2.1: Multiplication of Whole Numbers

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Multiplication of Whole Numbers

understand the process of multiplication
be able to multiply whole numbers
be able to simplify multiplications with numbers ending in zero
be able to use a calculator to multiply one whole number by another

Multiplication

Definition: Multiplication

Multiplication is a description of repeated addition.

In the addition of

$5 + 5 + 5$

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

$3 \times 5$

Thus,

$3 \times 5 = 5 + 5 + 5$

Definition: Multiplicand

In a multiplication, the repeated addend (number being added) is called the multiplicand. In $3 \times 5$, the 5 is the multiplicand.

Definition: Multiplier

Also, in a multiplication, the number that records the number of times the multiplicand is used is called the multiplier. In $3 \times 5$, the 3 is the multiplier.

Sample Set A

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

$7 + 7 + 7 + 7 + 7 + 7$

Solution

$6 \times 7$, Multiplier is 6. Multiplicand is 7.

Sample Set A

$18 + 18 + 18$

Solution

$3 \times 18$. 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.

$12 + 12 + 12 + 12$

. Multiplier is . Multiplicand is .

Answer: $4 \times 12$. Multiplier is 4. Multiplicand is 12.

Practice Set A

$36 + 36 + 36 + 36 + 36 + 36 + 36 + 36$

. Multiplier is . Multiplicand is .

Answer: $8 \times 36$. Multiplier is 8. Multiplicand is 36.

Practice Set A

$0 + 0 + 0 + 0 + 0$

. Multiplier is . Multiplicand is .

Answer: $5 \times 0$. Multiplier is 5. Multiplicand is 0.

Practice Set A

$\underbrace{1847 + 1847 + \cdots + 1847}_{12,000 \text{ times}}$

s. Multiplier is . Multiplicand is .me

Answer: $12,000 \times 1,847$. Multiplier is 12,000. Multiplicand is 1,847.

Definition: Factors

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

Definition: Products

The result of a multiplication is called the product. In $3 \times 5 = 15$, 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 $\times, \cdot, ()$

The multiplication symbol ($\times$) is not the only symbol used to indicate multiplication. Other symbols include the dot ($\cdot$) and pairs of parentheses ( ). The expressions

$3 \times 5, 3 \cdot 5, 3(5), (3)5, (3)(5)$

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 $7 \times 8 = 56$. Write the 6 in the ones column. Carry the 5. Then take $7 \times 3 = 21$. Add to 21 the 5 that was carried: $21 + 5 = 26$. The product is 266.

Sample Set B

Find the following products.

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

Solution

$\begin{array} {lcl} {3 \times 4 = 12} &\ \ \ \ & {\text{Write the 2, carry the 1.}} \\ {3 \times 6 = 18} &\ \ \ \ & {\text{Add to 18 the 1 that was carried: 18 + 1 = 19.}} \end{array}$

The product is 192.

Sample Set B

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

Solution

$\begin{array} {lcl} {5 \times 6 = 30} & \ \ \ \ & {\text{Write the 0, carry the 3.}} \\ {5 \times 2 = 10} & \ \ \ \ & {\text{Add to 10 the 3 that was carried: 10 + 3 = 13. Write the 3, carry the 1.}} \\ {5 \times 5 = 25} & \ \ \ \ & {\text{Add to 25 the 1 that was carried: 25 + 1 = 6.}} \end{array}$

The product is 2,630.

Sample Set B

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

Solution

$\begin{array} {lcl} {9 \times 4 = 36} & \ \ \ \ & {\text{Write the 6, carry the 3.}} \\ {9 \times 0 = 0} & \ \ \ \ & {\text{Add to 0 the 3 that was carried: 0 + 3 = 13. Write the 3.}} \\ {9 \times 8 = 72} & \ \ \ \ & {\text{Write the 2, carry the 7.}} \\ {} & \ \ \ \ & {\text{Since there are no more multiplications to perform, write both the 1 and 6.}} \end{array}$

The product is 16,236.

Practice Set B

Find the following products.

$\begin{array} {r} {37} \\ {\underline{\times \ \ 5}} \end{array}$

Answer: 185

Practice Set B

Find the following products.

$\begin{array} {r} {78} \\ {\underline{\times \ \ 8}} \end{array}$

Answer: 624

Practice Set B

Find the following products.

$\begin{array} {r} {537} \\ {\underline{\times \ \ \ \ 7}} \end{array}$

Answer: 3,752

Practice Set B

Find the following products.

$\begin{array} {r} {40,019} \\ {\underline{\times \ \ \ \ \ \ \ \ \ 8}} \end{array}$

Answer: 320,152

Practice Set B

Find the following products.

$\begin{array} {r} {301,599} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ \ 3}} \end{array}$

Answer: 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:

First Partial Product Multiply the multiplicand by the ones digit of the multiplier. This product is called the first partial product.
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.
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.
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

Multiply 326 by 48.

Solution

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.

