Introduction

In this lesson, we'll describe a procedure known as borrowing. Borrowing is used often when subtracting large numbers, as we'll soon discover.

Let's consider the place value chart of the number 30\mathbin{:}

tens ones
3 0

Since 3 \, \textrm{tens} = {\color{red}{2}} \, \textrm{tens} + {\color{blue}{1}} \, \textrm{tens}, we can write our place value chart as follows:

tens ones
{\color{red}2}+{\color{blue}1} 0

Now, since 1 ten is the same as 10 ones, we can borrow {\color{blue}1} from tens place and convert it into ones:

tens ones
{\color{red}2}+{\color{blue}1} 0
{\color{blue} 10 \, \textrm{ones} \rightarrow}

Therefore, we have now have \color{red}2 tens and {\color{blue}{10}}\,\textrm{ones} + 0\,\textrm{ones} = {\color{blue}{10}}\,\textrm{ones}.

tens ones
\color{red}2 {\color{blue}10}

Therefore, " 3 tens" is equivalent to " 2 tens and 10 ones."

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Example: Borrowing From Tens

What number should be inserted into the blank space to make the following statement true?

\qquad " 15 tens" is equivalent to " 14 tens and \bbox[2pt, white, border: 1pt solid black]{\phantom{AA}} ones"

EXPLANATION

Let's write " 15 tens" in a place value chart:

tens ones
15 0

Now, since 15 \, \textrm{tens} = {\color{red}{14}} \, \textrm{tens} + {\color{blue}{1}} \, \textrm{ten}, we can borrow \color{blue}1 from tens place and convert it into ones:

tens ones
{\color{red}{14}}+{\color{blue}1} 0
{\color{blue} 10 \, \textrm{ones} \rightarrow}

This gives the following place value chart:

tens ones
\color{red}14 {\color{blue}10}

Therefore, " 15 tens" is equivalent to " 14 tens and \bbox[2pt, white, border: 1pt solid black]{10} ones."

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Practice: Borrowing From Tens

What number should be inserted into the blank space to make the following statement true?

$\qquad$ "$7$ tens" is equivalent to "$6$ tens and $\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}$ ones"

a
 $70$
b
 $1$
c
 $10$
d
 $7$
e
 $9$
Practice: Borrowing From Tens

What number should be inserted into the blank space to make the following statement true?

$\qquad$ "$14$ tens" is equivalent to "$\,\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}$ tens and $10$ ones"

a
 $1$
b
 $11$
c
 $12$
d
 $10$
e
 $13$
Example: Borrowing From Tens With Addition

What number should be inserted into the blank space to make the following statement true?

\qquad " 5 tens and 5 ones" is equivalent to " 4 tens and \bbox[2pt, white, border: 1pt solid black]{\phantom{AA}} ones"

EXPLANATION

Let's write " 5 tens and 5 ones" in a place value chart:

tens ones
5 5

Now, since 5 \, \textrm{tens} = {\color{red}{4}} \, \textrm{tens} + {\color{blue}{1}} \, \textrm{ten}, we can borrow \color{blue}1 from tens place and convert it into ones:

tens ones
{\color{red}{4}}+{\color{blue}1} 5
{\color{blue} 10 \, \textrm{ones} \rightarrow}

This gives the following place value chart:

tens ones
{\color{red}{4}} {\color{blue}15}

Therefore, " 5 tens and 5 ones" is equivalent to " 4 tens and \bbox[2pt, white, border: 1pt solid black]{15} ones"

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Practice: Borrowing From Tens With Addition

What number should be inserted into the blank space to make the following statement true?

$\qquad$ "$3$ tens and $7$ ones" is equivalent to "$2$ tens and $\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}$ ones"

a
 $10$
b
 $7$
c
 $8$
d
 $1$
e
 $17$
Practice: Borrowing From Tens With Addition

What number should be inserted into the blank space to make the following statement true?

$\qquad$ "$7$ tens and $2$ ones" is equivalent to "$6$ tens and $\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}$ ones"

a
 $72$
b
 $12$
c
 $10$
d
 $2$
e
 $62$
Borrowing From Hundreds

We can also borrow from other place values.

