Multiply Whole Numbers

1.4 Multiply Whole Numbers

The topics covered in this section are:

  1. Use multiplication notation
  2. Model multiplication of whole numbers
  3. Multiply whole numbers
  4. Translate word phrases to math notation
  5. Multiply whole numbers in applications

1.4.1 Use Multiplication Notation

Suppose you were asked to count all these pennies shown in the figure below.

An image of 3 horizontal rows of pennies, each row containing 8 pennies.

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

$8+8+8$

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

$3\times 8$

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.

Operation Symbols for Multiplication

To describe multiplication, we can use symbols and words.

OperationNotationExpressionRead asResult
Multiplication$\times$
$\cdot$
$( \ )$
$3 \times 8$
$3 \cdot 8$
$3(8)$
three times eightthe product of $3$ and $8$

Example 1

Translate from math notation to words:

  • $7 \times 6$
  • $12 \cdot 14$
  • $6(13)$
Solution
  1. We read this as seven times six and the result is the product of seven and six.
  2. We read this as twelve times fourteen and the result is the product of twelve and fourteen.
  3. We read this as six times thirteen and the result is the product of six and thirteen.

1.4.2 Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used 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 2

Model: $3 \times 8$.

Solution

To model the product $3 \times 8$, we’ll start with a row of $8$ counters.

An image of a horizontal row of 8 counters.

The other factor is $3$, so we’ll make $3$ rows of $8$ counters.

An image of 3 horizontal rows of counters, each row containing 8 counters.

Now we can count the result. There are $24$ counters in all.

$3 \times 8 = 24$

If you look at the counters sideways, you’ll see that we could have also made $8$ rows of $3$ counters. The product would have been the same. We’ll get back to this idea later.

1.4.3 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.

X0123456789
00000000000
10123456789
2024681012141618
30369121518212427
404812162024283236
5051015202530354045
6061218243036424854
7071421283542495663
8081624324048566472
9091824364554637281
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 $0$ is $0$.
$a \cdot 0 = 0$
$0 \cdot a = 0$

Example 3

Multiply:

  • $0 \cdot 11$
  • $(42)0$
Solution

1.$0 \cdot 11 $
The product of any number and zero is zero.0
2.$(42)0$
Multiplying by zero results in zero.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.

IDENTITY PROPERTY OF MULTIPLICATION

The product of any number and $1$ is the number.
$1 \cdot a = a$
$a \cdot 1 = a$

Example 4

Multiply:

  • $(11)1$
  • $1 \cdot 42$
Solution

1.$(11)1$
The product of any number and one is the number.$11$
2.$1 \cdot 42$
Multiplying by one does not change the value.$42$

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.

$4 \cdot7=28$$7 \cdot 4 = 28$
$9 \cdot 7= 63$$7 \cdot 9 = 63$
$8 \cdot 9=72$$9 \cdot 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.
$a \cdot b = b \cdot a$

Example 5

Multiply:

  • $8 \cdot 7$
  • $7 \cdot 8$
Solution

1.$8 \cdot 7$
Multiply.$56$
2.$7 \cdot 8$
Multiply.$56$

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

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.

No Alt Text

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.

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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 6

Multiply: $15 \cdot 4 $.

Solution

Write the numbers so the digits $5$ and $4$ line up vertically.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.11.18-AM-1.png
Multiply $4$ by the digit in the ones place of $15$. $4 \cdot 5=20$
Write $0$ in the ones place of the product and carry the $2$ tens.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.11.22-AM.png
Multiply $4$ by the digit in the tens place of $15$. $4 \cdot 1=4$.
Add the $2$ tens we carried. $4+2=6$.
Write the $6$ in the tens place of the product.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.11.28-AM-1.png

Example 7

Multiply: $286 \cdot 5$.

Solution

Write the numbers so the digits $5$ and $6$ line up vertically.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.19.42-AM.png
Multiply $5$ by the digit in the ones place of $286$. $5 \cdot 6=30$
Write $0$ in the ones place of the product and carry the $3$ to the tens place. Multiply $5$ by the digit in the tens place of $286$. $5 \cdot 8=40$.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.19.48-AM.png
Add the $3$ tens we carried to get $40+3=43$.
Write the $3$ in the tens place of the product and carry the $4$ to the hundreds place.
This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.19.53-AM.png
Multiply $5$ by the digit in the hundreds place of $286$. $5 \cdot 2=10$.
Add the $4$ hundreds we carried to get $10+4=14$.
Write the $4$ in the hundreds place of the product and the $1$ into the thousands place.
This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.19.58-AM.png

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. Write the numbers so each place value lines up vertically.
  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 $9$ 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. Add the partial products.

Example 8

Multiply: $62(87)$.

Solution

Write the numbers so each place lines up vertically.CNX_BMath_Figure_01_04_020_img-02.png
Start by multiplying $7$ by $62$. Multiply $7$ by the digit in the ones place of $62$. $7 \cdot 2=14$. Write the $4$ in the ones place of the product and carry the $1$ to the tens place.CNX_BMath_Figure_01_04_020_img-03.png
Multiply $7$ by the digit in the tens place of $62$. $7 \cdot 2=42$. Add the $1$ ten we carried. $42+1=43$. Write the $3$ in the tens place of the product and the $4$ in the hundreds place.CNX_BMath_Figure_01_04_020_img-04.png
The first partial product is $434$.
Now, write a $0$ under the $4$ in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of $87$ by $62$. Multiply $8$ by the digit in the ones place of $62$. $8 \cdot 2=16$. Write the $6$ in the next place of the product, which is the tens place. Carry the $1$ to the tens place.CNX_BMath_Figure_01_04_020_img-05.png
Multiply $8$ by $6$, the digit in the tens place of $62$, then add the $1$ ten we carried to get $49$. Write the $9$ in the hundreds place of the product and the $4$ in the thousands place.CNX_BMath_Figure_01_04_020_img-06.png
The second partial product is $4960$. Add the partial products.CNX_BMath_Figure_01_04_020_img-07.png

The product is $5,394$.

