We can use the standard algorithm when we want to multiply a three-digit number by a two-digit number.
As an example, let's find
706 \times 25.
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & \\
& & & \!\!\!\!7\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\hline
&
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
\phantom{3,530 + 14,120 = {\color{red}17 , 650}}
\end{array}
\\[5pt]
&
\end{align*}
First, we multiply the top number (
706
\bbox[2px, lightgray]{5}
\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}3[/math]} } \!\!\!\! & \\
& & & \!\!\!\!7\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{5}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
706 \times \bbox[2px, lightgray]{5} = {\color{red}3,530}\phantom{000000,}
\end{array}
\end{align*}
Next, we multiply top number (
706
\bbox[2px, lightgray]{2}0.
Step 1 : We add a zero underneath the result of the first calculation, in the ones place:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 0 \!\!\!\! & \!\!\!\! 6 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\
\hline
& & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!0\!\!\!\! \\
& \!\!\!\!\color{red}\phantom{1}\!\!\!\! & \!\!\!\!\color{red}\phantom{4}\!\!\!\! & \!\!\!\!\color{red}\phantom{1}\!\!\!\! & \!\!\!\!\color{red}\phantom{2}\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
\phantom{706 \times \bbox[2px, lightgray]{2}0 = {\color{red}14,120}\phantom{0000,}}
\end{array}
\end{align*}
Step 2 : We continue to work out
706\times \bbox[2px, lightgray]{2}0,
\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}3[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 0 \!\!\!\! & \!\!\!\! 6 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\
\hline
& & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!0\!\!\!\! \\
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
706 \times \bbox[2px, lightgray]{2}0 = {\color{red}14,120}\phantom{0000,}
\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]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 0 \!\!\!\! & \!\!\!\! 6 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\hline
& & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
3,530 + 14,120 = {\color{red}17 , 650}
\end{array}
\\[5pt]
&
\end{align*}
Therefore,
706 \times 25 = 17 , 650.
What is
268
54?
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!8\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\hline
&
\end{array}
&\qquad\qquad&\;
\phantom{1,072 + 13,400 = {\color{red}14 , 472}}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \!\!\color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!8\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{4}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
268 \times \bbox[2px, lightgray]{4} = {\color{red}1,072\phantom{0000000}}
\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}3[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}4[/math]} \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 8 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{5} \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!2\!\!\!\! \\
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
268 \times \bbox[2px, lightgray]{5}0 = {\color{red}13,400\phantom{0000,}}
\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}3[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}4[/math]} \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 8 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
1,072 + 13,400 = {\color{red}14 , 472}
\end{array}
\\[5pt]
&
\end{align*}
Therefore,
268 \times 54 = 14 , 472.
What is $187$ multiplied by $42?$
a
$9,814$
b
$4,544$
c
$8,274$
d
$9,204$
e
$7,854$
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & \\
& & & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!7\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\hline
& & & & & &
\end{array}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!7\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{2}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
187 \times \bbox[2px, lightgray]{2} = {\color{red}374}
\end{array}
\end{align*}
Next, we multiply the tens:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}3$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
187 \times \bbox[2px, lightgray]{4}0 = {\color{red}7,480}
\end{array}
\end{align*}
Finally, we add the results:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}3$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
374 + 7,480 = {\color{red}7 , 854}
\end{array}
\\[5pt]
&
\end{align*}
Therefore, $187 \times 42 = 7 , 854.$
From left to right, what are the missing digits in the following multiplication problem?
\[
\begin{array}{ccccc}
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!3\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!\fbox{$\phantom{0}$}\!\!\!\! & \!\!\!\!8\!\!\!\! \\
\end{array}
\]
a
$4$ and $2$
b
$8$ and $4$
c
$8$ and $0$
d
$9$ and $0$
e
$3$ and $0$
We already have the ones and tens multiplied:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! \\
& \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\end{array}
\end{align*}
So, we add the two final rows:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}\fbox{$8$}\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}\fbox{$0$}\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
1,848 + 36,960 = {\color{red}38 , 808}
\end{array}
\\[5pt]
&
\end{align*}
Therefore, the missing digits are $8$ and $0.$
There is no limit to the size of the numbers that we can multiply using the standard algorithm.
For example, to calculate
6,132 \times 54
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & \\
& & & \!\!\!\!6\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\hline
&
\end{array}
&\qquad\qquad&\;
\phantom{4,859 + 97,180 = {\color{red}102 , 039}}
\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]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\
& & & \!\!\!\!6\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{4}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
6,132 \times \bbox[2px, lightgray]{4} = {\color{red}24,528\!\phantom{0000000}}
\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}1[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{5} \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\hline
& & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!8\!\!\!\! \\
& \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
6,132 \times \bbox[2px, lightgray]{5}0 = {\color{red}306,600\phantom{0\!0000}}
\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]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\! \!\!\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}\phantom{0}[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{5} \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\hline
& & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!8\!\!\!\! \\
& \!\!\!\!3\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
24,528 + 306,600 = {\color{red}331 , 128}
\end{array}
\\[5pt]
&
\end{align*}
Therefore,
6,132 \times 54 = 331 , 128.