Sample Set C

Multiply 5,369 by 842.

Solution

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.

Sample Set C

Multiply 1,508 by 206.

Solution

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

Multiply 73 by 14.

Answer: 1,022

Practice Set C

Multiply 86 by 52.

Answer: 4,472

Practice Set C

Multiply 419 by 85.

Answer: 35,615

Practice Set C

Multiply 2,376 by 613.

Answer: 1,456,488

Practice Set C

Multiply 8,107 by 304.

Answer: 2,464,528

Practice Set C

Multiply 66,260 by 1,008.

Answer: 66,790,080

Practice Set C

Multiply 209 by 501.

Answer: 104,709

Practice Set C

Multiply 24 by 10.

Answer: 240

Practice Set C

Multiply 3,809 by 1,000.

Answer: 3,809,000

Practice Set C

Multiply 813 by 10,000.

Answer: 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).

$(49,000)(1,200)$ = $\begin{array} {r} {49000} \\ {\underline{\times \ \ 1200}} \end{array}$

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

Multiply 1,800 by 90.

Answer: 162,000

Practice Set D

Multiply 420,000 by 300.

Answer: 126,000,000

Practice Set D

Multiply 20,500,000 by 140,000.

Answer: 2,870,000,000,000

Calculators

Most multiplications are performed using a calculator.

Sample Set E

Multiply 75,891 by 263.

Solution

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

The product is 19,959,333.

Sample Set E

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

Solution

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.

Sample Set E

Multiply 57,847,298 by 38,976.

Solution

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.

