {}

Exercise 26

8.0⋅5.98.0⋅5.9 size 12{8 "." 0 cdot 5 "." 9} {}

Solution

47.20

Exercise 27

6.1⋅76.1⋅7 size 12{6 "." 1 cdot 7} {}

Exercise 28

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

Solution

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{ \( "12" "." "52" \) \( 0 "." "37" \) } {}

Solution

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{ \( "31" "." "82" \) \( 0 "." 1 \) } {}

Solution

3.182

Exercise 33

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

Exercise 34

0.0021⋅0.0130.0021⋅0.013 size 12{0 "." "0021" cdot 0 "." "013"} {}

Solution

0.0000273

Exercise 35

1.0037⋅1.000371.0037⋅1.00037 size 12{1 "." "0037" cdot 1 "." "00037"} {}

Exercise 36

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

Solution

2.56

Exercise 37

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

Exercise 38

0.9⋅0.90.9⋅0.9 size 12{0 "." 9 cdot 0 "." 9} {}

Solution

0.81

Exercise 39

1.11⋅1.111.11⋅1.11 size 12{1 "." "11" cdot 1 "." "11"} {}

Exercise 40

6.815⋅4.36.815⋅4.3 size 12{6 "." "815" cdot 4 "." 3} {}

Solution

29.3045

Exercise 41

9.0168⋅1.29.0168⋅1.2 size 12{9 "." "0168" cdot 1 "." 2} {}

Exercise 42

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

Solution

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{ \( "10" \) \( 4 "." "96" \) } {}

Solution

49.6

Exercise 45

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

Exercise 46

10⋅421.884210⋅421.8842 size 12{"10" cdot "421" "." "8842"} {}

Solution

4,218.842

Exercise 47

10⋅8.010710⋅8.0107 size 12{"10" cdot 8 "." "0107"} {}

Exercise 48

100⋅0.19621100⋅0.19621 size 12{"100" cdot 0 "." "19621"} {}

Solution

19.621

Exercise 49

100⋅0.779100⋅0.779 size 12{"100" cdot 0 "." "779"} {}

Exercise 50

1000⋅3.5961681000⋅3.596168 size 12{"1000" cdot 3 "." "596168"} {}

Solution

3,596.168

Exercise 51

1000⋅42.71255711000⋅42.7125571 size 12{"1000" cdot "42" "." "7125571"} {}

Exercise 52

1000⋅25.011000⋅25.01 size 12{"1000" cdot "25" "." "01"} {}

Solution

25,010

Exercise 53

100,000⋅9.923100,000⋅9.923 size 12{"100","000" cdot 9 "." "923"} {}

Exercise 54

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

Table 5
Actual product Tenths Hundreds Thousandths
       

Solution

Table 6
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{ \( 8 "." "09" \) \( 7 "." 1 \) } {}

Table 7
Actual product Tenths Hundreds Thousandths
       

Exercise 56

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

Table 8
Actual product Tenths Hundreds Thousandths
       

Solution

Table 9
Actual product Tenths Hundreds Thousandths
134.216048 134.2 134.22 134.216

Exercise 57

0.0083⋅1.0909010.0083⋅1.090901 size 12{0 "." "0083" cdot 1 "." "090901"} {}

Table 10
Actual product Tenths Hundreds Thousandths
       

Exercise 58

7⋅26.5187⋅26.518 size 12{7 cdot "26" "." "518"} {}

Table 11
Actual product Tenths Hundreds Thousandths
       

Solution

Table 12
Actual product Tenths Hundreds Thousandths
185.626 185.6 185.63 185.626

For the following 15 problems, perform the indicated operations

Exercise 59

Find 5.2 of 3.7.

Exercise 60

Find 12.03 of 10.1

Solution

121.503

Exercise 61

Find 16 of 1.04

Exercise 62

Find 12 of 0.1

Solution

1.2

Exercise 63

Find 0.09 of 0.003

Exercise 64

Find 1.02 of 0.9801

Solution

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

Solution

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

Solution

2.9544

Exercise 69

If a person earns $8.55 an hour, how much does he earn in twenty-five hundredths of an hour?

Exercise 70

A man buys 14 items at $1.16 each. What is the total cost?

Solution

$16.24

Exercise 71

In the problem above, how much is the total cost if 0.065 sales tax is added?

Exercise 72

A river rafting trip is supposed to last for 10 days and each day 6 miles is to be rafted. On the third day a person falls out of the raft after only 2525 size 12{ { {2} over {5} } } {} of that day’s mileage. If this person gets discouraged and quits, what fraction of the entire trip did he complete?

Solution

0.24

Exercise 73

A woman starts the day with $42.28. She buys one item for $8.95 and another for $6.68. She then buys another item for sixty two-hundredths of the remaining amount. How much money does she have left?

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 0.019⋅0.321

Solution

0.006099

Exercise 75

0.261 ⋅ 1.96 0.261⋅1.96

Exercise 76

4.826 ⋅ 4.827 4.826⋅4.827

Solution

23.295102

Exercise 77

9.46 2 9.46 2

Exercise 78

0.012 2 0.012 2

Solution

0.000144

Exercise 79

0.00037 ⋅ 0.0065 0.00037⋅0.0065

Exercise 80

0.002 ⋅ 0.0009 0.002⋅0.0009

Solution

0.0000018

Exercise 81

0.1286 ⋅ 0.7699 0.1286⋅0.7699

Exercise 82

0.01 ⋅ 0.00000471 0.01⋅0.00000471

Solution

0.0000000471

Exercise 83

0.00198709 ⋅ 0.03 0.00198709⋅0.03

Exercises for Review

Exercise 84

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

Solution

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.

Solution

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.

Solution

1.78

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