Introduction

The standard algorithm for multiplication works in the same way when multiplying three-digit numbers by three-digit numbers.

To demonstrate, let's compute 237 \times 142. We start by writing it in the usual way:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & & & \\ & & & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!7\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \phantom{474 + 9,480 + 23 , 700 =} \end{array} \end{align*}

First, we multiply the ones:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\! & \\ & & & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!7\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{2}\!\!\! \\ \hline & & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 237 \times \bbox[2px, lightgray]{2} = {\color{red}474} \phantom{0000000,} \end{array} \end{align*}

Next, we multiply the tens. Remember that when multiplying the tens, the result will always end in a zero.

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}2[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! \\ & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 237 \times \bbox[2px, lightgray]{4}0 = {\color{red}9,480} \phantom{00000} \end{array} \end{align*}

Next, we multiply the hundreds. Since a hundred ( 100 ) has two zeros at the end, the result will end in two zeros! So, we write these first:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 3 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}2[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{1} \!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! \\ & & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}\phantom{2}\!\!\!\! & \!\!\!\!\color{red}\phantom{3}\!\!\!\! & \!\!\!\!\color{red}\phantom{7}\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \phantom{237 \times \bbox[2px, lightgray]{1}00 = {\color{red}23 , 700} \phantom{000}} \end{array} \end{align*}

Then, we proceed to multiply the rest of the hundreds as usual.

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 3 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}2[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{1} \!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! \\ & & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 237 \times \bbox[2px, lightgray]{1}00 = {\color{red}23 , 700} \phantom{000} \end{array} \end{align*}

Finally, we add the results:

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}2[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 474 + 9,480 + 23 , 700 = \\[2pt] \qquad = {\color{red}33 , 654} \end{array} \\[5pt] & \end{align*}

Therefore, 237 \times 142 = 33 , 654.

FLAG
Example: Three-Digit by Three-Digit Multiplication: Find the Missing Digits

What are the missing digits in the following multiplication problem?

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!\!\!\fbox{[math]\phantom{i}[/math]}\!\!\!\! & \!\!\!\!\!\fbox{[math]\phantom{i}[/math]}\!\!\!\! & \!\!\!\!\fbox{[math]\phantom{i}[/math]}\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \end{array} & \end{align*}

EXPLANATION

We write it this way:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & & & \\ & & & & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!1\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \phantom{180 + 1,800 + 90,000 =\!\!} \end{array} \end{align*}

First, we multiply the ones:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{1}\!\!\!\! \\ \hline & & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 180 \times \bbox[2px, lightgray]{1} = {\color{red}180} \phantom{00000,,} \end{array} \end{align*}

Next, we multiply the tens:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{1} \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 180 \times \bbox[2px, lightgray]{1}0 = {\color{red}1,800} \phantom{000,,} \end{array} \end{align*}

Next, we multiply the hundreds:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 3 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}4[/math]} \\ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{5} \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 180 \times \bbox[2px, lightgray]{5}00 = {\color{red}90 , 000} \phantom{0,,} \end{array} \end{align*}

Finally, we add the results:

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}4[/math]} \\ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 0 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\!\!\!\color{red}\fbox{1}\!\!\!\! & \!\!\!\!\!\!\color{red}\fbox{9}\!\!\!\! & \!\!\!\!\!\color{red}\fbox{8}\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 180 + 1,800 + 90 , 000 = \\[2pt] \qquad = {\color{red}91 , 980} \end{array} \\[5pt] & \end{align*}

Therefore, the missing digits are 1 , 9 and 8.

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Practice: Three-Digit by Three-Digit Multiplication: Find the Missing Digits

From left to right, what are the missing digits in the following multiplication problem?

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 0 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\ \hline & & & \!\!\!\!1\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!1\!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\!\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!1\!\!\!\! \\ \end{array} & \end{align*}

a
 $2$, $3$ and $5$
b
 $3$, $5$ and $5$
c
 $2$, $4$ and $5$
d
 $3$, $4$ and $5$
e
 $4$, $4$ and $5$
Practice: Three-Digit by Three-Digit Multiplication: Find the Missing Digits

From left to right, what are the missing digits in the following multiplication problem?

