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Multiplication of Whole Numbers

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 whole numbers. By the end of the module students should be able to understand the process of multiplication, multiply whole numbers, simplify multiplications with numbers ending in zero, and use a calculator to multiply one whole number by another.

Section Overview

  • Multiplication
  • The Multiplication Process With a Single Digit Multiplier
  • The Multiplication Process With a Multiple Digit Multiplier
  • Multiplication With Numbers Ending in Zero
  • Calculators

Multiplication

Multiplication is a description of repeated addition.

In the addition of

5 + 5 + 5 5 + 5 + 5 size 12{5+5+5} {}

the number 5 is repeated 3 times. Therefore, we say we have three times five and describe it by writing

3 × 5 3 × 5 size 12{3 times 5} {}

Thus,

3 × 5 = 5 + 5 + 5 3 × 5 = 5 + 5 + 5 size 12{3 times 5=5+5+5} {}

Multiplicand

In a multiplication, the repeated addend (number being added) is called the multi­plicand. In 3×53×5 size 12{3 times 5} {}, the 5 is the multiplicand.

Multiplier

Also, in a multiplication, the number that records the number of times the multiplicand is used is called the multiplier. In 3×53×5 size 12{3 times 5} {}, the 3 is the multiplier.

Sample Set A

Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand.

Example 1

7 + 7 + 7 + 7 + 7 + 7 7 + 7 + 7 + 7 + 7 + 7 size 12{7+7+7+7+7+7} {}

6×76×7 size 12{6 times 7} {}.
Multiplier is 6.
Multiplicand is 7.

Example 2

18 + 18 + 18 18 + 18 + 18 size 12{"18"+"18"+"18"} {}

3×183×18 size 12{3 times "18"} {}.
Multiplier is 3.
Multiplicand is 18.

Practice Set A

Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand.

Exercise 1

12 + 12 + 12 + 12 12 + 12 + 12 + 12 size 12{"12"+"12"+"12"+"12"} {}
. Multiplier is
. Multiplicand is
.

Solution

4×124×12 size 12{4 times "12"} {}. Multiplier is 4. Multiplicand is 12.

Exercise 2

36 + 36 + 36 + 36 + 36 + 36 + 36 + 36 36 + 36 + 36 + 36 + 36 + 36 + 36 + 36 size 12{"36"+"36"+"36"+"36"+"36"+"36"+"36"+"36"} {}
. Multiplier is
. Multiplicand is
.

Solution

8×368×36 size 12{8 times "36"} {}. Multiplier is 8. Multiplicand is 36.

Exercise 3

0 + 0 + 0 + 0 + 0 0 + 0 + 0 + 0 + 0 size 12{0+0+0+0+0} {}
. Multiplier is
. Multiplicand is
.

Solution

5×05×0 size 12{5 times 0} {}. Multiplier is 5. Multiplicand is 0.

Exercise 4

1847 + 1847 + ... + 1847 ︸ 12,000 times 1847 + 1847 + ... + 1847 ︸ 12,000 times
. Multiplier is
. Multiplicand is
.

Solution

12,000×1,84712,000×1,847 size 12{"12","000" times 1,"847"} {}. Multiplier is 12,000. Multiplicand is 1,847.

Factors

In a multiplication, the numbers being multiplied are also called factors.

Products

The result of a multiplication is called the product. In 3×5=153×5=15 size 12{3 times 5="15"} {}, the 3 and 5 are not only called the multiplier and multiplicand, but they are also called factors. The product is 15.

Indicators of Multiplication ××,⋅,( )

The multiplication symbol (××) is not the only symbol used to indicate multiplication. Other symbols include the dot ( ⋅ ) and pairs of parentheses ( ). The expressions

3×53×5 size 12{3 times 5} {},
3⋅53⋅5 size 12{3 cdot 5} {},
3(5)3(5) size 12{3 \( 5 \) } {},
(3)5(3)5 size 12{ \( 3 \) 5} {},
(3)(5)(3)(5) size 12{ \( 3 \) \( 5 \) } {}

all represent the same product.

The Multiplication Process With a Single Digit Multiplier

Since multiplication is repeated addition, we should not be surprised to notice that carrying can occur. Carrying occurs when we find the product of 38 and 7:

Vertical multiplication. 38 times 7 is 266. The 5 is carried on top of the 3.

