1.3 Subtract Whole Numbers

Learning objectives

Learning objectives

• Use subtraction notation

• Model subtraction of whole numbers

• Subtract whole numbers

• Translate word phrases to math notation

• Subtract whole numbers in applications

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

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

Example 1.26

Solution

1. Identify the first part. In $8-1$, the minuend is $8$ and the subtrahend is $1$, so the operation says to take $1$ away from $8$.
2. Translate the first expression. Read $8-1$ as eight minus one, and describe the result as the difference of eight and one.
3. Translate the second expression. In $26-14$, the minuend is $26$ and the subtrahend is $14$, so read the expression as twenty-six minus fourteen.
4. Complete the wording. The result of $26-14$ is the difference of twenty-six and fourteen, which gives both expressions their subtraction wording.

Takeaway: A subtraction expression names the minuend, operation, and subtrahend, so $8-1$ is eight minus one and $26-14$ is twenty-six minus fourteen.

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 $\mathrm{base-10}$ blocks. Remember a block represents 1 and a rod represents 10. Let’s start by modeling the subtraction expression we just considered, $7-3.$

Example 1.27

Solution

1. Model the minuend. Draw a row of $8$ ones blocks to represent $8$, as shown in the first diagram with the label $8$.
2. Remove the subtrahend. Take away $2$ of those blocks. The second diagram circles the two blocks that are removed.
3. Count what remains. The remaining row has $6$ blocks, as shown in the final diagram. Therefore $8-2=6$.

Takeaway: In a subtraction model, begin with the whole and remove exactly the number of blocks stated by the subtrahend.

Example 1.28

Solution

1. Model $13$. Use $1$ tens rod and $3$ ones blocks, as shown by the groups labeled $10$ and $3$ in the diagrams.
2. Prepare to remove $8$. Exchange the tens rod for $10$ ones blocks. Together with the original $3$ ones, this gives $13$ ones blocks and makes it possible to remove $8$.
3. Remove and count. Take away $8$ ones blocks. Five ones blocks remain, so $13-8=5$, agreeing with the final diagram of five separated squares.

Takeaway: Exchange one ten for ten ones when the minuend has too few ones to subtract the subtrahend directly.

Example 1.29

Solution

1. Model the minuend. Represent $43$ with $4$ tens rods and $3$ ones blocks, since $43$ consists of $4$ tens and $3$ ones.
2. Prepare to remove the subtrahend. The number $26$ consists of $2$ tens and $6$ ones, but the minuend currently has only $3$ ones to remove.
3. Exchange one ten. Regroup one of the $4$ tens rods into $10$ ones blocks, leaving $3$ tens and giving $13$ ones blocks in total.
4. Remove and count. Take away $2$ tens and $6$ ones. The remaining blocks are $1$ ten and $7$ ones, so $43-26=17$.

Takeaway: To subtract $26$ from $43$, exchange one ten for ten ones when necessary; after removing $2$ tens and $6$ ones, $43-26=17$.

Subtract Whole Numbers

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.

We know $7-3=4$ because $4+3=7.$ 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.

Example 1.30

Solution

1. Understand the check. A subtraction can be checked by adding the difference to the subtrahend because the two amounts must restore the original minuend.
2. Subtract the first pair. For part a, $9-7=2$. Adding the difference to the subtrahend gives $2+7=9$, which confirms the answer.
3. Subtract the second pair. For part b, $8-3=5$. Adding the difference to the subtrahend gives $5+3=8$, which also confirms the answer.
4. Summarize the results. The checked differences are $2$ for $9-7$ and $5$ for $8-3$, so both subtractions are correct.

Takeaway: Subtraction can be checked with addition because a difference plus the subtrahend recovers the minuend, so $9-7=2$ and $8-3=5$.

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 1.31

Solution

1. Align the place values. Write $89$ above $61$ so that the $8$ and $6$ are in the tens column and the $9$ and $1$ are in the ones column.
2. Subtract the ones. Begin at the rightmost column and calculate $9-1=8$, so write $8$ in the ones place of the difference.
3. Subtract the tens. Move left and calculate $8-6=2$, then write $2$ in the tens place to obtain the difference $28$.
4. Check with addition. Add $28+61=89$; because the sum restores the original minuend, the subtraction is correct.

Takeaway: Subtract within each aligned place-value column and check by adding, which confirms that $89-61=28$ because $28+61=89$.

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.