Find the value of
2,324\times 46.
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\hline
& & & & & &
\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}1[/math]} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{6}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
2,324 \times \bbox[2px, lightgray]{6} = {\color{red}13,944}
\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}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 6 \!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!4\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
2,324 \times \bbox[2px, lightgray]{4}0 = {\color{red}92,960}
\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]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
13,944 + 92,960 = {\color{red}106 , 904}
\end{array}
\\[5pt]
&
\end{align*}
Therefore,
2,324 \times 46 = 106 , 904.
What is $1,533$ multiplied by $53?$
a
$80,139$
b
$81,249$
c
$79,689$
d
$77,249$
e
$83,359$
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & \\
& & & \!\!\!\!1\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\hline
& & & & & &
\end{array}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\!1\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{3}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
1,533 \times \bbox[2px, lightgray]{3} = {\color{red}4,599}
\end{array}
\end{align*}
Next, we multiply the tens:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{5} \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!9\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
1,533 \times \bbox[2px, lightgray]{5}0 = {\color{red}76,650}
\end{array}
\end{align*}
Finally, we add the results:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!9\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
4,599 + 76,650 = {\color{red}81 , 249}
\end{array}
\\[5pt]
&
\end{align*}
Therefore, $1,533 \times 53 = 81 , 249.$
a
$87 , 912$
b
$82 , 719$
c
$89 , 127$
d
$89,272$
e
$89,712$
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\hline
& & & & & &
\end{array}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{2}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
2,136 \times \bbox[2px, lightgray]{2} = {\color{red}4,272}
\end{array}
\end{align*}
Next, we multiply the tens:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 6 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!2\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
2,136 \times \bbox[2px, lightgray]{4}0 = {\color{red}85,440}
\end{array}
\end{align*}
Finally, we add the results:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 6 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
4,272 + 85,440 = {\color{red}89 , 712}
\end{array}
\\[5pt]
&
\end{align*}
Therefore, $2,136 \times 42 = 89 , 712.$
A hospital orders
21
4,859
To calculate the total number of syringes the hospital ordered, we need to multiply
4,859
21.
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & \\
& & & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!9\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! \\
\hline
&
\end{array}
&\qquad\qquad&\;
\phantom{4,859 + 97,180 = {\color{red}102 , 039}}
\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]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\
& & & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!9\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{1}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
4,859 \times \bbox[2px, lightgray]{1} = {\color{red}4,859\!\phantom{0000000}}
\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}\phantom{0}[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}1[/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]} } \!\!\!\! & \\
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 9 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!9\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
4,859 \times \bbox[2px, lightgray]{2}0 = {\color{red}97,180\phantom{0\!0000}}
\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]} \\[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]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 9 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!1\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!9\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
4,859 + 97,180 = {\color{red}102 , 039}
\end{array}
\\[5pt]
&
\end{align*}
So,
4,859 \times 21 = 102 , 039.
Therefore, the hospital ordered
102,039
A soccer club offers all $8,254$ fans attending a game a free portion of fries. Each portion contains $31$ individual fries. How many individual fries must the club prepare?
a
$255,874$
b
$236,844$
c
$287,224$
d
$226,394$
e
$289,854$
To calculate the total number of individual fries, we need to multiply $8,254$ by $31.$
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & \\
& & & \!\!\!\!8\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!1\!\!\!\! \\
\hline
& & & & & &
\end{array}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\!8\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{1}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
8,254 \times \bbox[2px, lightgray]{1} = {\color{red}8,254}
\end{array}
\end{align*}
Next, we multiply the tens:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{3} \!\!\!\! & \!\!\!\! 1 \!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! \\
& \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
8,254 \times \bbox[2px, lightgray]{3}0 = {\color{red}247,620}
\end{array}
\end{align*}
Finally, we add the results:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!1\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!4\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
8,254 + 247,620 = {\color{red}255 , 874}
\end{array}
\\[5pt]
&
\end{align*}
So, $8,254 \times 31 = 255 , 874.$
Therefore, the soccer club must prepare $255,874$ individual fries.
A cookie factory produces $2,493$ batches of cookies in one day. There are $36$ cookies in each batch. How many cookies does the factory produce in a day?
a
$85,248$
b
$98,478$
c
$93,418$
d
$91,468$
e
$89,748$
To calculate the total number of cookies produced, we need to multiply $2,493$ by $36.$
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\hline
& & & & & &
\end{array}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}5$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{6}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
2,493 \times \bbox[2px, lightgray]{6} = {\color{red}14,958}
\end{array}
\end{align*}
Next, we multiply the tens:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}5$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 9 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{3} \!\!\!\! & \!\!\!\! 6 \!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!8\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
2,493 \times \bbox[2px, lightgray]{3}0 = {\color{red}74,790}
\end{array}
\end{align*}
Finally, we add the results:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}5$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 9 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!8\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
14,958 + 74,790 = {\color{red}89 , 748}
\end{array}
\\[5pt]
&
\end{align*}
So, $2,493 \times 36 = 89 , 748.$
Therefore, the factory produces a total of $89,748$ cookies each day.
