Subtract Whole Numbers

1.3 Subtract Whole Numbers

The topics covered in this section are:

  1. Use subtraction notation
  2. Model subtraction of whole numbers
  3. Subtract whole numbers
  4. Translate word phrases to math notation
  5. Subtract whole numbers in applications

1.3.1 Use Subtraction Notation

Suppose there are seven bananas in a bowl. Elana uses three of them to make a smoothie. How many bananas are left in the bowl? To answer the question, we subtract three from seven. When we subtract, we take one number away from another to find the difference. The notation we use to subtract $3$ from $7$ is

$7-3$

We read $7-3$ as seven minus three and the result is the difference of seven and three.

Subtraction Notation

To describe subtraction, we can use symbols and words.

OperationNotationExpressionRead asResult
Subtraction$-$$7-3$seven minus threethe difference of $7$ and $3$

Example 1

Translate from math notation to words:

  • $8-1$
  • $26-14$
Solution
  • We read this as eight minus one. The result is the difference of eight and one.
  • We read this as twenty-six minus fourteen. The result is the difference of twenty-six and fourteen.

1.3.2 Model Subtraction of Whole Numbers

A model can help us visualize the process of subtraction much as it did with addition. Again, we will use base-$10$ blocks. Remember a block represents $1$ and a rod represents $10$. Let’s start by modeling the subtraction expression we just considered, $7-3$.

We start by modeling the first number, $7$.
Now take away the second number, $3$. We’ll circle $3$ blocks to show that we are taking them away.
Count the number of blocks remaining.
There are $4$ ones blocks left.We have shown that $7-3=4$

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity Model Subtraction of Whole Numbers will help you develop a better understanding of subtracting whole numbers.

Example 2

Model the subtraction: $8-2$.

Solution
$8-2$ means the difference of $8$ and $2$.
Model the first, $8$.
Take away the second number, $2$.
Count the number of blocks remaining.
There are $6$ ones blocks left.We have shown that $8-2=6$.

Example 3

Model the subtraction: $13-8$.

Solution
Model the first number, $13$. We use $1$ ten and $3$ ones.
Take away the second number, $8$. However, there are not $8$ ones, so we will exchange the $1$ ten for $10$ ones.
Now we can take away $8$ ones.
Count the blocks remaining.
There are five ones left.We have shown that $13-8=5$.

As we did with addition, we can describe the models as ones blocks and tens rods, or we can simply say ones and tens.

Example 4

Model the subtraction: $43-26$.

Solution

Because $43-26$ means $43$ take away $26$, we begin by modeling the $43$.

Now, we need to take away $26$, which is $2$ tens and $6$ ones. We cannot take away $6$ ones from $3$ ones. So, we exchange $1$ ten for $10$ ones.

Now we can take away $2$ tens and $6$ ones.

Count the number of blocks remaining. There is $1$ ten and $7$ ones, which is $17$.

1.3.3 Subtract Whole Numbers

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition. We know $7-3=4$ and $4+3=7$. Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.

$7-3=4$because$4+3=7$
$13-8=5$because$5+8=13$
$43-26=17$because$17+26=43$

Example 5

Subtract then check by adding:

  • $9-7$
  • $8-3$
Solution

Part 1

$9-7$
Subtract $7$ from $9$.$2$
Check with addition.
$2+7=9$

Part 2

$8-3$
Subtract $3$ from $8$.$5$
Check with addition.
$5+3=8$

To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.

Example 6

Subtract then check by adding: $89-61$.

Solution
Write the numbers so the ones and tens digits line up vertically.$\begin{align*} 89&\\ \underline{- \ 61}& \end{align*}$
Subtract the digits in each place value.

Subtract the ones:$9-1=8$
Subtract the tens: $8-6=2$
$\begin{align*} 89&\\ \underline{- \ 61}&\\ 28& \end{align*}$
Check using addition.
$\begin{align*} 28&\\ \underline{+ \ 61}&\\ 89& \end{align*}$

Our answer is correct.