First, we compute 7×8=567×8=56 size 12{7 times 8="56"} {}. Write the 6 in the ones column. Carry the 5. Then take 7×3=217×3=21 size 12{7 times 3="21"} {}. Add to 21 the 5 that was carried: 21+5=2621+5=26 size 12{"21"+5="26"} {}. The product is 266.

Sample Set B

Find the following products.

Example 3

Vertical multiplication. 64 times 3 is 192. The 1 is carried on top of the 6.

3×4=12 Write the 2, carry the 1. 3×6=18 Add to 18 the 1 that was carried:18+1=19. 3×4=12 Write the 2, carry the 1. 3×6=18 Add to 18 the 1 that was carried:18+1=19.

The product is 192.

Example 4

Vertical multiplication. 526 times 5 is 2,630. The 2 is carried on top of the 2, and the 1 is carried on top of the 5.

5×6=30 Write the 0, carry the 3. 5×2=10 Add to 10 the 3 that was carried:10+3=13. Write the 3, carry the 1. 5×5=25 Add to 25 the 1 that was carried:25+1=6. 5×6=30 Write the 0, carry the 3. 5×2=10 Add to 10 the 3 that was carried:10+3=13. Write the 3, carry the 1. 5×5=25 Add to 25 the 1 that was carried:25+1=6.

The product is 2,630.

Example 5

Vertical multiplication. 1,804 times 9 is 16,236. The 3 is carried on top of the 0, and the 7 is carried on top of the 1.

9×4=36 Write the 6, carry the 3. 9×0=0 Add to the 0 the 3 that was carried:0+3=3. Write the 3. 9×8=72 Write the 2, carry the 7. 9×1=9 Add to the 9 the 7 that was carried: 9+7=16. Since there are no more multiplications to perform,write both the 1 and 6. 9×4=36 Write the 6, carry the 3. 9×0=0 Add to the 0 the 3 that was carried:0+3=3. Write the 3. 9×8=72 Write the 2, carry the 7. 9×1=9 Add to the 9 the 7 that was carried: 9+7=16. Since there are no more multiplications to perform,write both the 1 and 6.

The product is 16,236.

Practice Set B

Find the following products.

Exercise 5

37 ×   5 ̲ 37 ×   5 ̲

Solution

185

Exercise 6

78 ×   8 ̲ 78 ×   8 ̲

Solution

624

Exercise 7

536 ×     7 ̲ 536 ×     7 ̲

Solution

3,752

Exercise 8

40,019 ×         8 ̲ 40,019 ×         8 ̲

Solution

320,152

Exercise 9

301,599 ×           3 ̲ 301,599 ×           3 ̲

Solution

904,797

The Multiplication Process With a Multiple Digit Multiplier

In a multiplication in which the multiplier is composed of two or more digits, the multiplication must take place in parts. The process is as follows:

  • Part 1: First Partial Product Multiply the multiplicand by the ones digit of the multiplier. This product is called the first partial product.
  • Part 2: Second Partial Product Multiply the multiplicand by the tens digit of the multiplier. This product is called the second partial product. Since the tens digit is used as a factor, the second partial product is written below the first partial product so that its rightmost digit appears in the tens column.
  • Part 3: If necessary, continue this way finding partial products. Write each one below the previous one so that the rightmost digit appears in the column directly below the digit that was used as a factor.
  • Part 4: Total Product Add the partial products to obtain the total product.

Note:

It may be necessary to carry when finding each partial product.

Sample Set C

Example 6

Multiply 326 by 48.

  • Part 1:

    Vertical multiplication. 326 times 48 is 2608. The 4 is carried on top of the 2, and the 2 is carried on top of the 3. The product is labeled, first partial product.

  • Part 2:

    Vertical multiplication. 326 times 48, with the first part of the product, 2608, in the first line of the product space, and the second part of the product, 1304, in the second line of the product space. This number is naturally aligned with the tens column of the number above it. The second round of numbers are carried, with a 2 in the tens column and a 1 in the hundreds column. 1304 is labeled, second partial product.

  • Part 3: This step is unnecessary since all of the digits in the multiplier have been used.
  • Part 4: Add the partial products to obtain the total product.

    Vertical multiplication. 326 times 48, with the first part of the product, 2608, in the first line of the product space, and the second part of the product, 1304, in the second line of the product space. The two lines of the product space are added together to make a total product of 15648.

