1.6: Divide Whole Numbers

Prealgebra · LibreTexts

0%

Contents

    Text

    Size
    100%
    Theme
    Typeface
    Line length
    Spacing

    From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Learning Objectives

    By the end of this section, you will be able to:

    • Use division notation
    • Model division of whole numbers
    • Divide whole numbers
    • Translate word phrases to math notation
    • Divide whole numbers in applications

    Be Prepared 1.7

    Before you get started, take this readiness quiz.

    Multiply:
    If you missed this problem, review Example 1.44.

    Be Prepared 1.8

    Subtract:
    If you missed this problem, review Example 1.32

    Be Prepared 1.9

    Multiply:
    If you missed this problem, review Example 1.45.

    Use Division Notation

    So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the

    An image of three rows of four cookies to show twelve cookies.
    Figure 1.13

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

    An image of 3 bags of cookies, each bag containing 4 cookies.

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

    Instead of subtracting repeatedly, we can write

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

    In the past you may have used the notation , but this division also can be written as In each case the is the dividend and the is the divisor.

    Operation Symbols for Division

    To represent and describe division, we can use symbols and words.

    Operation Notation Expression Read as Result






    Division is performed on two numbers at a time. When translating from math notation to English words, or English words to math notation, look for the words of and and to identify the numbers.

    Example 1.56

    Translate from math notation to words.

    ⓐ ⓑ ⓒ

    Answer

    • ⓐ We read this as sixty-four divided by eight and the result is the quotient of sixty-four and eight.
    • ⓑ We read this as forty-two divided by seven and the result is the quotient of forty-two and seven.
    • ⓒ We read this as twenty-eight divided by four and the result is the quotient of twenty-eight and four.

    Try It 1.111

    Translate from math notation to words:

    ⓐ ⓑ ⓒ

    Try It 1.112

    Translate from math notation to words:

    ⓐ ⓑ ⓒ

    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.

    Manipulative Mathematics

    Doing the Manipulative Math Worksheets activity "Model Division of Whole Numbers" will help you develop a better understanding of dividing whole numbers.

    Example 1.57

    Model the division:

    Answer

    To find the quotient we want to know how many groups of are in

    Model the dividend. Start with counters.

    An image of 24 counters placed randomly.

    The divisor tell us the number of counters we want in each group. Form groups of counters.

    An image of 24 counters, all contained in 3 bubbles, each bubble containing 8 counters.

    Count the number of groups. There are groups.

    Try It 1.113

    Model:

    Try It 1.114

    Model:

    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 because 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 is correct because

    Example 1.58

    Divide. Then check by multiplying. ⓐ ⓑ ⓒ

    Answer

    • ⓐ
      Divide 42 by 6.
      Check by multiplying.
    • ⓑ
      Divide 72 by 9.
      Check by multiplying.
  1. ⓒ
    Divide 63 by 7.
    Check by multiplying.

    2. Try It 1.115

    Divide. Then check by multiplying:

    ⓐ ⓑ

    Try It 1.116

    Divide. Then check by multiplying:

    ⓐ ⓑ

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

    Division Properties of One

    Any number (except 0) divided by itself is one.
    Any number divided by one is the same number.
    Table 1.6

    Example 1.59

    Divide. Then check by multiplying:

    1. ⓐ
    2. ⓑ
    3. ⓒ
    Answer

    • ⓐ
      A number divided by itself is 1.
      Check by multiplying.
    • ⓑ
      A number divided by 1 equals itself.
      Check by multiplying.
  2. ⓒ
    A number divided by 1 equals itself.
    Check by multiplying.

    3. Try It 1.117

    Divide. Then check by multiplying:

    ⓐ ⓑ

    Try It 1.118

    Divide. Then check by multiplying:

    ⓐ ⓑ

    Suppose we have and want to divide it among people. How much would each person get? Each person would get Zero divided by any number is

    Now suppose that we want to divide by That means we would want to find a number that we multiply by to get This cannot happen because times any number is Division by zero is said to be undefined.

    These two ideas make up the Division Properties of Zero.

