Prealgebra

1.4Multiply Whole Numbers

Prealgebra1.4 Multiply Whole Numbers

Learning Objectives

By the end of this section, you will be able to:
• Use multiplication notation
• Model multiplication of whole numbers
• Multiply whole numbers
• Translate word phrases to math notation
• Multiply whole numbers in applications

Be Prepared 1.3

Before you get started, take this readiness quiz.

1. Add: $1,683+479.1,683+479.$
If you missed this problem, review Example 1.21.
2. Subtract: $605−321.605−321.$
If you missed this problem, review Example 1.33.

Use Multiplication Notation

Suppose you were asked to count all these pennies shown in Figure 1.11.

Figure 1.11

Would you count the pennies individually? Or would you count the number of pennies in each row and add that number $33$ times.

$8+8+88+8+8$

Multiplication is a way to represent repeated addition. So instead of adding $88$ three times, we could write a multiplication expression.

$3×83×8$

We call each number being multiplied a factor and the result the product. We read $3×83×8$ as three times eight, and the result as the product of three and eight.

There are several symbols that represent multiplication. These include the symbol $××$ as well as the dot, $··$, and parentheses $().().$

Operation Symbols for Multiplication

To describe multiplication, we can use symbols and words.

Operation Notation Expression Read as Result
$MultiplicationMultiplication$ $××$
$··$
$()()$
$3×83×8$
$3·83·8$
$3(8)3(8)$
$three times eightthree times eight$ $the product of 3 and 8the product of 3 and 8$

Example 1.39

Translate from math notation to words:

1. $7×67×6$
2. $12·1412·14$
3. $6(13)6(13)$

Try It 1.77

Translate from math notation to words:

1. $8×78×7$
2. $18·1118·11$

Try It 1.78

Translate from math notation to words:

1. $(13)(7)(13)(7)$
2. $5(16)5(16)$

Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used $base-10base-10$ blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.

Example 1.40

Model: $3×8.3×8.$

Try It 1.79

Model each multiplication: $4×6.4×6.$

Try It 1.80

Model each multiplication: $5×7.5×7.$

Multiply Whole Numbers

In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section.

Table 1.4 shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.

× 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 18 24 30 36 42 48 54
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
9 0 9 18 27 36 45 54 63 72 81
Table 1.4

What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.

Multiplication Property of Zero

The product of any number and $00$ is $0.0.$

$a·0=00·a=0a·0=00·a=0$

Example 1.41

Multiply:

1. $0·110·11$
2. $(42)0(42)0$

Try It 1.81

Find each product:

1. $0·190·19$
2. $(39)0(39)0$

Try It 1.82

Find each product:

1. $0·240·24$
2. $(57)0(57)0$

What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and $11$ is called the multiplicative identity.

Identity Property of Multiplication

The product of any number and $11$ is the number.

$1·a=aa·1=a1·a=aa·1=a$

Example 1.42

Multiply:

1. $(11)1(11)1$
2. $1·421·42$

Try It 1.83

Find each product:

1. $(19)1(19)1$
2. $1·391·39$

Try It 1.84

Find each product:

1. $(24)(1)(24)(1)$
2. $1×571×57$

Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that $8+9=178+9=17$ is the same as $9+8=17.9+8=17.$

Is this also true for multiplication? Let’s look at a few pairs of factors.

$4·7=287·4=284·7=287·4=28$
$9·7=637·9=639·7=637·9=63$
$8·9=729·8=728·9=729·8=72$

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

Commutative Property of Multiplication

Changing the order of the factors does not change their product.

$a·b=b·aa·b=b·a$

Example 1.43

Multiply:

1. $8·78·7$
2. $7·87·8$

Try It 1.85

Multiply:

1. $9·69·6$
2. $6·96·9$

Try It 1.86

Multiply:

1. $8·68·6$
2. $6·86·8$

To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.

$27×3___27×3___$

We start by multiplying $33$ by $7.7.$

$3×7=213×7=21$

We write the $11$ in the ones place of the product. We carry the $22$ tens by writing $22$ above the tens place.

Then we multiply the $33$ by the $2,2,$ and add the $22$ above the tens place to the product. So $3×2=6,3×2=6,$ and $6+2=8.6+2=8.$ Write the $88$ in the tens place of the product.

The product is $81.81.$

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.

