1.10: Divide Whole Numbers (Part 2)
- Page ID
- 5774
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\PageIndex{2}\).
| Operation | Word Phrase | Example | Expression |
|---|---|---|---|
| Division | divided by | 12 divided by 4 |
12 ÷ 4 \(\dfrac{12}{4}\) |
| quotient of | the quotient of 12 and 4 | 12/4 | |
| divided into | 4 divided into 12 | \(4 \overline{\smash{)}12}\) |
Translate and simplify: the quotient of \(51\) and \(17\).
Solution
The word quotient tells us to divide.
| Translate. | 51 ÷ 17 |
| Divide. | 3 |
We could just as correctly have translated the quotient of \(51\) and \(17\) using the notation \(17 \overline{\smash{)}51}\) or \(\dfrac{51}{17}\).
Translate and simplify: the quotient of \(91\) and \(13\).
- Answer
-
\(91 \div 13; 7\)
Translate and simplify: the quotient of \(52\) and \(13\).
- Answer
-
\(52 \div 13; 4\)
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.
Cecelia bought a \(160\)-ounce box of oatmeal at the big box store. She wants to divide the \(160\) ounces of oatmeal into \(8\)-ounce servings. She will put each serving into a plastic bag so she can take one bag to work each day. How many servings will she get from the big box?
Solution
We are asked to find the how many servings she will get from the big box.
| Write a phrase. | 160 ounces divided by 8 ounces |
| Translate to math notation. | 160 ÷ 8 |
| Simplify by dividing. | 20 |
| Write a sentence to answer the question. | Cecelia will get 20 servings from the big box. |
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?
- Answer
-
Marcus can fill \(15\) cups.
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?
- Answer
-
Andrea can make \(9\) bows.
Access Additional Online Resources
- Dividing Whole Numbers
- Dividing Whole Numbers No Remainder
- Dividing Whole Numbers With Remainder
Key Concepts
| Operation | Notation | Expression | Read as | Result |
|---|---|---|---|---|
| \(\begin{align*} \div &\\ \dfrac{a}{b} &\\ b\overline{\smash{)} a} &\\ a/b & \end{align*}\) | \(\begin{align*} 12\div 4 &\\ \dfrac{12}{4} &\\ 4\overline{\smash{)}12} &\\ 12/4 & \end{align*}\) |
- Division Properties of One
- Any number (except \(0\)) divided by itself is one. \(a÷a=1\)
- Any number divided by one is the same number. \(a÷1=a\)
- Division Properties of Zero
- Zero divided by any number is \(0\). \(0÷a=0\)
- Dividing a number by zero is undefined.
- Divide whole numbers.
- Divide the first digit of the dividend by the divisor. If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.
- Write the quotient above the dividend.
- Multiply the quotient by the divisor and write the product under the dividend.
- Subtract that product from the dividend.
- Bring down the next digit of the dividend.
- Repeat from Step 1 until there are no more digits in the dividend to bring down.
- Check by multiplying the quotient times the divisor.
Glossary
- dividend
-
When dividing two numbers, the dividend is the number being divided.
- divisor
-
When dividing two numbers, the divisor is the number dividing the dividend.
- quotient
-
The quotient is the result of dividing two numbers.
Practice Makes Perfect
Use Division Notation
In the following exercises, translate from math notation to words.
- 54 ÷ 9
- \(\dfrac{56}{7}\)
- \(\dfrac{32}{8}\)
- \(6 \overline{\smash{)}42}\)
- 48 ÷ 6
- \(\dfrac{63}{9}\)
- \(7 \overline{\smash{)}63}\)
- 72 ÷ 8
Model Division of Whole Numbers
In the following exercises, model the division.
- 15 ÷ 5
- 10 ÷ 5
- \(\dfrac{14}{7}\)
- \(\dfrac{18}{6}\)
- \(4 \overline{\smash{)}20}\)
- \(3 \overline{\smash{)}15}\)
- 24 ÷ 6
- 16 ÷ 4
Divide Whole Numbers
In the following exercises, divide. Then check by multiplying.
