Introduction

Let's use an area model to solve the following multiplication problem:

{43} \times 4

The number 43 in expanded form is {\color{blue}{43}} = {\color{blue}40}+{\color{blue}3}. So, we drawn an area model, place {\color{blue}40} and {\color{blue}3} above the left and right rectangles, and {\color{red}4} to the left of the big rectangle.

Next, we compute the areas of the blue and pink rectangles. This gives the following picture:

The area of the big rectangle is

160+12 = 172.

Therefore, we conclude that

{43} \times 4 = 172.

We'll now describe a similar process that allows us to carry out this multiplication without using models.

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Multiplication Without Models

Let's once again consider the following multiplication problem:

{43} \times {\color{red}{4}}

To evaluate this product without models, we follow a three-step process:

  • First, we write our two-digit number (43) in expanded form: 43 = {\color{blue}{40}} + {\color{blue}{3}}

  • Then, we multiply each of the numbers from the expanded form by {\color{red}{4}}\mathbin{:}

\begin{align*} {\color{blue}{40}} \times {\color{red}{4}} &= 160 \\[5pt] {\color{blue}{3}} \times {\color{red}{4}} &= 12 \end{align*}

  • Finally, we add the results:

\begin{array}{cccccccc} & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 6 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\! \!\!\!\! & \!\!\!\! 1 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \\ \hline & \!\!\!\! 1 \!\!\!\!& \!\!\!\! 7 \!\!\!\!& \!\!\!\! 2 \!\!\!\! \end{array}

So, we conclude that

43 \times 4 = 172.

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Example: Multiplying a Two-Digit Number by a One-Digit Number Given a Decomposition

Kyle wishes to find the value of 8\times 62. He proceeds as follows:

\begin{align*} 8\times 60 &= \fbox{[math]\phantom{000}[/math]}\\[5pt] 8\times 2 &= \fbox{[math]\phantom{00}[/math]}\\[5pt] \end{align*}

Find the missing numbers, and use your answers to find the value of 8\times 62.

EXPLANATION

Notice that 62 in expanded form is
62 = 60+2. So, we can compute 8\times 62 by filling in the missing numbers and adding the results.

First, let's find the missing values:

\begin{align*} 8 \times 60 &= \fbox{[math]\,480\,[/math]} \\[5pt] 8 \times 2 &= \fbox{[math]\,16\,[/math]} \end{align*}

Let's now add these results:

\begin{array}{cccccccc} & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 8 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\! \!\!\!\! & \!\!\!\! 1 \!\!\!\!& \!\!\!\! 6 \!\!\!\! \\ \hline & \!\!\!\! 4 \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 6 \!\!\!\! \end{array}

Therefore,

8 \times 62 = 496.

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Practice: Multiplying a Two-Digit Number by a One-Digit Number Given a Decomposition

Lou wishes to find the value of $76\times 5.$ He proceeds as follows:

\begin{align*} 70\times 5 &= \fbox{$\phantom{000}$}\\[5pt] 6\times 5 &= \fbox{$\phantom{00}$}\\[5pt] \end{align*}

Find the missing numbers, and use your answers to find the value of $76\times 5.$

a
 $360$
b
 $350$
c
 $390$
d
 $380$
e
 $370$
Practice: Multiplying a Two-Digit Number by a One-Digit Number Given a Decomposition

Seb wishes to find the value of $8\times 78.$ He proceeds as follows:

\begin{align*} 8\times 70 &= \fbox{$\phantom{000}$}\\[5pt] 8\times 8 &= \fbox{$\phantom{00}$}\\[5pt] \end{align*}

Find the missing numbers, and use your answers to find the value of $8\times 78.$

a
 $634$
b
 $614$
c
 $624$
d
 $654$
e
 $644$
Example: Multiplying a Two-Digit Number by a One-Digit Number

What is the value of 59 \times 5?

EXPLANATION

First, let's write the two-digit number (59) in expanded form:

59 = {\color{blue}{50}} + {\color{blue}{9}}

We now multiply both numbers in our expanded form by {\color{red}{5}}\mathbin{:}

\begin{align*} {\color{blue}{50}} \times {\color{red}{5}} &= 250 \\[5pt] {\color{blue}{9}} \times {\color{red}{5}} &= 45 \end{align*}

Let's now add these results:

\begin{array}{cccccccc} & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 5 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\! \!\!\!\! & \!\!\!\! 4 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \\ \hline & \!\!\!\! 2 \!\!\!\!& \!\!\!\! 9 \!\!\!\!& \!\!\!\! 5 \!\!\!\! \end{array}

Therefore,

59 \times 5 = 295.

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Practice: Multiplying a Two-Digit Number by a One-Digit Number

$64 \times 6 =$

a
 $314$
b
 $384$
c
 $348$
d
 $388$
e
 $344$
Practice: Multiplying a Two-Digit Number by a One-Digit Number

$8 \times 19 =$

a
 $172$
b
 $72$
c
 $149$
d
 $152$
e
 $160$
Example: Multiplying a Two-Digit Number by a One-Digit Number: Word Problems

A hospital has 5 floors. On each floor, there are 96 beds. How many beds does the hospital have in total?

EXPLANATION

To determine the total number of beds in the hospital, we need to compute the value of

{\color{red}{5}} \times 96.

First, let's write the two-digit number (96) in expanded form:

96 = {\color{blue}{90}} + {\color{blue}{6}}

We now multiply both numbers in our expanded form by {\color{red}{5}}\mathbin{:}

\begin{align*} {\color{red}{5}} \times {\color{blue}{90}} &= 450 \\[5pt] {\color{red}{5}} \times {\color{blue}{6}} &= 30 \end{align*}

Let's now add these results:

\begin{array}{cccccccc} & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 5 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\! \!\!\!\! & \!\!\!\! 3 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \\ \hline & \!\!\!\! 4 \!\!\!\!& \!\!\!\! 8 \!\!\!\!& \!\!\!\! 0 \!\!\!\! \end{array}

Therefore, there are 480 beds in the hospital.

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Practice: Multiplying a Two-Digit Number by a One-Digit Number: Word Problems

There are $24$ boxes of pencils in a container. If each box has $8$ pencils, how many pencils are in the container in total?

a
 $160$ pencils
b
 $188$ pencils
c
 $192$ pencils
d
 $164$ pencils
e
 $194$ pencils
Practice: Multiplying a Two-Digit Number by a One-Digit Number: Word Problems

Each copy of a recently published book on mathematics costs $\$28.$ How much would the school library have to pay to purchase $8$ copies of the book?

a
 $\$ 114$
b
 $\$142$
c
 $\$ 242$
d
 $\$ 222$
e
 $\$ 224$
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