Example 1.32

Solution

1. Set up the subtraction. Align $43$ above $26$ so that the ones and tens digits form matching columns, as in the first figure.
2. Regroup for the ones. There are only $3$ ones, so exchange one ten for ten ones. This changes $43$ to $3$ tens and $13$ ones, matching the crossed-out digits in the regrouping figure.
3. Subtract each place. The ones give $13-6=7$, and the tens give $3-2=1$, so the difference is $17$.
4. Check the result. Add $17+26=43$. The checkmark in the final figure confirms that the difference restores the original minuend.

Takeaway: A subtraction answer is correct when adding the subtrahend and difference returns the original minuend.

Example 1.33

Solution

1. Set up the subtraction. Align $64$ under the last two digits of $207$, placing the $7$ in the ones column and the $6$ in the tens column.
2. Regroup the tens. The ones need no regrouping because $7>4$, but the tens digit $0<6$, so borrow one hundred from the $2$ hundreds: the minuend becomes $1$ hundred, $10$ tens, and $7$ ones.
3. Subtract the ones. The ones column gives $7-4=3$, so write $3$ in the ones place.
4. Subtract the tens and hundreds. The tens give $10-6=4$, and the hundreds give $1-0=1$, so $207-64=143$.
5. Check by adding. Adding $143+64=207$ returns the original minuend, as shown by the checkmark in the final figure.

Takeaway: Borrow from the nearest higher place that can provide a ten, and include every regrouped digit in the subtraction.

Example 1.34

Solution

1. Set up the subtraction. Align $586$ beneath $910$ so that the $6$, $8$, and $5$ occupy the ones, tens, and hundreds places.
2. Regroup for the ones. Borrow one ten from the tens digit, changing $1$ ten to $0$ tens and $0$ ones to $10$ ones. Then $10-6=4$, so the ones digit of the difference is $4$.
3. Regroup for the tens. Since $0<8$, borrow one hundred from the $9$, leaving $8$ hundreds. Change the $0$ tens to $10$ tens, and then $10-8=2$.
4. Subtract the hundreds. The remaining hundreds give $8-5=3$, so $910-586=324$. The later figures correct the incomplete tens calculation and display the complete result.
5. Check by adding. Adding $324+586=910$ returns the dividend, and the final figure marks the check with a checkmark.

Takeaway: When a zero stands between two borrowing steps, regroup the place directly above it before continuing to the left.

Example 1.35

Solution

1. Set up the subtraction. Align $479$ under the last three digits of $2,162$, so the columns represent thousands, hundreds, tens, and ones.
2. Regroup the ones. Since $2<9$, borrow one ten from the $6$, changing the tens digit to $5$ and the ones to $12$, so $12-9=3$.
3. Regroup the tens. Because $5<7$, borrow one hundred from the $1$, leaving $0$ hundreds and changing $5$ tens to $15$ tens, so $15-7=8$.
4. Regroup the hundreds. Since $0<4$, borrow one thousand from the $2$, leaving $1$ thousand and changing the hundreds to $10$; the hundreds give $10-4=6$ and the thousands give $1-0=1$, so $2,162-479=1,683$.
5. Check by adding. Adding $1,683+479=2,162$ returns the original minuend, which the final figures confirm.

Takeaway: Regrouping changes the displayed digits for convenience, but the numerical value of the minuend must remain unchanged.

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.

Table

1.3

Example 1.36

Solution

1. Identify the operation and order. Both phrases describe subtraction, but each phrase determines the order of its minuend and subtrahend.
2. Handle the first part. The difference of $13$ and $8$ keeps the numbers in the given order, so it translates to $13-8$ and simplifies to $5$.
3. Handle the second part. Subtract $24$ from $43$ means that $24$ is removed from $43$, so it translates to $43-24$ and simplifies to $19$.
4. Compare the results. The first phrase gives $5$ and the second gives $19$, with the order of the numbers chosen correctly in each translation.

Takeaway: The phrase subtract $24$ from $43$ reverses the apparent order, so it becomes $43-24=19$, while the difference of $13$ and $8$ becomes $13-8=5$.

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 1.37

Solution

1. Choose the subtraction order. The morning temperature was $73$ degrees Fahrenheit and the noon temperature was $27$ degrees Fahrenheit, so subtract $27$ from $73$.
2. Subtract the ones place. Since $3<7$, regroup one ten, changing $7$ tens to $6$ tens and $3$ ones to $13$ ones. Then $13-7=6$.
3. Subtract the tens place. Now $6-2=4$, giving a difference of $46$, as shown in the figure.
4. State the answer with units. The temperature difference between morning and noon was $46$ degrees Fahrenheit.