$52 \times 27$

Answer: 1,404

Practice Set E

$1,448 \times 6,155$

Answer: 8,912,440

Practice Set E

$8,940,000 \times 205,000$

Answer: 1,832,700,000,000

Exercises

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

Exercise $\PageIndex{1}$

$\begin{array} {r} {8} \\ {\underline{\times 3}} \end{array}$

Answer: 24

Exercise $\PageIndex{2}$

$\begin{array} {r} {3} \\ {\underline{\times 5}} \end{array}$

Exercise $\PageIndex{3}$

$\begin{array} {r} {8} \\ {\underline{\times 6}} \end{array}$

Answer: 48

Exercise $\PageIndex{4}$

$\begin{array} {r} {5} \\ {\underline{\times 7}} \end{array}$

Exercise $\PageIndex{5}$

$6 \times 1$

Answer: 6

Exercise $\PageIndex{6}$

$4 \times 5$

Exercise $\PageIndex{7}$

$75 \times 3$

Answer: 225

Exercise $\PageIndex{8}$

$35 \times 5$

Exercise $\PageIndex{9}$

$\begin{array} {r} {45} \\ {\underline{\times \ \ 6}} \end{array}$

Answer: 270

Exercise $\PageIndex{10}$

$\begin{array} {r} {31} \\ {\underline{\times \ \ 7}} \end{array}$

Exercise $\PageIndex{11}$

$\begin{array} {r} {97} \\ {\underline{\times \ \ 6}} \end{array}$

Answer: 582

Exercise $\PageIndex{12}$

$\begin{array} {r} {75} \\ {\underline{\times 57}} \end{array}$

Exercise $\PageIndex{13}$

$\begin{array} {r} {64} \\ {\underline{\times 15}} \end{array}$

Answer: 960

Exercise $\PageIndex{14}$

$\begin{array} {r} {73} \\ {\underline{\times 15}} \end{array}$

Exercise $\PageIndex{15}$

$\begin{array} {r} {81} \\ {\underline{\times 95}} \end{array}$

Answer: 7,695

Exercise $\PageIndex{16}$

$\begin{array} {r} {31} \\ {\underline{\times 33}} \end{array}$

Exercise $\PageIndex{17}$

$57 \times 64$

Answer: 3,648

Exercise $\PageIndex{18}$

$76 \times 42$

Exercise $\PageIndex{19}$

$894 \times 52$

Answer: 46,488

Exercise $\PageIndex{20}$

$684 \times 38$

Exercise $\PageIndex{21}$

$\begin{array} {r} {115} \\ {\underline{\times \ \ 22}} \end{array}$

Answer: 2,530

Exercise $\PageIndex{22}$

$\begin{array} {r} {706} \\ {\underline{\times \ \ 81}} \end{array}$

Exercise $\PageIndex{23}$

$\begin{array} {r} {328} \\ {\underline{\times \ \ 21}} \end{array}$

Answer: 6,888

Exercise $\PageIndex{24}$

$\begin{array} {r} {550} \\ {\underline{\times \ \ 94}} \end{array}$

Exercise $\PageIndex{25}$

$930 \times 26$

Answer: 24,180

Exercise $\PageIndex{26}$

$318 \times 63$

Exercise $\PageIndex{27}$

$\begin{array} {r} {582} \\ {\underline{\times 127}} \end{array}$

Answer: 73,914

Exercise $\PageIndex{28}$

$\begin{array} {r} {247} \\ {\underline{\times 116}} \end{array}$

Exercise $\PageIndex{29}$

$\begin{array} {r} {305} \\ {\underline{\times 225}} \end{array}$

Answer: 68,625

Exercise $\PageIndex{30}$

$\begin{array} {r} {782} \\ {\underline{\times 547}} \end{array}$

Exercise $\PageIndex{31}$

$\begin{array} {r} {771} \\ {\underline{\times 663}} \end{array}$

Answer: 511,173

Exercise $\PageIndex{32}$

$\begin{array} {r} {638} \\ {\underline{\times 516}} \end{array}$

Exercise $\PageIndex{33}$

$1,905 \times 710$

Answer: 1,352,550

Exercise $\PageIndex{34}$

$5,757 \times 5,010$

Exercise $\PageIndex{35}$

$\begin{array} {r} {3,106} \\ {\underline{\times 1,752}} \end{array}$

Answer: 5,441,712

Exercise $\PageIndex{36}$

$\begin{array} {r} {9,300} \\ {\underline{\times 1,130}} \end{array}$

Exercise $\PageIndex{37}$

$\begin{array} {r} {7,057} \\ {\underline{\times 5,229}} \end{array}$

Answer: 36,901,053

Exercise $\PageIndex{38}$

$\begin{array} {r} {8,051} \\ {\underline{\times 5,580}} \end{array}$

Exercise $\PageIndex{39}$

$\begin{array} {r} {5,804} \\ {\underline{\times 4,300}} \end{array}$

Answer: 24,957,200

Exercise $\PageIndex{40}$

$\begin{array} {r} {357} \\ {\underline{\times \ \ 16}} \end{array}$

Exercise $\PageIndex{41}$

$\begin{array} {r} {724} \\ {\underline{\times \ \ \ \ 0}} \end{array}$

Answer: 0

Exercise $\PageIndex{42}$

$\begin{array} {r} {2,649} \\ {\underline{\times \ \ \ \ \ 41}} \end{array}$

Exercise $\PageIndex{43}$

$\begin{array} {r} {5,173} \\ {\underline{\times \ \ \ \ \ \ \ 8}} \end{array}$

Answer: 41,384

Exercise $\PageIndex{44}$

$\begin{array} {r} {1,999} \\ {\underline{\times \ \ \ \ \ \ \ 0}} \end{array}$

Exercise $\PageIndex{45}$

$\begin{array} {r} {1,666} \\ {\underline{\times \ \ \ \ \ \ \ 0}} \end{array}$

Answer: 0

Exercise $\PageIndex{46}$

$\begin{array} {r} {51,730} \\ {\underline{\times \ \ \ \ \ 142}} \end{array}$

Exercise $\PageIndex{47}$

$\begin{array} {r} {387} \\ {\underline{\times 190}} \end{array}$

Answer: 73,530

Exercise $\PageIndex{48}$

$\begin{array} {r} {3,400} \\ {\underline{\times \ \ \ \ \ 70}} \end{array}$

Exercise $\PageIndex{49}$

$\begin{array} {r} {460,000} \\ {\underline{\times \ \ 14,000}} \end{array}$

Answer: 6,440,000,000

Exercise $\PageIndex{50}$

$\begin{array} {r} {558,000,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ 81,000}} \end{array}$

Exercise $\PageIndex{51}$

$\begin{array} {r} {37,000} \\ {\underline{\times \ \ \ \ \ 120}} \end{array}$

Answer: 4,440,000

Exercise $\PageIndex{52}$

$\begin{array} {r} {498,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ \ 0}} \end{array}$

Exercise $\PageIndex{53}$

$\begin{array} {r} {4,585,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ 140}} \end{array}$

Answer: 641,900,000

Exercise $\PageIndex{54}$

$\begin{array} {r} {30,700,000} \\ {\underline{\times \ \ \ \ \ \ \ \ \ \ \ \ 180}} \end{array}$

Exercise $\PageIndex{55}$

$\begin{array} {r} {8,000} \\ {\underline{\times \ \ \ \ \ 10}} \end{array}$

Answer: 80,000

Exercise $\PageIndex{56}$

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 $\PageIndex{57}$

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?

Answer: 384 reports

Exercise $\PageIndex{58}$

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 $\PageIndex{59}$

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?

Answer: 3,330 problems

Exercise $\PageIndex{60}$

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 $\PageIndex{61}$

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?

Answer: 114 units

Exercise $\PageIndex{62}$

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

Exercise $\PageIndex{63}$

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

Answer: 3,600 seconds

Exercise $\PageIndex{64}$

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 $\PageIndex{65}$

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?)

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

Exercise $\PageIndex{66}$

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 $\PageIndex{67}$

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

Answer: $110,055

Exercises for Review

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

Round 448,062,187 to the nearest hundred thousand.

Answer: 448,100,000

Find the sum. 22,451 + 18,976.

Subtract 2,289 from 3,001.

Answer: 712

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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