- 18 ÷ 2
- 14 ÷ 2
- \(\dfrac{27}{3}\)
- \(\dfrac{30}{3}\)
- \(4 \overline{\smash{)}28}\)
- \(4 \overline{\smash{)}36}\)
- \(\dfrac{45}{5}\)
- \(\dfrac{35}{5}\)
- 72 / 8
- \(8 \overline{\smash{)}64}\)
- \(\dfrac{35}{7}\)
- 42 ÷ 7
- \(15 \overline{\smash{)}15}\)
- \(12 \overline{\smash{)}12}\)
- 43 ÷ 43
- 37 ÷ 37
- \(\dfrac{23}{1}\)
- \(\dfrac{29}{1}\)
- 19 ÷ 1
- 17 ÷ 1
- 0 ÷ 4
- 0 ÷ 8
- \(\dfrac{5}{0}\)
- \(\dfrac{9}{0}\)
- \(\dfrac{26}{0}\)
- \(\dfrac{32}{0}\)
- \(12 \overline{\smash{)}0}\)
- \(16 \overline{\smash{)}0}\)
- 72 ÷ 3
- 57 ÷ 3
- \(\dfrac{96}{8}\)
- \(\dfrac{78}{6}\)
- \(5\overline{\smash{)}465}\)
- \(4\overline{\smash{)}528}\)
- 924 ÷ 7
- 861 ÷ 7
- \(\dfrac{5,226}{6}\)
- \(\dfrac{3,776}{8}\)
- \(4\overline{\smash{)}31,324}\)
- \(5\overline{\smash{)}46,855}\)
- 7,209 ÷ 3
- 4,806 ÷ 3
- 5,406 ÷ 6
- 3,208 ÷ 4
- \(4\overline{\smash{)}2,816}\)
- \(6 \overline{\smash{)}3624}\)
- \(\dfrac{91,881}{9}\)
- \(\dfrac{83,256}{8}\)
- 2,470 ÷ 7
- 3,741 ÷ 7
- \(8\overline{\smash{)}55,305}\)
- \(9\overline{\smash{)}51,492}\)
- \(\dfrac{431,174}{5}\)
- \(\dfrac{297,277}{4}\)
- 130,016 ÷ 3
- 105,609 ÷ 2
- \(15\overline{\smash{)}5,735}\)
- \(\dfrac{4,933}{21}\)
- 56,883 ÷ 67
- 43,725 / 75
- \(\dfrac{30,144}{314}\)
- 26,145 ÷ 415
- \(273\overline{\smash{)}542,195}\)
- 816,243 ÷ 462
Mixed Practice
In the following exercises, simplify.
- 15(204)
- 74 • 391
- 256 − 184
- 305 − 262
- 719 + 341
- 647 + 528
- \(25\overline{\smash{)}875}\)
- 1104 ÷ 23
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate and simplify.
- the quotient of 45 and 15
- the quotient of 64 and 16
- the quotient of 288 and 24
- the quotient of 256 and 32
Divide Whole Numbers in Applications
In the following exercises, solve.
- 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?
- Crackers Evie bought a 42 ounce box of crackers. She wants to divide it into bags with 3 ounces of crackers in each bag. How many bags can Evie fill?
- 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?
- Flower shop Melissa’s flower shop got a shipment of 152 roses. She wants to make bouquets of 8 roses each. How many bouquets can Melissa make?
- 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?
- Dental floss One package of dental floss is 54 feet long. Brian uses 2 feet of dental floss every day. How many days will one package of dental floss last Brian?
Mixed Practice
In the following exercises, solve.
- 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?
- Distance Mayra lives 53 miles from her mother’s house and 71 miles from her motherin-law’s house. How much farther is Mayra from her mother-in-law’s house than from her mother’s house?
- 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?
- Potting soil Aki bought a 128 ounce bag of potting soil. How many 4 ounce pots can he fill from the bag?
- 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?
- Reading Last night Emily read 6 pages in her Business textbook, 26 pages in her History text, 15 pages in her Psychology text, and 9 pages in her math text. What is the total number of pages Emily read?
- 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?
- Scouts There are 14 boys in Dave’s scout troop. At summer camp, each boy earned 5 merit badges. What was the total number of merit badges earned by Dave’s scout troop at summer camp?
Writing Exercises
- Explain how you use the multiplication facts to help with division.
- Oswaldo divided 300 by 8 and said his answer was 37 with a remainder of 4. How can you check to make sure he is correct?
Everyday Math
- 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?
- Cat food One bag of cat food feeds Lara’s cat for 25 days. How many bags of cat food does Lara need for 365 days?
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?
Contributors and Attributions
- Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.