From left to right, what are the missing digits in the following multiplication problem?
\begin{array}{ccccc}
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 9 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! \\
\hline
& \!\!\!\!2\!\!\!\! & \!\!\!\!\!\fbox{[math]\phantom{i}[/math]}\!\!\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!\!\fbox{[math]\phantom{i}[/math]}\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!\!\fbox{[math]\phantom{i}[/math]}\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\end{array}
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & & \\
& & & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!9\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! \\
\hline
& & & & & & &
\end{array}
&\qquad\qquad& \,\phantom{42,689 \times \bbox[2px, lightgray]{4}0 = {\color{red}1,707,560}}
\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}4[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}6[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}6[/math]} } \!\!\!\! & \\
& & & \!\!\!\!4\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!9\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{7}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
42,689 \times \bbox[2px, lightgray]{7} = {\color{red}298,823}\phantom{00,}
\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}2[/math]} \\[2pt] \fbox{[math]\color{blue}4[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}3[/math]} \\[2pt] \fbox{[math]\color{blue}6[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}3[/math]} \\[2pt] \fbox{[math]\color{blue}6[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 9 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\
\hline
& & \!\!\!\!2\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! \\
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
42,689 \times \bbox[2px, lightgray]{4}0 = {\color{red}1,707,560}
\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]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}2[/math]} \\[2pt] \fbox{[math]\color{blue}4[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}3[/math]} \\[2pt] \fbox{[math]\color{blue}6[/math]} } \!\!\!\! & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}3[/math]} \\[2pt] \fbox{[math]\color{blue}6[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 6 \!\!\!\! & \!\!\!\! 8 \!\!\!\! & \!\!\!\! 9 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! \\
\hline
& & \!\!\!\!2\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\!\!\color{red}\fbox{[math]0[/math]}\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\!\!\color{red}\fbox{[math]6[/math]}\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\!\!\color{red}\fbox{[math]8[/math]}\!\!\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
298,823 + 1,707,560 = \phantom{0,,} \\[2pt]
\qquad={\color{red}2 , 006 , 383}
\end{array}
\\[5pt]
&
\end{align*}
Therefore, the missing digits are
0,6
8.
A museum's admittance fee is $\$43.$ In January, the museum had $23,772$ visitors. How many dollars did the museum make in admittance fees in January in total?
a
$\$1,022,196$
b
$\$1,028,726$
c
$\$1,035,086$
d
$\$1,129,116$
e
$\$1,122,286$
To calculate the total amount of dollars in admittance fees the museum made, we need to multiply $23,772$ by $43.$
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\hline
& & & & & & &
\end{array}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{3}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
23,772 \times \bbox[2px, lightgray]{3} = {\color{red}71,316}
\end{array}
\end{align*}
Next, we multiply the tens:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}3$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!6\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
23,772 \times \bbox[2px, lightgray]{4}0 = {\color{red}950,880}
\end{array}
\end{align*}
Finally, we add the results:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}3$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
71,316 + 950,880 = {\color{red}1 , 022 , 196}
\end{array}
\\[5pt]
&
\end{align*}
So, $23,772 \times 43 = 1 , 022 , 196.$
Therefore, the museum made $\$1,022,196$ in admittance fees in January.
a
$412,896$
b
$341,126$
c
$520,666$
d
$394,966$
e
$251,426$
We write it this way:
\begin{align*}
%%%%%%%%%%
%%% Step 0 %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & & & & & \\
& & & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\hline
& & & & & & &
\end{array}
\end{align*}
First, we multiply the ones:
\begin{align*}
&
%%%%%%%%%%
%%% Step 1 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!2\!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!\bbox[2px, lightgray]{3}\!\!\!\! \\
\hline
& & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\!
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
17,952 \times \bbox[2px, lightgray]{3} = {\color{red}53,856}
\end{array}
\end{align*}
Next, we multiply the tens:
\begin{align*}
\require{cancel}
&
%%%%%%%%%%
%%% Step 2 %%%
%%%%%%%%%%
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 9 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!6\!\!\!\! \\
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
17,952 \times \bbox[2px, lightgray]{2}0 = {\color{red}359,040}
\end{array}
\end{align*}
Finally, we add the results:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}1$} } \!\!\!\! & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}\phantom{0}$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 7 \!\!\!\! & \!\!\!\! 9 \!\!\!\! & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!8\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!6\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \\
\end{array}
&\qquad\qquad&
%%%% Explanations %%%
\begin{array}{l}
53,856 + 359,040 = {\color{red} 412 , 896}
\end{array}
\\[5pt]
&
\end{align*}
Therefore, $17,952 \times 23 = 412 , 896.$
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