  • The product is 15,648.

Example 7

Multiply 5,369 by 842.

  • Part 1:

    Vertical multiplication. 5369 times 842, with the first part of the product, 10738, in the first line of the product space. A 1 is carried above the 6, and a 1 is carried above the 3. 10738 is labeled, first partial product.

  • Part 2:

    Vertical multiplication. 5639 times 842, with the first part of the product, 10738, in the first line of the product space, and the second part of the product, 21476, in the second line of the product space. This number is aligned with the tens column of the number above it. A second round of numbers are carried, with a 3 in the tens column, a 2 in the hundreds column, and a 1 in the thousands column. 21476 is labeled, second partial product.

  • Part 3:

    Vertical multiplication. 5639 times 842, with the first part of the product, 10738, in the first line of the product space, and the second part of the product, 21476, in the second line of the product space. This number is aligned with the tens column of the number above it. The third partial of the product, 42952, is below it, and is aligned with the hundreds column. A third round of numbers are carried, with a 7 in the tens column, a 5 in the hundreds column, and a 2 in the thousands column. Adding the partial products together makes a total product of 4520698, labeled Part 4.

  • The product is 4,520,698.

Example 8

Multiply 1,508 by 206.

  • Part 1:

    Vertical multiplication. 1508 times 206, with the first part of the product, 9048, in the first line of the product space. A 4 is carried in the tens column, and a 3 is carried in the thousands column. 9048 is labeled, first partial product.

  • Part 2:

    Vertical multiplication. 1508 times 206, with the first part of the product, 9048, in the first line of the product space. A 4 is carried in the tens column, and a 3 is carried in the thousands column.
    Since 0 times 1508 is 0, the partial product will not change the identity of the total product (which is obtained by addition). Go to the next partial product.

  • Part 3:

    Vertical multiplication. 1508 times 206, with the first part of the product, 9048, in the first line of the product space, and the third part of the product, 3016, which is aligned in the hundreds column. A second round of numbers are carried, with a 1 in the tens column and a 1 in the thousands column. Adding the partial products together makes a total product of 310648, labeled Part 4.

  • The product is 310,648

Practice Set C

Exercise 10

Multiply 73 by 14.

Solution

1,022

Exercise 11

Multiply 86 by 52.

Solution

4,472

Exercise 12

Multiply 419 by 85.

Solution

35,615

Exercise 13

Multiply 2,376 by 613.

Solution

1,456,488

Exercise 14

Multiply 8,107 by 304.

Solution

2,464,528

Exercise 15

Multiply 66,260 by 1,008.

Solution

66,790,080

Exercise 16

Multiply 209 by 501.

Solution

104,709

Exercise 17

Multiply 24 by 10.

Solution

240

Exercise 18

Multiply 3,809 by 1,000.

Solution

3,809,000

Exercise 19

Multiply 813 by 10,000.

Solution

8,130,000

Multiplications With Numbers Ending in Zero

Often, when performing a multiplication, one or both of the factors will end in zeros. Such multiplications can be done quickly by aligning the numbers so that the rightmost nonzero digits are in the same column.

Sample Set D

Perform the multiplication (49,000)(1,200)(49,000)(1,200) size 12{ \( "49","000" \) \( 1,"200" \) } {}.

(49,000)(1,200) = 49000 ×    1200 ̲ (49,000)(1,200) = 49000 ×    1200 ̲

Since 9 and 2 are the rightmost nonzero digits, put them in the same column.

49000 times 1200, with the 1200 aligned one space to the left.

Draw (perhaps mentally) a vertical line to separate the zeros from the nonzeros.

49000 times 1200, with the 1200 aligned one space to the left. A vertical line is drawn to separate the zeros in both numbers from the nonzero digits.

Multiply the numbers to the left of the vertical line as usual, then attach to the right end of this product the total number of zeros.

49000 times 1200, with the 1200 aligned one space to the left. A vertical line is drawn to separate the zeros in both numbers from the nonzero digits. 98 is the first partial product, and 49 is the second partial product. The final product is 588, and the 5 zeros are then attached to the end of the product, making a total product of 58800000.

The product is 58,800,000

Practice Set D

Exercise 20

Multiply 1,800 by 90.

Solution

162,000

Exercise 21

Multiply 420,000 by 300.