For example, let's consider the equation below.

\textrm{[math]2[/math] hundreds} = \textrm{[math]1[/math] hundred} + \textrm{[math]\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}[/math] tens}

In this case, we want to borrow 1 from the hundreds place and convert it to tens.

First, notice that we can write " 2 hundreds" as follows:

\begin{align*} \textrm{[math]2[/math] hundreds} = \textrm{[math]\color{red}1[/math] hundred} + \textrm{[math]\color{blue}1[/math] hundred} \end{align*}

Now, we convert 1 hundred into 10 tens:

\begin{align*} \textrm{[math]\color{red}1[/math] hundred} + {\textrm{[math]\color{blue}1[/math] hundred}} &=\\[5pt] \textrm{[math]\color{red}1[/math] hundred} + {\textrm{[math]\color{blue}10[/math] tens}} &\\[5pt] \end{align*}

Therefore,

\textrm{[math]2[/math] hundreds} = \textrm{[math]1[/math] hundred} + \textrm{[math]\bbox[2pt, white, border: 1pt solid black]{10}[/math] tens} .

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Example: Borrowing From Hundreds With and Without Addition

What is the missing number in the equation below?

\textrm{[math]2[/math] hundreds} + \textrm{[math]1[/math] ten} + \textrm{[math]1[/math] one} = \textrm{[math]1[/math] hundred} + \textrm{[math]\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}[/math] tens} + \textrm{[math]1[/math] one}

EXPLANATION

We can borrow 1 from the hundreds place and convert it to tens.

First, notice that

\begin{align*} \textrm{[math]2[/math] hundreds} + \textrm{[math]1[/math] ten} + \textrm{[math]1[/math] one} = (\textrm{[math]\color{red}1[/math] hundred} + \textrm{[math]\color{blue}1[/math] hundred}) + \textrm{[math]1[/math] ten} + \textrm{[math]1[/math] one} . \end{align*}

Now, we convert 1 hundred into 10 tens:

\begin{align*} (\textrm{[math]\color{red}1[/math] hundred} + {\textrm{[math]\color{blue}1[/math] hundred}}) + \textrm{[math]1[/math] ten} + \textrm{[math]1[/math] one} &=\\[5pt] (\textrm{[math]\color{red}1[/math] hundred} + {\textrm{[math]\color{blue}10[/math] tens}}) + \textrm{[math]1[/math] ten} + \textrm{[math]1[/math] one} &=\\[5pt] \textrm{[math]\color{red}1[/math] hundred} + (\textrm{[math]\color{blue}10[/math] tens} + \textrm{[math]1[/math] ten}) + \textrm{[math]1[/math] one} &=\\[5pt] \textrm{[math]\color{red}1[/math] hundred} + \textrm{[math]\color{blue}11[/math] tens} + \textrm{[math]1[/math] one} & \end{align*}

Therefore,

\textrm{[math]2[/math] hundreds} + \textrm{[math]1[/math] ten} + \textrm{[math]1[/math] one} = \textrm{[math]1[/math] hundred} + \textrm{[math]\bbox[2pt, white, border: 1pt solid black]{11}[/math] tens} + \textrm{[math]1[/math] one}.

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Practice: Borrowing From Hundreds With and Without Addition

What is the missing number in the equation below?

\[ \textrm{$8$ hundreds $+$ $8$ ones} = \textrm{$7$ hundreds} + \textrm{$\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}$ tens} + \textrm{$8$ ones} \]

a
 $1$
b
 $2$
c
 $9$
d
 $20$
e
 $10$
Practice: Borrowing From Hundreds With and Without Addition

What is the missing number in the equation below?

\[ \textrm{$7$ hundreds} + \textrm{$7$ tens} + \textrm{$5$ ones} = \textrm{$6$ hundreds} + \textrm{$\bbox[2pt, white, border: 1pt solid black]{\phantom{AA}}$ tens} + \textrm{$5$ ones} \]

a
 $70$
b
 $7$
c
 $17$
d
 $0$
e
 $170$
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