1.4 Multiply Whole Numbers

Learning objectives

Learning objectives

• Use multiplication notation

• Model multiplication of whole numbers

• Multiply whole numbers

• Translate word phrases to math notation

• Multiply whole numbers in applications

Use Multiplication Notation

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

Would you count the pennies individually? Or would you count the number of pennies in each row and add that number $3$ times.

Multiplication is a way to represent repeated addition. So instead of adding $8$ three times, we could write a multiplication expression.

We call each number being multiplied a factor and the result the product. We read $3\,\times\,8$ 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 $\times$ as well as the dot, $\cdot$, and parentheses $(\,\mathrm{).}$

Example 1.39

Solution

1. Frame the translation task. Identify the two factors in each expression and use the word times to name the multiplication operation.
2. Translate the first expression. The expression $7\,\times\,6$ names seven times six, and its result is the product of seven and six.
3. Translate the second expression. The expression $12\cdot{}14$ names twelve times fourteen, and its result is the product of twelve and fourteen.
4. Translate the third expression. The expression $6(13)$ names six times thirteen, and its result is the product of six and thirteen.
5. Summarize the notation. All three expressions describe repeated equal groups even though they use different multiplication symbols.

Takeaway: All multiplication notations describe repeated equal groups: $7\times6$, $12\cdot14$, and $6(13)$ are read as seven times six, twelve times fourteen, and six times thirteen.

Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used $\mathrm{base-10}$ 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

Solution

1. Choose the first factor. Begin with one row containing $8$ counters so that the row length represents $8$.
2. Choose the second factor. Use the other factor, $3$, to make $3$ equal rows of $8$ counters in the array model.
3. Count the whole array. Multiplying the number of rows by the number in each row gives $3\cdot{}8=24$ counters in all.
4. Check the rotated model. Turning the array sideways gives $8$ rows of $3$ counters, which also contains $24$ counters.

Takeaway: A factor model for $3\times8$ has $3$ rows of $8$ counters, and rotating the model to $8$ rows of $3$ still gives $24$.

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.

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.

Example 1.41

Solution

1. Frame the zero property. A product containing a factor of $0$ always has value $0$, regardless of the other factor.
2. Evaluate the first product. In $0\cdot{}11$, one factor is zero, so the product is $0$.
3. Evaluate the second product. In $(42)0$, the other factor is also zero, so the product is $0$.
4. State the common conclusion. Both expressions equal $0$ because multiplying any number by zero produces zero.

Takeaway: The zero property of multiplication states that a product containing a factor of $0$ is $0$, so $0\cdot11=0$ and $42\cdot0=0$.

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 $1$ is called the multiplicative identity.

Example 1.42

Solution

1. Frame the identity property. Multiplying a whole number by $1$ does not change the value of that number.
2. Evaluate the first product. In $(11)1$, the factor $1$ leaves $11$ unchanged, so the product is $11$.
3. Evaluate the second product. In $1\cdot{}42$, the factor $1$ leaves $42$ unchanged, so the product is $42$.
4. State the common conclusion. The multiplicative identity is $1$, because $11\cdot{}1=11$ and $1\cdot{}42=42$.

Takeaway: The multiplicative identity is $1$, so multiplying any number by $1$ leaves it unchanged: $11\cdot1=11$ and $1\cdot42=42$.

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 $8+9=17$ is the same as $9+8=17.$

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

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

Example 1.43

Solution

1. Frame the comparison. Both expressions contain the same two factors, so use multiplication to find each product.
2. Evaluate the first product. Multiplying $8$ by $7$ gives $56$, so $8\cdot{}7=56$.
3. Evaluate the second product. Multiplying $7$ by $8$ gives $56$, so $7\cdot{}8=56$.
4. Compare the results. The products are equal because reversing the order of the factors does not change multiplication.

Takeaway: Multiplication is commutative because changing the order of the factors does not change the product, so $8\cdot7=7\cdot8=56$.

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.