Solution

126,000,000

Exercise 22

Multiply 20,500,000 by 140,000.

Solution

2,870,000,000,000

Calculators

Most multiplications are performed using a calculator.

Sample Set E

Example 9

Multiply 75,891 by 263.

Table 1
Display Reads
Type 75891 75891
Press × 75891
Type 263 263
Press = 19959333

The product is 19,959,333.

Example 10

Multiply 4,510,000,000,000 by 1,700.

Table 2
Display Reads
Type 451 451
Press × 451
Type 17 17
Press = 7667

The display now reads 7667. We'll have to add the zeros ourselves. There are a total of 12 zeros. Attaching 12 zeros to 7667, we get 7,667,000,000,000,000.

The product is 7,667,000,000,000,000.

Example 11

Multiply 57,847,298 by 38,976.

Table 3
Display Reads
Type 57847298 57847298
Press × 57847298
Type 38976 38976
Press = 2.2546563 12

The display now reads 2.2546563 12. What kind of number is this? This is an example of a whole number written in scientific notation. We'll study this concept when we get to decimal numbers.

Practice Set E

Use a calculator to perform each multiplication.

Exercise 23

52×2752×27 size 12{"52" times "27"} {}

Solution

1,404

Exercise 24

1,448×6,1551,448×6,155 size 12{1,"448" times 6,"155"} {}

Solution

8,912,440

Exercise 25

8,940,000×205,0008,940,000×205,000 size 12{8,"940","000" times "205","000"} {}

Solution

1,832,700,000,000

Exercises

For the following problems, perform the multiplications. You may check each product with a calculator.