When we modeled subtracting $26$ from $43$, we exchanged $1$ ten for $10$ ones. When we do this without the model, we say we borrow $1$ from the tens place and add $10$ to the ones place.

How to: Find the Difference of Whole Numbers

  1. Write the numbers so each place value lines up vertically.
  2. Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.
  3. Continue subtracting each place value from right to left, borrowing if needed.
  4. Check by adding.

Example 7

Subtract: $43-26$.

Solution
Write the numbers so each place value lines up vertically.$\begin{align*} 43&\\ \underline{- \ 26}& \end{align*}$
Subtract the ones. We cannot subtract $6$ from $3$, so we borrow $1$ ten. This makes $3$ tens and $13$ ones. We write these numbers above each place and cross out the original digits.$\begin{align*} \overset{3}{\cancel{4}} \overset{13}{\cancel{3}}&\\ \underline{- \ 2 \ \ \ 6}& \end{align*}$
Now we can subtract the ones. $13−6=7$. We write the $7$ in the ones place in the difference.$\begin{align*} \overset{3}{\cancel{4}}\overset{13}{\cancel{3}}&\\ \underline{- \ 2 \ \ \ 6}&\\ 7& \end{align*}$
Now we subtract the tens. $3−2=1$. We write the $1$ in the tens place in the difference.$\begin{align*} \overset{3}{\cancel{4}}\overset{13}{\cancel{3}}&\\ \underline{- \ 2 \ \ \ 6}&\\ 1 \ 7&\end{align*}$
Check by adding.
$\begin{align*} 17&\\ \underline{+ \ 26}&\\ 43& \end{align*}$
Our answer is correct.

Example 8

Subtract then check by adding: $207-64$.

Solution
Write the numbers so each place value lines up vertically.$\begin{align*} 2 \ 0 \ 7&\\ \underline{- \ 6 \ 4}& \end{align*}$
Subtract the ones. $7−4=3$.
Write the $3$ in the ones place in the difference. Write the $3$ in the ones place in the difference.
$\begin{align*} 2 \ 0 \ 7&\\ \underline{- \ 6 \ 4}&\\ 3& \end{align*}$
Subtract the tens. We cannot subtract $6$ from $0$ so we borrow $1$ hundred and add $10$ tens to the $0$ tens we had. This makes a total of $10$ tens. We write $10$ above the tens place and cross out the $0$. Then we cross out the $2$ in the hundreds place and write 1 above it.$\begin{align*} \overset{1}{\cancel{2}} \overset{10}{\cancel{0}}7&\\ \underline{- \ 6 \ 4}&\\ 3& \end{align*}$
Now we subtract the tens. $10−6=4$.
We write the 4 in the tens place in the difference.
$\begin{align*} \overset{1}{\cancel{2}} \overset{10}{\cancel{0}}7&\\ \underline{- \ 6 \ 4}&\\ 4 \ 3& \end{align*}$
Finally, subtract the hundreds. There is no digit in the hundreds place in the bottom number so we can imagine a $0$ in that place. Since $1−0=1$, we write $1$ in the hundreds place in the difference.$\begin{align*} \overset{1}{\cancel{2}} \overset{10}{\cancel{0}}7&\\ \underline{- \ 6 \ 4}&\\ 1 \ 4 \ 3& \end{align*}$
Check by adding.
$\begin{align*} \overset{1}{1} \ 4 \ 3&\\ \underline{+ \ 6 \ 4}&\\ 2 \ 0 \ 7&\\ \end{align*}$
Our answer is correct.

Example 9

Subtract then check by adding: $910-586$.