    Division Properties of Zero

    Zero divided by any number is 0.
    Dividing a number by zero is undefined. undefined
    Table 1.7

    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 from Because subtracting will never change the total, we will never get an answer. So we cannot divide a number by

    Example 1.60

    Divide. Check by multiplying: ⓐ ⓑ

    Answer

    • ⓐ
      Zero divided by any number is zero.
      Check by multiplying.
    • ⓑ
      Division by zero is undefined. undefined

    Try It 1.119

    Divide. Then check by multiplying:

    ⓐ ⓑ

    Try It 1.120

    Divide. Then check by multiplying:

    ⓐ ⓑ

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

    Divide the first digit of the dividend, 7, by the divisor, 3.
    The divisor 3 can go into 7 two times since . Write the 2 above the 7 in the quotient. CNX_BMath_Figure_01_05_043_img-02.png
    Multiply the 2 in the quotient by 3 and write the product, 6, under the 7. CNX_BMath_Figure_01_05_043_img-03.png
    Subtract that product from the first digit in the dividend. Subtract . Write the difference, 1, under the first digit in the dividend. CNX_BMath_Figure_01_05_043_img-04.png
    Bring down the next digit of the dividend. Bring down the 8. CNX_BMath_Figure_01_05_043_img-05.png
    Divide 18 by the divisor, 3. The divisor 3 goes into 18 six times. CNX_BMath_Figure_01_05_043_img-06.png
    Write 6 in the quotient above the 8.
    Multiply the 6 in the quotient by the divisor and write the product, 18, under the dividend. Subtract 18 from 18. CNX_BMath_Figure_01_05_043_img-07.png

    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 to make sure that product equals the dividend,

    It does, so our answer is correct.

    How To

    Divide whole numbers.

    1. Step 1. 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.
    2. Step 2. Write the quotient above the dividend.
    3. Step 3. Multiply the quotient by the divisor and write the product under the dividend.
    4. Step 4. Subtract that product from the dividend.
    5. Step 5. Bring down the next digit of the dividend.
    6. Step 6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
    7. Step 7. Check by multiplying the quotient times the divisor.

    Example 1.61

    Divide Check by multiplying:

    Answer

    Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_044_img-01.png
    Divide the first digit of the dividend, 2, by the divisor, 4. CNX_BMath_Figure_01_05_044_img-02.png
    Since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. The divisor 4 goes into 25 six times.
    We write the 6 in the quotient above the 5. CNX_BMath_Figure_01_05_044_img-03.png
    Multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend. CNX_BMath_Figure_01_05_044_img-04.png
    Subtract that product from the first two digits in the dividend. Subtract . Write the difference, 1, under the second digit in the dividend. CNX_BMath_Figure_01_05_044_img-05.png
    Now bring down the 9 and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19. CNX_BMath_Figure_01_05_044_img-06.png
    Bring down the 6 and repeat these steps. There are 9 fours in 36. Write the 9 over the 6. Multiply the 9 by 4 and subtract this product from 36. CNX_BMath_Figure_01_05_044_img-07.png
    So .
    Check by multiplying.
    CNX_BMath_Figure_01_05_044_img-08.png

    It equals the dividend, so our answer is correct.

    Try It 1.121

    Divide. Then check by multiplying:

    Try It 1.122

    Divide. Then check by multiplying:

    Example 1.62

    Divide Check by multiplying:

    Answer

    Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_045_img-01.png
    First we try to divide 6 into 4. CNX_BMath_Figure_01_05_045_img-02.png
    Since that won't work, we try 6 into 45.
    There are 7 sixes in 45. We write the 7 over the 5.    		 CNX_BMath_Figure_01_05_045_img-03.png
    Multiply the 7 by 6 and subtract this product from 45. CNX_BMath_Figure_01_05_045_img-04.png
    Now bring down the 0 and repeat these steps. There are 5 sixes in 30.
    Write the 5 over the 0. Multiply the 5 by 6 and subtract this product from 30.    		 CNX_BMath_Figure_01_05_045_img-05.png
    Now bring down the 6 and repeat these steps. There is 1 six in 6.
    Write the 1 over the 6. Multiply 1 by 6 and subtract this product from 6.    		 CNX_BMath_Figure_01_05_045_img-06.png
    Check by multiplying.
    CNX_BMath_Figure_01_05_045_img-07.png

    It equals the dividend, so our answer is correct.