We start by multiplying $3$ by $7.$

We write the $1$ in the ones place of the product. We carry the $2$ tens by writing $2$ above the tens place.

Then we multiply the $3$ by the $2,$ and add the $2$ above the tens place to the product. So $3\,\times\,2=6,$ and $6+2=8.$ Write the $8$ in the tens place of the product.

The product is $81.$

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

Solution

1. Align the digits. Write $15$ above $4$ with the $5$ in the ones column and the $1$ in the tens column.
2. Multiply the ones digit. Compute $4\cdot{}5=20$, write $0$ in the ones place, and carry $2$ tens to the tens column.
3. Multiply the tens digit. Compute $4\cdot{}1=4$ and add the carried $2$, giving $4+2=6$ in the tens place.
4. Read the completed product. The digits $60$ give the final answer, so $15\cdot{}4=60$.

Takeaway: Multiply each digit by $4$ according to its place value and carry the tens, so $4\cdot5=20$ gives a ones digit of $0$ and $4\cdot1+2=6$ gives $15\cdot4=60$.

Example 1.45

Solution

1. Align the digits. Write $286$ above $5$ so that the $6$, $8$, and $2$ occupy the ones, tens, and hundreds places.
2. Multiply the ones digit. Compute $5\cdot{}6=30$, write $0$ in the ones place, and carry $3$ tens.
3. Multiply the tens digit. Compute $5\cdot{}8=40$ and add the carried $3$ to obtain $43$. Write $3$ in the tens place and carry $4$ hundreds.
4. Multiply the hundreds digit. Compute $5\cdot{}2=10$ and add the carried $4$ to obtain $14$. Write $4$ in the hundreds place and $1$ in the thousands place.
5. Read the completed product. The resulting digits $1,430$ show that $286\cdot{}5=1,430$.

Takeaway: When multiplying by a single digit, carry each group of ten leftward, giving $5\cdot6=30$, $5\cdot8+3=43$, and $5\cdot2+4=14$, so $286\cdot5=1,430$.

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.

Example 1.46

Solution

1. Set up the multiplication. Align $62$ above $87$ so that the ones and tens digits form matching columns, as in the first figure.
2. Multiply by the ones digit. Compute $62\cdot{}7=434$. The red $4$ in the ones place shows the carry from $7\cdot{}2=14$.
3. Multiply by the tens digit. Compute $62\cdot{}8=496$, which represents $62\cdot{}80$ after shifting one place to the left. The second partial product is therefore $4,960$.
4. Add the partial products. Adding $434+4,960$ gives $5,394$, which is the completed product shown in the final figure.

Takeaway: In two-digit multiplication, shift the tens partial product one place left before adding it to the ones partial product.

Example 1.47

Solution

1. Frame the powers-of-ten pattern. Multiplying by $10$ shifts each digit one place left, while multiplying by $100$ shifts each digit two places left.
2. Compute the first product. Multiplying $47$ by $10$ appends one zero, giving $47\cdot{}10=470$.
3. Compute the second product. Multiplying $47$ by $100$ appends two zeros, giving $47\cdot{}100=4{,}700$.
4. Check the pattern. Each additional zero in the power of ten shifts the original digits one more place to the left.

Takeaway: Multiplying by $10$ appends one zero, while multiplying by $100$ appends two, so $47\cdot10=470$ and $47\cdot100=4{,}700$.

Example 1.48

Solution

1. Multiply by the ones digit. Compute $354\cdot{}8=2,832$ and record this as the first partial product.
2. Multiply by the tens digit. Compute $354\cdot{}3=1,062$. Since $3$ occupies the tens place, shift this partial product one place left to represent $10,620$.
3. Multiply by the hundreds digit. Compute $354\cdot{}4=1,416$. Because $4$ occupies the hundreds place, shift this product two places left to represent $141,600$.
4. Add the partial products. Add $2,832+10,620+141,600=155,052$, which is the value of $(354)(438)$.

