1.5 Divide Whole Numbers

Learning objectives

Learning objectives

• Use division notation

• Model division of whole numbers

• Divide whole numbers

• Translate word phrases to math notation

• Divide whole numbers in applications

Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the $12$ cookies in Figure 1.13 and want to package them in bags with $4$ cookies in each bag. How many bags would we need?

You might put $4$ cookies in first bag, $4$ in the second bag, and so on until you run out of cookies. Doing it this way, you would fill $3$ bags.

In other words, starting with the $12$ cookies, you would take away, or subtract, $4$ cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.

Instead of subtracting $4$ repeatedly, we can write

We read this as twelve divided by four and the result is the quotient of $12$ and $4.$ The quotient is $3$ because we can subtract $4$ from $12$ exactly $3$ times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is $12$ and the divisor is $4.$

In the past you may have used the notation $412$, but this division also can be written as $12\mathrm{÷}4,\,12/4,\frac{12}{4}.$ In each case the $12$ is the dividend and the $4$ is the divisor.

Example 1.56

Solution

1. Identify the operation in each expression. A division expression names a dividend, a divisor, and the quotient that results from dividing the first quantity by the second.
2. Read the first expression in words. The expression $64\div8$ is read as sixty-four divided by eight, and its result is the quotient of sixty-four and eight.
3. Read the fraction in words. The fraction $\frac{42}{7}$ is read as forty-two divided by seven, and its result is the quotient of forty-two and seven.
4. Read the third expression in words. The third expression is read as twenty-eight divided by four, and its result is the quotient of twenty-eight and four.
5. Summarize the common reading. In each form, the dividend comes first, the divisor comes second, and the quotient names the answer to the division.

Takeaway: A division expression can be read using its operation word or by naming the two quantities, with the result called the quotient.

Model Division of Whole Numbers

As we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly.

Example 1.57

Solution

1. Identify the dividend and divisor. In $24\div8$, the dividend is $24$ and the divisor is $8$, so the model must ask how many groups of $8$ can be formed.
2. Model the dividend. Represent the dividend with $24$ counters, keeping all of the counters together before separating them into equal groups.
3. Form equal groups. The divisor $8$ gives the size of each group, so partition the $24$ counters into groups containing $8$ counters each.
4. Count the groups. There are $3$ complete groups, with no counters left over, because $8+8+8=24$.
5. State the quotient. The number of equal groups is the quotient, so $24\div8=3$.

Takeaway: Dividing $24$ by $8$ asks how many groups of $8$ fit into $24$, so $24\div8=3$.

Divide Whole Numbers

We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know $12\mathrm{÷}4=3$ because $3\cdot{}4=12.$ Knowing all the multiplication number facts is very important when doing division.

We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. In Example 1.57, we know $24\mathrm{÷}8=3$ is correct because $3\cdot{}8=24.$

Example 1.58

Solution

1. Choose a division method. For each expression, divide the dividend by the divisor, then multiply the quotient by the divisor to check that the original dividend is recovered.
2. Divide and check the first expression. Since $42\div6=7$, the quotient is $7$. The check $7\cdot6=42$ confirms that the division is correct.
3. Divide and check the second expression. Since $\frac{72}{9}=8$, the quotient is $8$. The check $8\cdot9=72$ confirms that the division is correct.
4. Divide and check the third expression. Since $63\div7=9$, the quotient is $9$. The check $9\cdot7=63$ confirms that the division is correct.
5. Summarize the verification pattern. Multiplying each quotient by its divisor produces the dividend, so all three divisions have the correct answers.

Takeaway: Every division answer can be checked by multiplying the quotient by the divisor, and the products $7\cdot6=42$, $8\cdot9=72$, and $9\cdot7=63$ verify the quotients.

What is the quotient when you divide a number by itself?

Dividing any number $\mathrm{(except 0)}$ by itself produces a quotient of $1.$ Also, any number divided by $1$ produces a quotient of the number. These two ideas are stated in the Division Properties of One.

Example 1.59

Solution

1. Apply the division properties of one. Dividing a nonzero number by itself gives $1$, while dividing any number by $1$ leaves the number unchanged.
2. Divide and check the first expression. Since $11\div11=1$, the quotient is $1$. The check $1\cdot11=11$ matches the dividend.
3. Divide and check the second expression. Since $\frac{19}{1}=19$, the quotient is $19$. The check $19\cdot1=19$ matches the dividend.
4. Divide and check the third expression. Since $7\div1=7$, the quotient is $7$. The check $7\cdot1=7$ matches the dividend.
5. Summarize the two properties. Division by the same nonzero number produces $1$, whereas division by $1$ produces the original number itself.