Example 1.44

Multiply: $15·4.15·4.$

Try It 1.87

Multiply: $64·8.64·8.$

Try It 1.88

Multiply: $57·6.57·6.$

Example 1.45

Multiply: $286·5.286·5.$

Try It 1.89

Multiply: $347·5.347·5.$

Try It 1.90

Multiply: $462·7.462·7.$

When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.

How To

Multiply two whole numbers to find the product.

1. Step 1. Write the numbers so each place value lines up vertically.
2. Step 2.
Multiply the digits in each place value.
• Work from right to left, starting with the ones place in the bottom number.
• Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
• If a product in a place value is more than $9,9,$ carry to the next place value.
• Write the partial products, lining up the digits in the place values with the numbers above.
• Repeat for the tens place in the bottom number, the hundreds place, and so on.
• Insert a zero as a placeholder with each additional partial product.
3. Step 3. Add the partial products.

Example 1.46

Multiply: $62(87).62(87).$

Try It 1.91

Multiply: $43(78).43(78).$

Try It 1.92

Multiply: $64(59).64(59).$

Example 1.47

Multiply:

1. $47·1047·10$
2. $47·100.47·100.$

Try It 1.93

Multiply:

1. $54·1054·10$
2. $54·100.54·100.$

Try It 1.94

Multiply:

1. $75·1075·10$
2. $75·100.75·100.$

Example 1.48

Multiply: $(354)(438).(354)(438).$

Try It 1.95

Multiply: $(265)(483).(265)(483).$

Try It 1.96

Multiply: $(823)(794).(823)(794).$

Example 1.49

Multiply: $(896)201.(896)201.$

Try It 1.97

Multiply: $(718)509.(718)509.$

Try It 1.98

Multiply: $(627)804.(627)804.$

When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:

 to multiply $8⋅3⋅28⋅3⋅2$ first multiply $8⋅38⋅3$ $24⋅224⋅2$ then multiply $24⋅224⋅2$. $4848$

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process and translate word phrases into math notation. Some of the words that indicate multiplication are given in Table 1.5.

Operation Word Phrase Example Expression
Multiplication times
product
twice
$33$ times $88$
the product of $33$ and $88$
twice $44$
$3×8,3·8,(3)(8),3×8,3·8,(3)(8),$
$(3)8,or3(8)(3)8,or3(8)$
$2·42·4$
Table 1.5

Example 1.50

Translate and simplify: the product of $1212$ and $27.27.$

Try It 1.99

Translate and simplify the product of $1313$ and $28.28.$

Try It 1.100

Translate and simplify the product of $4747$ and $14.14.$

Example 1.51

Translate and simplify: twice two hundred eleven.

Try It 1.101

Translate and simplify: twice one hundred sixty-seven.

Try It 1.102

Translate and simplify: twice two hundred fifty-eight.

Multiply Whole Numbers in Applications

We will use the same strategy we used previously to solve applications of multiplication. First, we need to determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify to get the answer. Finally, we write a sentence to answer the question.

Example 1.52

Humberto bought $44$ sheets of stamps. Each sheet had $2020$ stamps. How many stamps did Humberto buy?

Try It 1.103

Valia donated water for the snack bar at her son’s baseball game. She brought $66$ cases of water bottles. Each case had $2424$ water bottles. How many water bottles did Valia donate?

Try It 1.104

Vanessa brought $88$ packs of hot dogs to a family reunion. Each pack has $1010$ hot dogs. How many hot dogs did Vanessa bring?

Example 1.53

When Rena cooks rice, she uses twice as much water as rice. How much water does she need to cook $44$ cups of rice?

Try It 1.105

Erin is planning her flower garden. She wants to plant twice as many dahlias as sunflowers. If she plants 14 sunflowers, how many dahlias does she need?

Try It 1.106

A college choir has twice as many women as men. There are 18 men in the choir. How many women are in the choir?

Example 1.54

Van is planning to build a patio. He will have $88$ rows of tiles, with $1414$ tiles in each row. How many tiles does he need for the patio?

Try It 1.107

Jane is tiling her living room floor. She will need 16 rows of tile, with 20 tiles in each row. How many tiles does she need for the living room floor?

Try It 1.108

Yousef is putting shingles on his garage roof. He will need 24 rows of shingles, with 45 shingles in each row. How many shingles does he need for the garage roof?