Takeaway: Each partial product must be shifted left according to the place value of the multiplier digit that produced it.

Example 1.49

Solution

1. Set up three place-value partial products. The multiplier $201$ has ones, tens, and hundreds digits, so form a partial product for each place.
2. Multiply by the ones digit. The ones digit is $1$, giving $896\cdot{}1=896$ in the ones place.
3. Multiply by the tens digit. The tens digit is $0$, so $896\cdot{}0=0$, which contributes nothing to the sum.
4. Multiply by the hundreds digit. The hundreds digit is $2$, and $896\cdot{}2=1,792$; written in the hundreds place this partial product is $179,200$.
5. Add the partial products. Adding $896+0+179,200$ gives $180,096$, so $(896)(201)=180,096$.

Takeaway: In a three-digit multiplier each partial product shifts one place left, so $896\cdot201=896+0+179{,}200=180{,}096$.

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

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.

Table

1.5

Example 1.50

Solution

1. Identify the operation. The phrase product tells us that the two given quantities must be multiplied.
2. Identify the factors. The quantities $12$ and $27$ supply the two factors in the multiplication expression.
3. Translate and simplify. The expression is $12\cdot{}27$, and multiplying the partial products $12\cdot{}7=84$ and $12\cdot{}20=240$ gives $324$.

Takeaway: The word product signals multiplication, so the product of $12$ and $27$ is $12\cdot27=324$.

Example 1.51

Solution

1. Identify the multiplier. The word twice tells us to multiply the stated number by $2$, rather than adding it to itself.
2. Identify the stated number. Two hundred eleven supplies the other factor, which is written as $211$ in standard form.
3. Translate and simplify. The expression is $2(211)$, and doubling $211$ gives $2\cdot{}211=422$ as the simplified value.

Takeaway: The word twice means multiply by $2$, so twice $211$ is $2\cdot211=422$.

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

Solution

1. Identify the number of groups. Humberto bought $4$ sheets of stamps, so one factor is $4$.
2. Identify the amount in each group. Each sheet contained $20$ stamps, so the other factor is $20$. The diagram shows $20$ multiplied by $4$.
3. Multiply and state the answer. Compute $20\cdot{}4=80$. Humberto bought $80$ stamps in all.

Takeaway: For equal groups, multiply the number of groups by the number of items in each group.

Example 1.53

Solution

1. Identify the relationship. Twice as much water as rice means the amount of water is twice the amount of rice.
2. Translate to notation. For $4$ cups of rice, the required water is represented by $2\cdot{}4$ cups.
3. Multiply and label the result. Compute $2\cdot{}4=8$, and retain cups as the unit of measure.
4. Answer the question. Rena needs $8$ cups of water to cook $4$ cups of rice.

Takeaway: A phrase such as twice as much means multiply the given amount by $2$, so $4$ cups of rice require $2\cdot4=8$ cups of water.

Example 1.54

Solution

1. Identify the array dimensions. There are $8$ rows of tiles, with $14$ tiles in each row.
2. Form the multiplication expression. The total number of tiles is the product of the two dimensions, $8\cdot{}14$.
3. Compute the partial products. Multiply $14$ by $8$ to obtain $112$, using the units and tens digits of the first factor.
4. Answer with the unit. Van needs $112$ tiles for his patio.

Takeaway: An array with $8$ rows of $14$ tiles has $8\cdot14=112$ tiles in total.

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 $2$ feet and a width of $3$ feet. Each square is $1$ foot wide by $1$ foot long, or $1$ square foot. The rug is made of $6$ squares. The area of the rug is $6$ square feet.

Example 1.55

Solution

1. Identify the dimensions. The ceiling is $9$ feet long and $12$ feet wide, so both measurements are needed.
2. Form the area expression. The area of a rectangle is length times width, represented by $9\cdot{}12$.
3. Compute the product. Multiplying the partial products $9\cdot{}2=18$ and $9\cdot{}10=90$ gives $108$.
4. Use square units in the answer. Jen's kitchen ceiling has an area of $108$ square feet because area measures two-dimensional space.