Exercise 26

8 × 3 ̲ 8 × 3 ̲

Solution

24

Exercise 27

3 × 5 ̲ 3 × 5 ̲

Exercise 28

8 × 6 ̲ 8 × 6 ̲

Solution

48

Exercise 29

5 × 7 ̲ 5 × 7 ̲

Exercise 30

6 × 16 × 1

Solution

6

Exercise 31

4 × 54 × 5

Exercise 32

75 × 375 × 3

Solution

225

Exercise 33

35 × 535 × 5

Exercise 34

45 ×   6 ̲ 45 ×   6 ̲

Solution

270

Exercise 35

31 ×   7 ̲ 31 ×   7 ̲

Exercise 36

97 ×   6 ̲ 97 ×   6 ̲

Solution

582

Exercise 37

75 × 57 ̲ 75 × 57 ̲

Exercise 38

64 × 15 ̲ 64 × 15 ̲

Solution

960

Exercise 39

73 × 15 ̲ 73 × 15 ̲

Exercise 40

81 × 95 ̲ 81 × 95 ̲

Solution

7,695

Exercise 41

31 × 33 ̲ 31 × 33 ̲

Exercise 42

57 × 6457 × 64

Solution

3,648

Exercise 43

76 × 4276 × 42

Exercise 44

894 × 52894 × 52

Solution

46,488

Exercise 45

684 × 38684 × 38

Exercise 46

115 ×   22 ̲ 115 ×   22 ̲

Solution

2,530

Exercise 47

706 ×   81 ̲ 706 ×   81 ̲

Exercise 48

328 ×   21 ̲ 328 ×   21 ̲

Solution

6,888

Exercise 49

550 ×   94 ̲ 550 ×   94 ̲

Exercise 50

930 × 26930 × 26

Solution

24,180

Exercise 51

318 × 63318 × 63

Exercise 52

582 × 127 ̲ 582 × 127 ̲

Solution

73,914

Exercise 53

247 × 116 ̲ 247 × 116 ̲

Exercise 54

305 × 225 ̲ 305 × 225 ̲

Solution

68,625

Exercise 55

782 × 547 ̲ 782 × 547 ̲

Exercise 56

771 × 663 ̲ 771 × 663 ̲

Solution

511,173

Exercise 57

638 × 516 ̲ 638 × 516 ̲

Exercise 58

1,905 × 7101,905 × 710

Solution

1,352,550

Exercise 59

5,757 × 5,0105,757 × 5,010

Exercise 60

3,106 × 1,752 ̲ 3,106 × 1,752 ̲

Solution

5,441,712

Exercise 61

9,300 × 1,130 ̲ 9,300 × 1,130 ̲

Exercise 62

7,057 × 5,229 ̲ 7,057 × 5,229 ̲

Solution

36,901,053

Exercise 63

8,051 × 5,580 ̲ 8,051 × 5,580 ̲

Exercise 64

5,804 × 4,300 ̲ 5,804 × 4,300 ̲

Solution

24,957,200

Exercise 65

357 × 16 ̲ 357 × 16 ̲

Exercise 66

724 ×     0 ̲ 724 ×     0 ̲

Solution

0

Exercise 67

2,649 ×     41 ̲ 2,649 ×     41 ̲

Exercise 68

5,173 ×       8 ̲ 5,173 ×       8 ̲

Solution

41,384

Exercise 69

1,999 ×       0 ̲ 1,999 ×       0 ̲

Exercise 70

1,666 ×       0 ̲ 1,666 ×       0 ̲

Solution

0

Exercise 71

51,730 ×     142 ̲ 51,730 ×     142 ̲

Exercise 72

387 × 190 ̲ 387 × 190 ̲

Solution

73,530

Exercise 73

3,400 ×     70 ̲ 3,400 ×     70 ̲

Exercise 74

460,000 ×   14,000 ̲ 460,000 ×   14,000 ̲

Solution

6,440,000,000

Exercise 75

558,000,000 ×         81,000 ̲ 558,000,000 ×         81,000 ̲

Exercise 76

37,000 ×       120 ̲ 37,000 ×       120 ̲

Solution

4,440,000

Exercise 77

498,000 ×           0 ̲ 498,000 ×           0 ̲

Exercise 78

4,585,000 ×         140 ̲ 4,585,000 ×         140 ̲

Solution

641,900,000

Exercise 79

30,700,000 ×             180 ̲ 30,700,000 ×             180 ̲

Exercise 80

8,000 ×      10 ̲ 8,000 ×      10 ̲

Solution

80,000

Exercise 81

Suppose a theater holds 426 people. If the theater charges $4 per ticket and sells every seat, how much money would they take in?

Exercise 82

In an English class, a student is expected to read 12 novels during the semester and prepare a report on each one of them. If there are 32 students in the class, how many reports will be prepared?

Solution

384 reports

Exercise 83

In a mathematics class, a final exam consists of 65 problems. If this exam is given to 28 people, how many problems must the instructor grade?

Exercise 84

A business law instructor gives a 45 problem exam to two of her classes. If each class has 37 people in it, how many problems will the instruc­tor have to grade?

Solution

3,330 problems

Exercise 85

An algebra instructor gives an exam that consists of 43 problems to four of his classes. If the classes have 25, 28, 31, and 35 students in them, how many problems will the instructor have to grade?

Exercise 86

In statistics, the term "standard deviation" refers to a number that is calculated from certain data. If the data indicate that one standard de­viation is 38 units, how many units is three stan­dard deviations?

Solution

114 units

Exercise 87

Soft drinks come in cases of 24 cans. If a supermarket sells 857 cases during one week, how many individual cans were sold?

Exercise 88

There are 60 seconds in 1 minute and 60 minutes in 1 hour. How many seconds are there in 1 hour?

Solution

3,600 seconds

Exercise 89

There are 60 seconds in 1 minute, 60 minutes in one hour, 24 hours in one day, and 365 days in one year. How many seconds are there in 1 year?

Exercise 90

Light travels 186,000 miles in one second. How many miles does light travel in one year? (Hint: Can you use the result of the previous problem?)

Solution

5,865,696,000,000 miles per year

Exercise 91

An elementary school cafeteria sells 328 lunches every day. Each lunch costs $1. How much money does the cafeteria bring in in 2 weeks?

Exercise 92

A computer company is selling stock for $23 a share. If 87 people each buy 55 shares, how much money would be brought in?

Solution

$110,055

Exercises for Review

Exercise 93

((Reference)) In the number 421,998, how may ten thousands are there?

Exercise 94

((Reference)) Round 448,062,187 to the nearest hundred thousand.

Solution

448,100,000

Exercise 95

((Reference)) Find the sum. 22,451 + 18,976.

Exercise 96

((Reference)) Subtract 2,289 from 3,001.

Solution

712

Exercise 97

((Reference)) Specify which property of addition justifies the fact that (a first whole number + a second whole number) = (the second whole number + the first whole number)

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