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \!\!\!\! 9 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\ \hline & & & \!\!\!\!2\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!3\!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!1\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!3\!\!\!\! \\ \end{array} & \end{align*}

a
 $6$, $2$ and $7$
b
 $6$, $0$ and $7$
c
 $5$, $1$ and $7$
d
 $5$, $2$ and $7$
e
 $6$, $1$ and $7$
Example: Three-Digit by Three-Digit Multiplication

Find the value of 435 \times 435.

EXPLANATION

We write it this way:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & & & \\ & & & & \!\!\!\!4\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \phantom{2,175 + 13,050 + 174,000 =} \end{array} \end{align*}

First, we multiply the ones:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\ & & & & \!\!\!\!4\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{5}\!\!\!\! \\ \hline & & & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 435 \times \bbox[2px, lightgray]{5} = {\color{red}2,175} \phantom{000000000,} \end{array} \end{align*}

Next, we multiply the tens:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{3} \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\ \hline & & & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! \\ & & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 435 \times \bbox[2px, lightgray]{3}0 = {\color{red}13,050} \phantom{00000000,} \end{array} \end{align*}

Next, we multiply the hundreds:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 3 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}2[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\ \hline & & & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! \\ & & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 435 \times \bbox[2px, lightgray]{4}00 = {\color{red}174 , 000} \phantom{000000,} \end{array} \end{align*}

Finally, we add the results:

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}2[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\ \hline & & & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 2,175 + 13,050 + 174 , 000 = \\[2pt] \qquad = {\color{red}189 , 225} \end{array} \\[5pt] & \end{align*}

Therefore, 435 \times 435 = 189 , 225.

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Practice: Three-Digit by Three-Digit Multiplication

Find the value of $246 \times 332.$

a
 $71,872$
b
 $76,937$
c
 $94,659$
d
 $81, 672$
e
 $83,972$
Practice: Three-Digit by Three-Digit Multiplication

Calculate $127 \times 325 .$

a
 $34, 876$
b
 $51, 285$
c
 $42, 455$
d
 $41, 275$
e
 $23, 564$
Example: Three-Digit by Three-Digit Multiplication: Word Problem

Richard has 504 boxes stored in a warehouse. Each box contains 240 plates. How many plates in total are there in the warehouse?

EXPLANATION

To find the total number of plates stored in the warehouse, we need to calculate 504 \times 240.

We write it this way:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & & & \\ & & & & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!4\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& \phantom{0 + 20,160 + 100 , 800 = } \end{align*}

First, we multiply the ones:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!4\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{0}\!\!\!\! \\ \hline & & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 504 \times \bbox[2px, lightgray]{0} = {\color{red}000\phantom{00000000}} \end{array} \end{align*}

Next, we multiply the tens:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 0 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 0 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 504 \times \bbox[2px, lightgray]{4}0 = {\color{red}20,160\phantom{0000}} \end{array} \end{align*}

Next, we multiply the hundreds:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 3 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 0 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 0 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & & \!\!\!\!2\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\ & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 504 \times \bbox[2px, lightgray]{2}00 = {\color{red}100 , 800\phantom{00}} \end{array} \end{align*}

Finally, we add the results:

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 0 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 0 \!\!\!\! \\ \hline & & & \!\!\!\!\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & & \!\!\!\!2\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 0 + 20,160 + 100 , 800 = \\[2pt] \qquad = {\color{red}120 , 960} \end{array} \\[5pt] & \end{align*}

Hence, 504 \times 240 = 120 , 960.

Therefore, Richard has 120,960 plates in the warehouse.

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Practice: Three-Digit by Three-Digit Multiplication: Word Problem

Proust Orchard has $891$ apple trees. Each tree grows $295$ apples. How many apples in total are there in the orchard?

a
 $262,785$
b
 $263,805$
c
 $264,185$
d
 $262,845$
e
 $263,195$
Practice: Three-Digit by Three-Digit Multiplication: Word Problem

Tickets for a professional basketball game are selling for $\$146$ per ticket. There are $926$ tickets for sale. How much money will be made if all the tickets are sold?

a
 $\$137,216$
b
 $\$139,426$
c
 $\$133,116$
d
 $\$ 135,196$
e
 $\$132,436$
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