Takeaway: The area of a rectangle is length times width, so a $9$-foot by $12$-foot ceiling has area $9\cdot12=108$ square feet.

Section 1.4 Exercises

Section 1.4 Exercises

Use Multiplication Notation

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

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

Multiply Whole Numbers

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

In the following exercises, multiply.

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

Mixed Practice

In the following exercises, simplify.

In the following exercises, translate and simplify.

Multiply Whole Numbers in Applications

In the following exercises, solve.

Everyday Math

Writing Exercises

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?

1. Translate from math notation to words:

1. ⓐ $8\,\times\,7$

2. ⓑ $18\cdot{}11$

2. Translate from math notation to words:

1. ⓐ $(13)(7)$

2. ⓑ $5(16)$

3. Find each product:

1. ⓐ $0\cdot{}19$

2. ⓑ $(39)0$

4. Find each product:

1. ⓐ $0\cdot{}24$

2. ⓑ $(57)0$

5. Find each product:

1. ⓐ $(19)1$

2. ⓑ $1\cdot{}39$

1. Find each product:

1. ⓐ $(24)(1)$

2. ⓑ $1\,\times\,57$

2. Multiply:

1. ⓐ $9\cdot{}6$

2. ⓑ $6\cdot{}9$

3. Multiply:

1. ⓐ $8\cdot{}6$

2. ⓑ $6\cdot{}8$

4. Multiply: $64\cdot{}8.$

5. Multiply: $57\cdot{}6.$

1. Multiply: $347\cdot{}5.$

2. Multiply: $462\cdot{}7.$

3. Multiply: $43(78).$

4. Multiply: $64(59).$

5. Multiply:

1. ⓐ $54\cdot{}10$

2. ⓑ $54\cdot{}100.$

1. Multiply:

1. ⓐ $75\cdot{}10$

2. ⓑ $75\cdot{}100.$

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

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

4. Multiply: $(718)509.$

5. Multiply: $(627)804.$

1. Translate and simplify: the product of $13$ and $28.$

2. Translate and simplify: the product of $47$ and $14.$

3. Translate and simplify: twice one hundred sixty-seven.

4. Translate and simplify: twice two hundred fifty-eight.

5. Valia donated water for the snack bar at her son’s baseball game. She brought $6$ cases of water bottles. Each case had $24$ water bottles. How many water bottles did Valia donate?

1. Vanessa brought $8$ packs of hot dogs to a family reunion. Each pack has $10$ hot dogs. How many hot dogs did Vanessa bring?

2. 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?

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

4. 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?

5. 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?

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

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

3. $4\,\times\,7$

4. $5\cdot{}12$

5. $(10)(25)$

1. $42(33)$

2. $1(240,055)$

3. 1. ⓐ $7\cdot{}6$

2. ⓑ $6\cdot{}7$

4. $275\cdot{}6$

5. $3,421\,\times\,7$

1. $52(38)$

2. $96\cdot{}73$

3. $27\,\times\,85$

4. $(100)(36)$

5. $1,000(88)$

1. $50\,\times\,1,000,000$

2. $247\,\times\,139$

3. $586(721)$

4. $915\cdot{}879$

5. $(104)(256)$

1. $348(705)$

2. $2,719\,\times\,543$

3. the product of $18$ and $33$

4. fifty-one times sixty-seven

5. twice $249$

1. ten times three hundred seventy-five

2. $38\,\times\,37$

3. $6,251+4,749$

4. $(56)(204)$

5. $15,382+1$

1. the difference of 50 and 18

2. twice 35

3. 20 more than 980

4. the product of 12 and 875

5. subtract 74 from 89

1. the sum of 3,075 and 95

2. 366 less than 814

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

4. 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?

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

1. 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?

2. 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?

3. 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?

4. 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?

5. 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?

1. Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price rose $12 per share. How much money did Javier’s portfolio gain?

2. 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?