If we want to know the size of a wall that needs to be painted or a floor that needs to be carpeted, we will need to find its area. The area is a measure of the amount of surface that is covered by the shape. Area is measured in square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm.) on a side. A square inch is a square that is one inch on each side, and so on.

For a rectangular figure, the area is the product of the length and the width. Figure 1.12 shows a rectangular rug with a length of $22$ feet and a width of $33$ feet. Each square is $11$ foot wide by $11$ foot long, or $11$ square foot. The rug is made of $66$ squares. The area of the rug is $66$ square feet.

Figure 1.12 The area of a rectangle is the product of its length and its width, or $66$ square feet.

Example 1.55

Jen’s kitchen ceiling is a rectangle that measures 9 feet long by 12 feet wide. What is the area of Jen’s kitchen ceiling?

Try It 1.109

Zoila bought a rectangular rug. The rug is 8 feet long by 5 feet wide. What is the area of the rug?

Try It 1.110

Rene’s driveway is a rectangle 45 feet long by 20 feet wide. What is the area of the driveway?

Section 1.4 Exercises

Practice Makes Perfect

Use Multiplication Notation

In the following exercises, translate from math notation to words.

225.

$4 × 7 4 × 7$

226.

$8 × 6 8 × 6$

227.

$5 · 12 5 · 12$

228.

$3 · 9 3 · 9$

229.

$( 10 ) ( 25 ) ( 10 ) ( 25 )$

230.

$( 20 ) ( 15 ) ( 20 ) ( 15 )$

231.

$42 ( 33 ) 42 ( 33 )$

232.

$39 ( 64 ) 39 ( 64 )$

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

233.

$3 × 6 3 × 6$

234.

$4 × 5 4 × 5$

235.

$5 × 9 5 × 9$

236.

$3 × 9 3 × 9$

Multiply Whole Numbers

In the following exercises, fill in the missing values in each chart.

237.
238.
239.
240.
241.
242.
243.
244.

In the following exercises, multiply.

245.

$0 · 15 0 · 15$

246.

$0 · 41 0 · 41$

247.

$( 99 ) 0 ( 99 ) 0$

248.

$( 77 ) 0 ( 77 ) 0$

249.

$1 · 43 1 · 43$

250.

$1 · 34 1 · 34$

251.

$( 28 ) 1 ( 28 ) 1$

252.

$( 65 ) 1 ( 65 ) 1$

253.

$1 ( 240,055 ) 1 ( 240,055 )$

254.

$1 ( 189,206 ) 1 ( 189,206 )$

255.
1. $7·67·6$
2. $6·76·7$
256.
1. $8×98×9$
2. $9×89×8$
257.

$( 79 ) ( 5 ) ( 79 ) ( 5 )$

258.

$( 58 ) ( 4 ) ( 58 ) ( 4 )$

259.

$275 · 6 275 · 6$

260.

$638 · 5 638 · 5$

261.

$3,421 × 7 3,421 × 7$

262.

$9,143 × 3 9,143 × 3$

263.

$52 ( 38 ) 52 ( 38 )$

264.

$37 ( 45 ) 37 ( 45 )$

265.

$96 · 73 96 · 73$

266.

$89 · 56 89 · 56$

267.

$27 × 85 27 × 85$

268.

$53 × 98 53 × 98$

269.

$23 · 10 23 · 10$

270.

$19 · 10 19 · 10$

271.

$( 100 ) ( 36 ) ( 100 ) ( 36 )$

272.

$( 100 ) ( 25 ) ( 100 ) ( 25 )$

273.

$1,000 ( 88 ) 1,000 ( 88 )$

274.

$1,000 ( 46 ) 1,000 ( 46 )$

275.

$50 × 1,000,000 50 × 1,000,000$

276.

$30 × 1,000,000 30 × 1,000,000$

277.

$247 × 139 247 × 139$

278.

$156 × 328 156 × 328$

279.

$586 ( 721 ) 586 ( 721 )$

280.

$472 ( 855 ) 472 ( 855 )$

281.

$915 · 879 915 · 879$

282.

$968 · 926 968 · 926$

283.

$( 104 ) ( 256 ) ( 104 ) ( 256 )$

284.

$( 103 ) ( 497 ) ( 103 ) ( 497 )$

285.

$348 ( 705 ) 348 ( 705 )$

286.

$485 ( 602 ) 485 ( 602 )$

287.