Takeaway: Dividing a nonzero number by itself gives $1$, while dividing a number by $1$ leaves it unchanged, so $11\div11=1$ and $19\div1=19$.

Suppose we have $\text{\textdollar 0},$ and want to divide it among $3$ people. How much would each person get? Each person would get $\text{\textdollar 0}.$ Zero divided by any number is $0.$

Now suppose that we want to divide $\text{\textdollar 1}0$ by $0.$ That means we would want to find a number that we multiply by $0$ to get $10.$ This cannot happen because $0$ times any number is $0.$ Division by zero is said to be undefined.

These two ideas make up the Division Properties of Zero.

Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away $0$ from $10?$ Because subtracting $0$ will never change the total, we will never get an answer. So we cannot divide a number by $0.$

Example 1.60

Solution

1. Distinguish the two division cases. The first expression divides zero by a nonzero number, while the second divides $10$ by zero, so the divisor determines whether a quotient exists.
2. Divide and check the first expression. Since $0\div3=0$, the quotient is $0$. The check $0\cdot3=0$ confirms that the division is correct.
3. Analyze the second expression. The expression $10\div0$ would require a number that multiplied by $0$ gives $10$, but the product of $0$ with every number is $0$.
4. State the undefined quotient. No quotient can satisfy the second division, so $10\div0$ is undefined rather than equal to any ordinary number.

Takeaway: Zero divided by a nonzero number is $0$, but division by zero is undefined because no number multiplied by $0$ equals $10$.

When the divisor or the dividend has more than one digit, it is usually easier to use the $412$ notation. This process is called long division. Let’s work through the process by dividing $78$ by $3.$

We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.

Check by multiplying the quotient times the divisor to get the dividend. Multiply $26\,\times\,3$ to make sure that product equals the dividend, $78.$

It does, so our answer is correct.

Example 1.61

Solution

1. Choose the first quotient digit. Since $4$ does not go into $2$, use the first two digits. Because $4$ goes into $25$ six times, write $6$ above the $5$ and note the remainder $1$.
2. Bring down the next digit. Appending $9$ to the remainder gives $19$. Since $4$ goes into $19$ four times, write $4$ in the next quotient position and subtract $16$, leaving $3$.
3. Finish the division. Bring down the final $6$, giving $36$. Since $4$ goes into $36$ nine times, write $9$ as the last quotient digit. The quotient is $649$ with remainder $0$, as shown in the completed division.
4. Check by multiplying. Multiply $649\cdot{}4=2,596$, which equals the dividend and confirms the result in the final figure.

Takeaway: At each long-division stage, choose the largest quotient digit whose product does not exceed the current dividend portion.

Example 1.62

Solution

1. Find the first quotient digit. Because $4<6$, begin with $45$. The largest multiple of $6$ not exceeding $45$ is $42$, so write $7$ and subtract to obtain a remainder of $3$.
2. Bring down the next digit. Appending $0$ gives $30$. Since $6$ goes into $30$ five times, write $5$ and subtract $30$, leaving remainder $0$.
3. Bring down the final digit. Appending $6$ gives $6$, which is one group of $6$. Write $1$ as the last quotient digit and leave remainder $0$.
4. State and check the quotient. The quotient is $751$. Multiplying $751\cdot{}6=4,506$, which returns the dividend and agrees with the final check figure.

Takeaway: If a brought-down portion is smaller than the divisor, write $0$ in the quotient and continue until the remainder can be extended.

Example 1.63

Solution

1. Find the first quotient digit. Since $7<9$, begin with $72$. Because $9$ goes into $72$ eight times, write $8$ and subtract $72$, leaving remainder $0$.
2. Continue with the next digit. Bring down $6$. Because $6<9$, write $0$ in the quotient, multiply $9$ by $0$, and retain $6$ as the remainder.
3. Bring down the final digit. Appending $3$ gives $63$, and $9$ goes into $63$ seven times. Write $7$ and subtract $63$, leaving remainder $0$. The quotient is $807$.
4. Check by multiplying. The product $807\cdot{}9=7,263$ equals the dividend, confirming the quotient shown in the final figure.
5. Interpret the remainder example. The cookie diagrams show $28$ cookies divided into groups of $8$, leaving $4$ cookies. Thus the quotient is $3$ with remainder $4$, and $3\cdot{}8+4=28$.

Takeaway: A division that does not come out evenly has a remainder smaller than the divisor.