Solution
Write the numbers so each place value lines up vertically.$\begin{align*} 9 \ 1 \ 0&\\ \underline{- \ 5 \ 8 \ 6}& \end{align*}$
Subtract the ones. We cannot subtract $6$ from $0$, so we borrow $1$ ten and add $10$ ones to the $10$ ones we had. This makes $10$ ones. We write a $0$ above the tens place and cross out the $1$. We write the $10$ above the ones place and cross out the $0$. Now we can subtract the ones. $10−6=4$.$\begin{align*} 9 \overset{0}{\cancel{1}} \overset{10}{\cancel{0}}&\\ \underline{- \ 5 \ \ 8 \ \ 6}& \end{align*}$
Write the $4$ in the ones place of the difference.$\begin{align*} 9 \overset{0}{\cancel{1}} \overset{10}{\cancel{0}}&\\ \underline{- \ 5 \ \ 8 \ \ 6}&\\ 4& \end{align*}$
Subtract the tens. We cannot subtract $8$ from $0$, so we borrow $1$ hundred and add $10$ tens to the $0$ tens we had, which gives us $10$ tens. Write $8$ above the hundreds place and cross out the $9$. Write $10$ above the tens place.$\begin{align*} \overset{8}{\cancel{9}} \overset{10}{\cancel{1}} \overset{10}{\cancel{0}}&\\ \underline{- \ 5 \ \ \ 8 \ \ \ 6}&\\ 4& \end{align*}$
Now we can subtract the tens. $10−8=2$.$\begin{align*} \overset{8}{\cancel{9}} \overset{10}{\cancel{1}} \overset{10}{\cancel{0}}&\\ \underline{- \ 5 \ \ \ 8 \ \ \ 6}&\\ 2 \ \ \ 4& \end{align*}$
Subtract the hundreds place. $8−5=3$.$\begin{align*} \overset{8}{\cancel{9}} \overset{10}{\cancel{1}} \overset{10}{\cancel{0}}&\\ \underline{- \ 5 \ \ \ 8 \ \ \ 6}&\\ 3 \ \ \ 2 \ \ \ 4& \end{align*}$
Check by adding.
$\begin{align*} \overset{1}{3} \ \overset{1}{2} \ 4&\\ \underline{+ \ 5 \ 8 \ 6}&\\ 9 \ 1 \ 0& \end{align*}$

Example 10

Subtract then check by adding: $2,162-479$.

Solution
Write the numbers so each place values line up vertically.$\begin{align*} 2, \ 1 \ 6 \ 2&\\ \underline{- \ 4 \ 7 \ 9}& \end{align*}$
Subtract the ones. Since we cannot subtract $9$ from $2$, borrow $1$ ten and add $10$ ones to the $2$ ones to make $12$ ones. Write $5$ above the tens place and cross out the $6$. Write $12$ above the ones place and cross out the $2$.$\begin{align*} 2, \ 1 \overset{5}{\cancel{6}} \overset{12}{\cancel{2}}&\\ \underline{- \ 4 \ \ 7 \ \ 9}& \end{align*}$
Now we can subtract the ones.$12-9=3$
Write $3$ in the ones place in the difference.$\begin{align*} 2, \ 1 \overset{5}{\cancel{6}} \overset{12}{\cancel{2}}&\\ \underline{- \ 4 \ \ 7 \ \ 9}&\\ 3& \end{align*}$
Subtract the tens. Since we cannot subtract $7$ from $5$, borrow $1$ hundred and add $10$ tens to the $5$ tens to make $15$ tens. Write $0$ above the hundreds place and cross out the $1$. Write $15$ above the tens place.$\begin{align*} 2, \overset{0}{\cancel{1}} \overset{15}{\overset{\cancel{5}}{\cancel{6}}} \overset{12}{\cancel{2}}&\\ \underline{- \ 4 \ \ 7 \ \ \ 9}&\\ 3& \end{align*}$
Now we can subtract the tens.$15-7=8$
Write $8$ in the tens place in the difference.$\begin{align*} 2, \overset{0}{\cancel{1}} \overset{15}{\cancel{6}} \overset{12}{\cancel{2}}&\\ \underline{- \ 4 \ \ 7 \ \ \ 9}&\\ 8 \ \ \ 3& \end{align*}$
Now we can subtract the hundreds.$\begin{align*} \overset{1}{\cancel{2}}, \overset{10}{\overset{\cancel{0}}{\cancel{1}}} \overset{15}{\cancel{6}} \overset{12}{\cancel{2}}&\\ \underline{- \ 4 \ \ \ 7 \ \ \ 9}&\\ 8 \ \ \ 3& \end{align*}$
Write $6$ in the hundreds place in the difference.$\begin{align*} \overset{1}{\cancel{2}}, \overset{10}{\cancel{1}} \overset{15}{\cancel{6}} \overset{12}{\cancel{2}}&\\ \underline{- \ 4 \ \ 7 \ \ \ 9}&\\ 6 \ \ 8 \ \ \ 3& \end{align*}$
Subtract the thousands. There is no digit in the thousands place of the bottom number, so we imagine a $0$. $1−0=1$. Write $1$ in the thousands place of the difference.$\begin{align*} \overset{1}{\cancel{2}}, \overset{10}{\cancel{1}} \overset{15}{\cancel{6}} \overset{12}{\cancel{2}}&\\ \underline{- \ 4 \ \ 7 \ \ \ 9}&\\1, \ \ 6 \ \ 8 \ \ \ 3& \end{align*}$
Check by adding.
$\begin{align*} \overset{1}{1}, \overset{1}{6}\overset{1}{8}3&\\ \underline{+ \ 479}&\\ 2,162& \end{align*}$