Example 9

Multiply:

  • $47 \cdot 10$.
  • $47 \cdot 100$.
Solution

1. $47 \cdot 10$.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.51.05-AM.png
2. $47 \cdot 100$.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-10.51.11-AM.png

When we multiplied $47$ times $10$, the product was $470$. Notice that $10$ has one zero, and we put one zero after $47$ to get the product. When we multiplied $47$ times $100$, the product was $4,700$. Notice that $100$ has two zeros and we put two zeros after $47$ to get the product.

Do you see the pattern? If we multiplied $47$ times $10,000$ which has four zeros, we would put four zeros after $47$ to get the product $470,000$.

Example 10

Multiply: $(354)(438)$.

Solution

There are three digits in the factors so there will be $3$ partial products. We do not have to write the $0$ as a placeholder as long as we write each partial product in the correct place.

An image of the multiplication problem “354 times 438” worked out vertically. 354 is the top number, 438 is the second number. Below 438 is a multiplication bar. Below the bar is the number 2,832. 2832 has the label “Multiply 8 times 354”. Below 2832 is the number 1,062;  1062 has the label “Multiply 3 times 354”.  Below 1062 is the number 1,416; 1416 has the label “Multiply 4 times 354”.  Below this is a bar and below the bar is the number “155,052”, with the label “Add the partial products”.

Example 11

Multiply: $(896)201$.

Solution

There should be $3$ partial products. The second partial product will be the result of multiplying $896$ by $0$.

An image of the multiplication problem “896 times 201” worked out vertically. 896 is the top number, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 201 is the second number,  the 2 in the hundreds place, the 0 in the tens place, the 1 in the ones place. Below 201 is a multiplcation bar. Below the bar is the number 896, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 896 has the label “Multiply 1 times 896”. Below 896 is the number “000”, the 0 in the thousands place, the 0 in the hundreds place, and the 0 in the tens place. “000” has the label “Multiply 0 times 896”.  Below “000” is the number 1792, the 1 in the hundred thousands place, the 7 in the ten thousands place, the 9 in the thousands place, and the 2 in the hundreds place. 1792 has the label “Multiply 2 times 896”.  Below this is a bar and below the bar is the number “180,096”, with the label “Add the partial products”.

Notice that the second partial product of all zeros doesn’t really affect the result. We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the $2$ in the hundreds place, as shown.

Multiply by $10$ but insert only one zero as a placeholder in the tens place. Multiply by $200$, putting the $2$ from the $12$. $2 \cdot 6 = 12$ in the hundreds place.

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$8 \cdot 3 \cdot 2$
first multiply $8 \cdot 3$$24 \cdot 2$
then multiply $24 \cdot 2$$48$

1.4.4 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 the table below.

OperationWord PhraseExampleExpression
Multiplicationtimes
product
twice
$3$ times $8$
the product of $3$ and $8$
twice $4$
$3 \times 8$, $3 \cdot 8$, $(3)(8)$,
$(3)8$, or $3(8)$
$2 \cdot 4$

Example 12

Translate and simplify: the product of $12$ and $27$.

Solution

The word product tells us to multiply. The words of $12$ and $27$ tell us the two factors.

the product of $12$ and $27$
Translate.$12 \cdot 27$
Multiply.$324$

Example 13

Translate and simplify: twice two hundred eleven.

Solution

The word twice tells us to multiply by 2.

twice two hundred eleven
Translate.$2(211)$
Multiply.$422$

1.4.5 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 14

Humberto bought $4$ sheets of stamps. Each sheet had $20$ stamps. How many stamps did Humberto buy?

Solution

We are asked to find the total number of stamps.

Write a phrase for the total.the product of $4$ and $20$
Translate to math notation.$4 \cdot 20$
Multiply..
Write a sentence to answer the question.Humberto bought $80$ stamps.

Example 15

When Rena cooks rice, she uses twice as much water as rice. How much water does she need to cook $4$ cups of rice?

Solution

We are asked to find how much water Rena needs.

Write as a phrase.twice as much as $4$ cups
Translate to math notation.$2 \cdot 4$
Multiply to simplify.$8$
Write a sentence to answer the question.Rena needs $8$ cups of water for $4$ cups of rice.

Example 16

Van is planning to build a patio. He will have $8$ rows of tiles, with $14$ tiles in each row. How many tiles does he need for the patio?

Solution

We are asked to find the total number of tiles.

Write a phrase.the product of $8$ and $14$
Translate to math notation.$8 \cdot 14$
Multiply to simplify.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-11.22.37-AM.png
Write a sentence to answer the question.Van needs $112$ tiles for his patio.

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.

An image of two squares, one larger than the other. The smaller square is 1 centimeter by 1 centimeter and has the label “1 square centimeter”. The larger square is 1 inch by 1 inch and has the label “1 square inch”.

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.

An image of a rectangle containing 6 blocks, 2 feet tall and 3 feet wide. This image has the label “2 times 3 = 6 feet squared”.
Figure 1.12– The areas of a rectangle is the product of its length and its width, or $6$ square feet.

Example 17

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?

Solution

We are asked to find the area of the kitchen ceiling.

Write a phrase for the area.the product of $9$ and $12$
Translate to math notation.$9 \cdot 12$
Multiply.This image has an empty alt attribute; its file name is Screen-Shot-2021-04-12-at-11.28.46-AM.png
Answer with a sentence.The area of Jen’s kitchen ceiling is $108$ square feet.
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