So far all the division problems have worked out evenly. For example, if we had $24$ cookies and wanted to make bags of $8$ cookies, we would have $3$ bags. But what if there were $28$ cookies and we wanted to make bags of $8?$ Start with the $28$ cookies as shown in Figure 1.14.

Try to put the cookies in groups of eight as in Figure 1.15.

There are $3$ groups of eight cookies, and $4$ cookies left over. We call the $4$ cookies that are left over the remainder and show it by writing R4 next to the $3.$ (The R stands for remainder.)

To check this division we multiply $3$ times $8$ to get $24,$ and then add the remainder of $4.$

Example 1.64

Solution

1. Find the first quotient digit. Divide $14$ by $4$. The largest quotient digit is $3$, since $4\cdot{}3=12$, leaving remainder $2$.
2. Bring down the next digit. Appending $3$ to the remainder gives $23$. Since $4$ goes into $23$ five times, write $5$ and subtract $20$, leaving remainder $3$.
3. Finish the division. Bring down the final $9$, giving $39$. The largest quotient digit is $9$, because $4\cdot{}9=36$, leaving remainder $3$. The result is $359$ with remainder $3$.
4. Check the division. Multiply $359\cdot{}4=1,436$, then add the remainder $3$ to obtain $1,439$, as shown in the final figure.

Takeaway: To check a quotient with remainder, multiply the quotient by the divisor and then add the remainder.

Example 1.65

Solution

1. Choose the first quotient digit. Since $13$ does not go into $1$, begin with $14$. The digit $1$ fits once, and $14-13=1$, leaving remainder $1$.
2. Bring down the next digit. Appending $6$ gives $16$. The digit $1$ fits once, leaving remainder $3$, so the partial quotient is $11$.
3. Bring down the final digit. Appending $1$ gives $31$. The digit $2$ fits twice because $13\cdot{}2=26$, leaving remainder $5$. Thus the quotient is $112$ with remainder $5$.
4. Check the result. Multiply $112\cdot{}13=1,456$ and add the remainder $5$ to obtain $1,461$, confirming the division shown in the final figure.

Takeaway: A final remainder must be smaller than the divisor, and it must be added after multiplication when checking.

Example 1.66

Solution

1. Find the first quotient digit. The divisor $241$ fits into $745$ three times, because $241\cdot{}3=723$, but four times would be too large. Write $3$ and subtract to obtain remainder $22$.
2. Bring down the next digit. Appending $2$ gives $222$. Since $222<241$, write $0$ in the quotient and keep $222$ as the remainder.
3. Bring down the final digit. Appending $1$ gives $2,221$. The digit $9$ fits because $241\cdot{}9=2,169$, leaving remainder $52$. Thus the quotient is $309$ with remainder $52$.
4. Check the result. Multiply $309\cdot{}241=74,469$ and add $52$ to obtain $74,521$, agreeing with the final figure.

Takeaway: When estimating a quotient digit, multiply to confirm that the trial digit fits without making the product too large.

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation for division into words. Now we’ll translate word phrases into math notation. Some of the words that indicate division are given in Table 1.8.

Table

1.8

Example 1.67

Solution

1. Identify the operation in the phrase. The word quotient signals division, and the two quantities keep their stated order: $51$ is the dividend and $17$ is the divisor.
2. Translate the phrase into notation. Write the division as $51\div17$, which can also be represented by the fraction $\frac{51}{17}$.
3. Divide to find the quotient. Since $17\cdot3=51$, dividing $51$ by $17$ gives the quotient $3$.
4. Check the translated expression. Multiplication reverses division, so $3\cdot17=51$ confirms that the quotient is correct.

Takeaway: The phrase quotient of $51$ and $17$ means division, so $51\div17=3$.

Divide Whole Numbers in Applications

We will use the same strategy we used in previous sections to solve applications. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify it to get the answer. Finally, we write a sentence to answer the question.

Example 1.68

Solution

1. Identify the total amount and serving size. The box contains $160$ ounces of oatmeal, and each plastic bag must hold one $8$-ounce serving.
2. Write a division phrase. The number of servings is the total ounces divided by the ounces in each serving, so write $160$ ounces divided by $8$ ounces.
3. Translate and simplify the expression. Cancel the compatible ounce units to obtain $160\div8=20$, because $20\cdot8=160$.
4. Answer the application question. The number of equal servings is $20$, so Cecelia will get $20$ servings and can put one serving in each of $20$ bags.

Takeaway: Dividing the total amount by the amount in each serving gives the number of servings, so $160\div8=20$ servings.

