Introduction

We can use the standard algorithm to multiply two-digit numbers.

To illustrate, let's use the standard algorithm to compute 12\times 13.

Recall that {\color{blue}1}{\color{red}3} in expanded form is {\color{blue}10} + {\color{red}3}. The idea is to first multiply 12 by {\color{red}3}, then by {\color{blue}10}, and add the results.

We write one number above the other as follows:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \\ & & & \!\!\!\!1\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& \phantom{36 + 120 = {\color{red}156}} \end{align*}

First, we multiply the top number ( 12 ) by the value in the ones place of the second number ( \bbox[2px, lightgray]{3} ), writing the result under the line:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \\ & & & \!\!\!\!1\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{3}\!\!\!\! \\ \hline & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 12 \times \bbox[2px, lightgray]{3} = {\color{red}36}\phantom{000} \end{array} \end{align*}

Next, we multiply 12 by \bbox[2px, lightgray]{1}0. Let's do this in steps:

Step 1: Any number multiplied by 10 always gives a zero at the end. Therefore, we place a zero in the ones column underneath our first result.

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \\ & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{1} \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \phantom{12 \times \bbox[2px, lightgray]{1}0 = {\color{red}120}\phantom{0}} \end{array} \end{align*}

Step 2: Now, we work out \bbox[2px, lightgray]{2}\times \bbox[2px, lightgray]{1}. We place the answer next to our zero:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \\ & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{1} \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \bbox[2px, lightgray]{2} \times \bbox[2px, lightgray]{1} = {\color{red}2}\phantom{0} \end{array} \end{align*}

Step 3: Next, we we work out \bbox[2px, lightgray]{1}\times \bbox[2px, lightgray]{1} :

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \\ & & & \!\!\!\! \bbox[2px, lightgray]{1} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{1} \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \bbox[2px, lightgray]{1} \times \bbox[2px, lightgray]{1} = {\color{red}1}\phantom{0} \end{array} \end{align*}

Finally, we add the results:

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \\ & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!3\!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!6\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 36 + 120 = {\color{red} 156} \end{array} \\[5pt] & \end{align*}

Therefore, 12 \times 13 = 156.

FLAG
Example: Multiplying a Two-Digit Number by a Two-Digit Number

What is 23 multiplied by 22?

EXPLANATION

We write it this way:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \\ & & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& \phantom{46 + 460 = {\color{red}506}} \end{align*}

First, we multiply the ones:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \\ & & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{2}\!\!\!\! \\ \hline & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 23 \times \bbox[2px, lightgray]{2} = {\color{red}46}\phantom{000} \end{array} \end{align*}

Next, we multiply the tens:

\begin{align*} \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & & \\ & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!6\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 23 \times \bbox[2px, lightgray]{2}0 = {\color{red}460}\phantom{0} \end{array} \end{align*}

Finally, we add the results:

\begin{align*} \require{cancel} %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \\ & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 3 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!6\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!6\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 46 + 460 = {\color{red} 506} \end{array} \\[5pt] & \end{align*}

Therefore, 23 \times 22 = 506.

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

$31\times 21=$

a
 $641$
b
 $651$
c
 $631$
d
 $661$
e
 $671$
Practice: Multiplying a Two-Digit Number by a Two-Digit Number

$13\times 22=$

a
 $288$
b
 $286$
c
 $274$
d
 $276$
e
 $282$
Multiplying Two-Digit Numbers by Two-Digit Numbers with Carries

Just like when multiplying a number by a one-digit number, we will sometimes have carries. Let's see how to deal with this by multiplying 24 by 32.

We write it this way:

\begin{align*} \quad\quad %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \\ & & & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& \phantom{48 + 720 = {\color{red}768}} \end{align*}

First, we multiply the ones:

\begin{align*} \quad\quad & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & \!\!\!\! \!\! \color{lightgray} \substack{ \phantom{0} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{2}\!\!\!\! \\ \hline & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 24 \times \bbox[2px, lightgray]{2} = {\color{red}48}\phantom{000} \end{array} \end{align*}

On this occasion, there were no carries when multiplying the ones, and so the carry box \color{lightgray} \fbox{[math]\color{blue}\phantom{0}[/math]} remains empty.

Next, we multiply the tens. Let's do this in steps.

Step 1: We start by writing a zero:

\begin{align*} \quad\quad \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & \!\!\!\! \!\! \color{lightgray} \substack{ \phantom{0} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{3} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \phantom{24 \times \bbox[2px, lightgray]{3}0 = {\color{red}720}\phantom{0}} \end{array} \end{align*}

Step 2: We now work out \bbox[2px, lightgray]{4}\times \bbox[2px, lightgray]{3} = {\color{blue}1}{\color{red}2}. We carry the {\color{blue}1} and write down \color{red}2 :

\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]} } \!\!\!\! & \\ & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! \bbox[2px, lightgray]{4} \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{3} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \bbox[2px, lightgray]{4} \times \bbox[2px, lightgray]{3} = 12\\ \textrm{Carry:}\, {\color{blue}{1}}\\ \textrm{Write:}\, {\color{red}{2}}\\ \end{array} \end{align*}

Step 3: We multiply \bbox[2px, lightgray]{2}\times \bbox[2px, lightgray]{3} = 6, and we add the carry from the previous multiplication to make 6+{\color{blue}{1}} = {\color{red}{7}}.