Our answer is correct.

1.3.4 Translate Word Phrases to Math Notation

As with addition, word phrases can tell us to operate on two numbers using subtraction. To translate from a word phrase to math notation, we look for key words that indicate subtraction. Some of the words that indicate subtraction are listed in Table 1.3 below.

OperationWord PhraseExampleExpression
Subtractionminus
difference
decreased by
less than
subtracted from
$5$ minus $1$
the difference of $9$ and $4$
$7$ decreased by $3$
$5$ less than $8$
$1$ subtracted from $6$
$5-1$
$9-4$
$7-3$
$8-5$
$6-1$

Example 11

Translate and then simplify:

  • the difference of $13$ and $8$
  • subtract $24$ from $43$
Solution

Part 1
The word difference tells us to subtract the two numbers. The numbers stay in the same order as in the phrase.

the difference of $13$ and $8$
Translate.$13-8$
Simplify.$5$

Part 2.
The words subtract from tells us to take the second number away from the first. We must be careful to get the order correct.

subtract $24$ from $43$
Translate.$43-24$
Simplify.$19$

1.3.5 Subtract Whole Numbers in Applications

To solve applications with subtraction, we will use the same plan that we used with addition. First, we need to determine what we are asked to find. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question, using the appropriate units.

Example 12

The temperature in Chicago one morning was $73$ degrees Fahrenheit. A cold front arrived and by noon the temperature was $27$ degrees Fahrenheit. What was the difference between the temperature in the morning and the temperature at noon?

Solution

We are asked to find the difference between the morning temperature and the noon temperature.

Write a phrase.the difference of $73$ and $27$
Translate to math notation. Difference tells us to subtract.$73-27$
Then we do the subtraction.$\begin{align*} \overset{6}{\cancel{7}} \overset{13}{\cancel{3}}&\\ \underline{- \ 2 \ \ 7} &\\ 4 \ \ 6 \end{align*}$
Write a sentence to answer the question.The difference in temperatures was $46$ degrees Fahrenheit.

Example 13

A washing machine is on sale for $\$399$.  Its regular price is $\$588$. What is the difference between the regular price and the sale price?

Solution

We are asked to find the difference between the regular price and the sale price.

Write a phrase.the difference between $588$ and $399$
Translate to math notation.$588-399$
Subtract.$\begin{align*} \overset{4}{\cancel{5}}\overset{17}{\cancel{8}}\overset{18}{\cancel{8}}&\\ \underline{- \ 3 \ \ 9 \ \ 9}&\\ 1 \ \ 8 \ \ 9& \end{align*}$
Write a sentence to answer the question.The difference between the regular price and the sale price is $\$189$.
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