Section 1.5 Exercises

Section 1.5 Exercises

Use Division Notation

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

Model Division of Whole Numbers

In the following exercises, model the division.

Divide Whole Numbers

In the following exercises, divide. Then check by multiplying.

Mixed Practice

In the following exercises, simplify.

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

Divide Whole Numbers in Applications

In the following exercises, solve.

Mixed Practice

In the following exercises, solve.

Writing Exercises

Everyday Math

Self Check

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

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

1. Translate and simplify: the quotient of $91$ and $13.$

2. Translate and simplify: the quotient of $52$ and $13.$

3. Marcus is setting out animal crackers for snacks at the preschool. He wants to put $9$ crackers in each cup. One box of animal crackers contains $135$ crackers. How many cups can he fill from one box of crackers?

1. Andrea is making bows for the girls in her dance class to wear at the recital. Each bow takes $4$ feet of ribbon, and $36$ feet of ribbon are on one spool. How many bows can Andrea make from one spool of ribbon?

2. $54\mathrm{÷}9$

3. $\frac{32}{8}$

1. $48\mathrm{÷}6$

2. $763$

3. $\frac{5}{0}$

1. $\frac{26}{0}$

2. $431,324$

3. $7,209\mathrm{÷}3$

1. $\frac{91,881}{9}$

2. $2,470\mathrm{÷}7$

3. $855,305$

1. $\frac{431,174}{5}$

2. $130,016\mathrm{÷}3$

3. $155,735$

1. $273542,195$

2. $15(204)$

3. $719+341$

1. the quotient of $45$ and $15$

2. the quotient of $288$ and $24$

3. Trail mix Ric bought $64$ ounces of trail mix. He wants to divide it into small bags, with $2$ ounces of trail mix in each bag. How many bags can Ric fill?

1. Astronomy class There are $125$ students in an astronomy class. The professor assigns them into groups of $5.$ How many groups of students are there?

2. Baking One roll of plastic wrap is $48$ feet long. Marta uses $3$ feet of plastic wrap to wrap each cake she bakes. How many cakes can she wrap from one roll?

3. Miles per gallon Susana’s hybrid car gets $45$ miles per gallon. Her son’s truck gets $17$ miles per gallon. What is the difference in miles per gallon between Susana’s car and her son’s truck?

1. Field trip The $45$ students in a Geology class will go on a field trip, using the college’s vans. Each van can hold $9$ students. How many vans will they need for the field trip?

2. Hiking Bill hiked $8$ miles on the first day of his backpacking trip, $14$ miles the second day, $11$ miles the third day, and $17$ miles the fourth day. What is the total number of miles Bill hiked?

3. Patients LaVonne treats $12$ patients each day in her dental office. Last week she worked $4$ days. How many patients did she treat last week?

1. Translate from math notation to words:

ⓐ $84\mathrm{÷}7$ ⓑ $\frac{18}{6}$ ⓒ $824$

2. Translate from math notation to words:

ⓐ $72\mathrm{÷}9$ ⓑ $\frac{21}{3}$ ⓒ $654$

3. Divide. Then check by multiplying:

ⓐ $54\mathrm{÷}6$ ⓑ $\frac{27}{9}$

1. Divide. Then check by multiplying:

ⓐ $\frac{36}{9}$ ⓑ $840$

2. Divide. Then check by multiplying:

ⓐ $14\mathrm{÷}14$ ⓑ $\frac{27}{1}$

3. Divide. Then check by multiplying:

ⓐ $\frac{16}{1}$ ⓑ $14$

1. Divide. Then check by multiplying:

ⓐ $0\mathrm{÷}2$ ⓑ $17/0$

2. Divide. Then check by multiplying:

ⓐ $0\mathrm{÷}6$ ⓑ $13/0$

3. Divide. Then check by multiplying: $4,928\mathrm{÷}7.$

1. Divide. Then check by multiplying: $3,812\mathrm{÷}8.$

2. Divide. Then check by multiplying: $4,319\mathrm{÷}8.$

3. Divide. Then check by multiplying: $1,493\mathrm{÷}13.$

1. Divide. Then check by multiplying: $1,461\mathrm{÷}12.$

2. Divide. Then check by multiplying: $78,641\mathrm{÷}256.$

3. Divide. Then check by multiplying: $76,461\mathrm{÷}248.$

1. Contact lenses Jenna puts in a new pair of contact lenses every $14$ days. How many pairs of contact lenses does she need for $365$ days?

2. Explain how you use the multiplication facts to help with division.