    Try It 1.123

    Divide. Then check by multiplying:

    Try It 1.124

    Divide. Then check by multiplying:

    Example 1.63

    Divide Check by multiplying.

    Answer

    Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_046_img-01.png
    First we try to divide 9 into 7. CNX_BMath_Figure_01_05_046_img-02.png
    Since that won't work, we try 9 into 72. There are 8 nines in 72.
    We write the 8 over the 2.    		 CNX_BMath_Figure_01_05_046_img-03.png
    Multiply the 8 by 9 and subtract this product from 72. CNX_BMath_Figure_01_05_046_img-04.png
    Now bring down the 6 and repeat these steps. There are 0 nines in 6.
    Write the 0 over the 6. Multiply the 0 by 9 and subtract this product from 6.    		 CNX_BMath_Figure_01_05_046_img-05.png
    Now bring down the 3 and repeat these steps. There are 7 nines in 63. Write the 7 over the 3.
    Multiply the 7 by 9 and subtract this product from 63.    		 CNX_BMath_Figure_01_05_046_img-06.png
    Check by multiplying.
    CNX_BMath_Figure_01_05_046_img-07.png

    It equals the dividend, so our answer is correct.

    Try It 1.125

    Divide. Then check by multiplying:

    Try It 1.126

    Divide. Then check by multiplying:

    So far all the division problems have worked out evenly. For example, if we had

    An image of 28 cookies placed at random.
    Figure 1.14

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

    An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles.
    Figure 1.15

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

    To check this division we multiply times to get and then add the remainder of

    Example 1.64

    Divide Check by multiplying.

    Answer

    Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_047_img-01.png
    First we try to divide 4 into 1. Since that won't work, we try 4 into 14.
    There are 3 fours in 14. We write the 3 over the 4.    		 CNX_BMath_Figure_01_05_047_img-02.png
    Multiply the 3 by 4 and subtract this product from 14. CNX_BMath_Figure_01_05_047_img-03.png
    Now bring down the 3 and repeat these steps. There are 5 fours in 23.
    Write the 5 over the 3. Multiply the 5 by 4 and subtract this product from 23.    		 CNX_BMath_Figure_01_05_047_img-04.png
    Now bring down the 9 and repeat these steps. There are 9 fours in 39.
    Write the 9 over the 9. Multiply the 9 by 4 and subtract this product from 39.
    There are no more numbers to bring down, so we are done.
    The remainder is 3.    		 CNX_BMath_Figure_01_05_047_img-05.png
    Check by multiplying.
    CNX_BMath_Figure_01_05_047_img-06.png

    So is with a remainder of Our answer is correct.

    Try It 1.127

    Divide. Then check by multiplying:

    Try It 1.128

    Divide. Then check by multiplying:

    Example 1.65

    Divide and then check by multiplying:

    Answer

    Let's rewrite the problem to set it up for long division.
    First we try to divide 13 into 1. Since that won't work, we try 13 into 14.
    There is 1 thirteen in 14. We write the 1 over the 4.    		 CNX_BMath_Figure_01_05_048_img-02.png
    Multiply the 1 by 13 and subtract this product from 14. CNX_BMath_Figure_01_05_048_img-03.png
    Now bring down the 6 and repeat these steps. There is 1 thirteen in 16.
    Write the 1 over the 6. Multiply the 1 by 13 and subtract this product from 16.    		 CNX_BMath_Figure_01_05_048_img-04.png
    Now bring down the 1 and repeat these steps. There are 2 thirteens in 31.
    Write the 2 over the 1. Multiply the 2 by 13 and subtract this product from 31. There are no more numbers to bring down, so we are done.
    The remainder is 5. is 112 with a remainder of 5.    		 CNX_BMath_Figure_01_05_048_img-05.png
    Check by multiplying.
    CNX_BMath_Figure_01_05_048_img-06.png

    Our answer is correct.