$2,719 × 543 2,719 × 543$

288.

$3,581 × 724 3,581 × 724$

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

289.

the product of $1818$ and $3333$

290.

the product of $1515$ and $2222$

291.

fifty-one times sixty-seven

292.

forty-eight times seventy-one

293.

twice $249249$

294.

twice $589589$

295.

ten times three hundred seventy-five

296.

ten times two hundred fifty-five

Mixed Practice

In the following exercises, simplify.

297.

$38 × 37 38 × 37$

298.

$86 × 29 86 × 29$

299.

$415 − 267 415 − 267$

300.

$341 − 285 341 − 285$

301.

$6,251 + 4,749 6,251 + 4,749$

302.

$3,816 + 8,184 3,816 + 8,184$

303.

$( 56 ) ( 204 ) ( 56 ) ( 204 )$

304.

$( 77 ) ( 801 ) ( 77 ) ( 801 )$

305.

$947 · 0 947 · 0$

306.

$947 + 0 947 + 0$

307.

$15,382 + 1 15,382 + 1$

308.

$15,382 · 1 15,382 · 1$

In the following exercises, translate and simplify.

309.

the difference of 50 and 18

310.

the difference of 90 and 66

311.

twice 35

312.

twice 140

313.

20 more than 980

314.

65 more than 325

315.

the product of 12 and 875

316.

the product of 15 and 905

317.

subtract 74 from 89

318.

subtract 45 from 99

319.

the sum of 3,075 and 950

320.

the sum of 6,308 and 724

321.

366 less than 814

322.

388 less than 925

Multiply Whole Numbers in Applications

In the following exercises, solve.

323.

Party supplies Tim brought 9 six-packs of soda to a club party. How many cans of soda did Tim bring?

324.

Sewing Kanisha is making a quilt. She bought 6 cards of buttons. Each card had four buttons on it. How many buttons did Kanisha buy?

325.

Field trip Seven school busses let off their students in front of a museum in Washington, DC. Each school bus had 44 students. How many students were there?

326.

Gardening Kathryn bought 8 flats of impatiens for her flower bed. Each flat has 24 flowers. How many flowers did Kathryn buy?

327.

Charity Rey donated 15 twelve-packs of t-shirts to a homeless shelter. How many t-shirts did he donate?

328.

School There are 28 classrooms at Anna C. Scott elementary school. Each classroom has 26 student desks. What is the total number of student desks?

329.

Recipe Stephanie is making punch for a party. The recipe calls for twice as much fruit juice as club soda. If she uses 10 cups of club soda, how much fruit juice should she use?

330.

Gardening Hiroko is putting in a vegetable garden. He wants to have twice as many lettuce plants as tomato plants. If he buys 12 tomato plants, how many lettuce plants should he get?

331.

Government The United States Senate has twice as many senators as there are states in the United States. There are 50 states. How many senators are there in the United States Senate?

332.

Recipe Andrea is making potato salad for a buffet luncheon. The recipe says the number of servings of potato salad will be twice the number of pounds of potatoes. If she buys 30 pounds of potatoes, how many servings of potato salad will there be?

333.

Painting Jane is painting one wall of her living room. The wall is rectangular, 13 feet wide by 9 feet high. What is the area of the wall?

334.

Home décor Shawnte bought a rug for the hall of her apartment. The rug is 3 feet wide by 18 feet long. What is the area of the rug?

335.

Room size The meeting room in a senior center is rectangular, with length 42 feet and width 34 feet. What is the area of the meeting room?

336.

Gardening June has a vegetable garden in her yard. The garden is rectangular, with length 23 feet and width 28 feet. What is the area of the garden?

337.

NCAA basketball According to NCAA regulations, the dimensions of a rectangular basketball court must be 94 feet by 50 feet. What is the area of the basketball court?

338.

NCAA football According to NCAA regulations, the dimensions of a rectangular football field must be 360 feet by 160 feet. What is the area of the football field?

Everyday Math

339.

Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price rose $1212$ per share. How much money did Javier’s portfolio gain?

340.

Salary Carlton got a $200200$ raise in each paycheck. He gets paid 24 times a year. How much higher is his new annual salary?

Writing Exercises

341.

How confident do you feel about your knowledge of the multiplication facts? If you are not fully confident, what will you do to improve your skills?

342.

How have you used models to help you learn the multiplication facts?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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