Takeaway: For a decrease from one measurement to another, subtract the later measurement from the earlier measurement.

Example 1.38

Solution

1. Identify the values to subtract. The regular price is $588$ dollars and the sale price is $399$ dollars, so compute $588-399$.
2. Regroup and subtract the ones. Since $8<9$, exchange one ten, leaving $5$ tens and $18$ ones. Then $18-9=9$.
3. Subtract the tens and hundreds. The tens give $5-9<0$, so regroup one hundred from the $5$ hundreds, leaving $4$ hundreds and $15$ tens. Then $15-9=6$, and $4-3=1$.
4. State the price difference. The regular price is $189$ dollars more than the sale price, which agrees with the answer in the diagram.

Takeaway: To find a discount, subtract the sale price from the regular price and attach the currency unit to the result.

Section 1.3 Exercises

Section 1.3 Exercises

Use Subtraction Notation

In the following exercises, translate from math notation to words.

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

Subtract Whole Numbers

In the following exercises, subtract and then check by adding.

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

Mixed Practice

In the following exercises, simplify.

In the following exercises, translate and simplify.

Subtract Whole Numbers in Applications

In the following exercises, solve.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Translate and simplify:

1. ⓐ $11$ decreased by $6$

2. ⓑ $18$ less than $67$

2. The high temperature on $\mathrm{June}\,{1}^{\mathrm{st}}$ in Boston was $77$ degrees Fahrenheit, and the low temperature was $58$ degrees Fahrenheit. What was the difference between the high and low temperatures?

3. The weather forecast for June $2$ in St Louis predicts a high temperature of $90$ degrees Fahrenheit and a low of $73$ degrees Fahrenheit. What is the difference between the predicted high and low temperatures?

1. A television set is on sale for $\text{\textdollar 499}.$ Its regular price is $\text{\textdollar 648}.$ What is the difference between the regular price and the sale price?

2. A patio set is on sale for $\text{\textdollar 149}.$ Its regular price is $\text{\textdollar 285}.$ What is the difference between the regular price and the sale price?

3. $15-9$

1. $42-35$

2. $675-350$

3. $5,946-4,625$

1. $6,318-2,799$

2. $2,150-964$

3. $43,650-8,982$

1. The difference of $10$ and $3$

2. The difference of $15$ and $4$

3. Subtract $6$ from $9$

1. Subtract $28$ from $75$

2. $45$ decreased by $20$

3. $92$ decreased by $67$

1. $12$ less than $16$

2. $38$ less than $61$

3. $719+341$

1. Seventy-five more than thirty-five

2. $13$ less than $41$

3. The difference of $100$ and $76$

1. Temperature The high temperature on June $2$ in Las Vegas was $80$ degrees and the low temperature was $63$ degrees. What was the difference between the high and low temperatures?

2. Class size Olivia’s third grade class has $35$ children. Last year, her second grade class had $22$ children. What is the difference between the number of children in Olivia’s third grade class and her second grade class?

3. Shopping A mountain bike is on sale for $\text{\textdollar 399}.$ Its regular price is $\text{\textdollar 650}.$ What is the difference between the regular price and the sale price?

1. 1. ⓐ $12-4$

2. ⓑ $29-11$

2. 1. ⓐ $11-2$

2. ⓑ $29-12$

3. Subtract and then check by adding:

$7-0$

1. Subtract and then check by adding:

$6-2$

2. Subtract and then check by adding: $86-54.$

3. Subtract and then check by adding: $99-74.$

1. Subtract and then check by adding: $93-58.$

2. Subtract and then check by adding: $81-39.$

3. Subtract and then check by adding: $439-52.$

1. Subtract and then check by adding: $318-75.$

2. Subtract and then check by adding: $832-376.$

3. Subtract and then check by adding: $847-578.$

1. Subtract and then check by adding: $4,585-697.$

2. Subtract and then check by adding: $5,637-899.$

3. Translate and simplify:

1. ⓐ the difference of $14$ and $9$

2. ⓑ subtract $21$ from $37$

1. Savings John wants to buy a laptop that costs $\text{\textdollar 840}.$ He has $\text{\textdollar 685}$ in his savings account. How much more does he need to save in order to buy the laptop?

2. Road trip Noah was driving from Philadelphia to Cincinnati, a distance of $502$ miles. He drove $115$ miles, stopped for gas, and then drove another $230$ miles before lunch. How many more miles did he have to travel?

3. Explain how subtraction and addition are related.