\begin{align*} \quad\quad\quad\quad\quad\quad\quad \require{cancel} & %%%%%%%%%% %%% Step 2 %%% %%%%%%%%%% \begin{array}{ccccc} & & & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & \!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{3} \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} \bbox[2px, lightgray]{2} \times \bbox[2px, lightgray]{3} = 6\quad \textrm{and}\quad 6+{\color{blue}{1}} = {\color{red}{7}}\\ \textrm{Carry:}\, -\\ \textrm{Write:}\, {\color{red}{7}}\\ \end{array} \end{align*}

Finally, we add the results:

\begin{align*} \require{cancel} \quad %%%%%%%%%% %%% Step A %%% %%%%%%%%%% & \begin{array}{ccccc} & & & \!\!\!\! \!\!\color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}\phantom{0}[/math]} } \!\!\!\! & \\ & & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 4 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!8\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}7\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}8\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 48 + 720 = {\color{red} 768} \end{array} \\[5pt] & \end{align*}

Therefore, 24 \times 32 = 768.

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Example: Multiplying a Two-Digit Number by a Two-Digit Number With Carried Digits

Calculate 47 \times 35.

EXPLANATION

As seen before, we start by writing one number on top of the other:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \\ & & & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& \phantom{00000000000000000} \end{align*}

First, we multiply the ones:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & \!\!\!\!\!\! \color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\ & & & \!\!\!\!4\!\!\!\! & \!\!\!\!7\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{5}\!\!\!\! \\ \hline & & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 47 \times \bbox[2px, lightgray]{5} = {\color{red}235}\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}2[/math]} \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\ & & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{3} \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\ \hline & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\ & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 47 \times \bbox[2px, lightgray]{3}0 = {\color{red}1,410}\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}2[/math]} \\[2pt] \fbox{[math]\color{blue}3[/math]} } \!\!\!\! & \\ & & & \!\!\!\! 4 \!\!\!\! & \!\!\!\! 7 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\ \hline & & \!\!\!\!2\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 235 + 1410 = {\color{red}1 , 645} \end{array} \\[5pt] & \end{align*}

Therefore, 47 \times 35 = 1 , 645.

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Practice: Multiplying a Two-Digit Number by a Two-Digit Number With Carried Digits

What is $43$ multiplied by $49?$

a
 $3,007$
b
 $2,107$
c
 $2,897$
d
 $1,997$
e
 $2,477$
Practice: Multiplying a Two-Digit Number by a Two-Digit Number With Carried Digits

Calculate $23 \times 27.$

a
 $501$
b
 $590$
c
 $621$
d
 $451$
e
 $730$
Example: Multiplying a Two-Digit Number by a Two-Digit Number: Word Problems

Jessica runs 12 kilometers every morning. How many kilometers does Jessica run in 28 days?

EXPLANATION

To calculate the total number of kilometers that Jessica runs in 28 days, we need to multiply 12 by 28.

So, we write it this way:

\begin{align*} %%%%%%%%%% %%% Step 0 %%% %%%%%%%%%% & \begin{array}{ccccc} & & & & \\ & & & \!\!\!\!1\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!8\!\!\!\! \\ \hline & \end{array} &\qquad\qquad& \phantom{12 \times \bbox[2px, lightgray]{2}0 = {\color{red}240}} \end{align*}

First, we multiply the ones:

\begin{align*} & %%%%%%%%%% %%% Step 1 %%% %%%%%%%%%% \begin{array}{ccccc} & & & \!\!\!\! \!\!\color{lightgray} \substack{ \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \\ & & & \!\!\!\!1\!\!\!\! & \!\!\!\!2\!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!\!\bbox[2px, lightgray]{8}\!\!\!\! \\ \hline & & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}9\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 12 \times \bbox[2px, lightgray]{8} = {\color{red}96} \phantom{000} \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}1[/math]} } \!\!\!\! & \\ & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\! \bbox[2px, lightgray]{2} \!\!\!\! & \!\!\!\! 8 \!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!6\!\!\!\! \\ & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}4\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 12 \times \bbox[2px, lightgray]{2}0 = {\color{red}240} \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}\phantom{0}[/math]} \\[2pt] \fbox{[math]\color{blue}1[/math]} } \!\!\!\! & \\ & & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 2 \!\!\!\! \\ \!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!8\!\!\!\! \\ \hline & & \!\!\!\!\!\!\!\! & \!\!\!\!9\!\!\!\! & \!\!\!\!6\!\!\!\! \\ \!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!4\!\!\!\! & \!\!\!\!0\!\!\!\! \\ \hline & \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! \\ \end{array} &\qquad\qquad& %%%% Explanations %%% \begin{array}{l} 96 + 240 = {\color{red} 336} \end{array} \\[5pt] & \end{align*}

So, 12 \times 28 = 336.

Therefore, Jessica runs a total of 336 kilometers in 28 days.

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

There are $16$ teams participating in an international soccer competition. If there are $22$ players on each team, how many players are participating in the competition in total?

a
 $320$
b
 $280$
c
 $352$
d
 $510$
e
 $420$
Practice: Multiplying a Two-Digit Number by a Two-Digit Number: Word Problems

In a particular section of a certain football stadium, there are $14$ rows with $25$ seats each. How many seats are there in the section in total?

a
 $310$
b
 $240$
c
 $430$
d
 $350$
e
 $280$
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