    Try It 1.129

    Divide. Then check by multiplying:

    Try It 1.130

    Divide. Then check by multiplying:

    Example 1.66

    Divide and check by multiplying:

    Answer

    Let's rewrite the problem to set it up for long division.
    First we try to divide 241 into 7. Since that won’t work, we try 241 into 74. That still won’t work, so we try 241 into 745. Since 2 divides into 7 three times, we try 3.
    Since , we write the 3 over the 5 in 745.
    Note that 4 would be too large because , which is greater than 745.
    Multiply the 3 by 241 and subtract this product from 745. CNX_BMath_Figure_01_05_049_img-02.png
    Now bring down the 2 and repeat these steps. 241 does not divide into 222.
    We write a 0 over the 2 as a placeholder and then continue.    		 CNX_BMath_Figure_01_05_049_img-03.png
    Now bring down the 1 and repeat these steps. Try 9. Since ,
    we write the 9 over the 1. Multiply the 9 by 241 and subtract this product from 2,221.    		 CNX_BMath_Figure_01_05_049_img-04.png
    There are no more numbers to bring down, so we are finished. The remainder is 52. So
    is 309 with a remainder of 52.
    Check by multiplying.
    CNX_BMath_Figure_01_05_049_img-05.png

    Sometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them.

    Try It 1.131

    Divide. Then check by multiplying:

    Try It 1.132

    Divide. Then check by multiplying:

    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.

    Operation Word Phrase Example Expression
    Division divided by
    quotient of
    divided into    		 divided by
    the quotient of and
    divided into


    Table 1.8

    Example 1.67

    Translate and simplify: the quotient of and

    Answer

    The word quotient tells us to divide.

    We could just as correctly have translated the quotient of and using the notation

    Try It 1.133

    Translate and simplify: the quotient of and

    Try It 1.134

    Translate and simplify: the quotient of and

    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

    Cecelia bought a box of oatmeal at the big box store. She wants to divide the ounces of oatmeal into 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?

    Answer

    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.
    Simplify by dividing.
    Write a sentence to answer the question. Cecelia will get 20 servings from the big box.

    Try It 1.135

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

    Try It 1.136

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

    Media

    ACCESS ADDITIONAL ONLINE RESOURCES

    Section 1.5 Exercises

    Practice Makes Perfect

    Use Division Notation

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

    343.

    344.

    345.

    346.

    347.

    348.

    349.

    350.

    Model Division of Whole Numbers

    In the following exercises, model the division.

    351.

    352.

    353.

    354.

    355.

    356.

    357.

    358.

    Divide Whole Numbers

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

    359.

    360.

    361.

    362.

    363.

    364.

    365.

    366.

    367.

    368.

    369.

    370.

    371.

    372.

    373.

    374.

    375.

    376.

    377.

    378.

    379.

    380.

    381.

    382.

    383.

    384.

    385.

    386.

    387.

    388.

    389.

    390.

    391.

    392.

    393.

    394.

    395.

    396.

    397.

    398.

    399.

    400.

    401.

    402.

    403.

    404.

    405.

    406.

    407.

    408.

    409.

    410.

    411.

    412.

    413.

    414.

    415.

    416.

    417.

    418.

    419.

    420.

    421.

    422.

    Mixed Practice

    In the following exercises, simplify.

    423.

    424.

    425.

    426.

    427.

    428.

    429.

    430.

    Translate Word Phrases to Algebraic Expressions

    In the following exercises, translate and simplify.

    431.

    the quotient of and

    432.

    the quotient of and

    433.

    the quotient of and

    434.

    the quotient of and

    Divide Whole Numbers in Applications

    In the following exercises, solve.

    435.

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

    436.

    Crackers Evie bought a ounce box of crackers. She wants to divide it into bags with ounces of crackers in each bag. How many bags can Evie fill?

    437.

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

    438.

    Flower shop Melissa’s flower shop got a shipment of roses. She wants to make bouquets of roses each. How many bouquets can Melissa make?

    439.

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

    440.

    Dental floss One package of dental floss is feet long. Brian uses feet of dental floss every day. How many days will one package of dental floss last Brian?

    Mixed Practice

    In the following exercises, solve.

    441.

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

    442.

    Distance Mayra lives miles from her mother’s house and miles from her mother-in-law’s house. How much farther is Mayra from her mother-in-law’s house than from her mother’s house?

    443.

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

    444.

    Potting soil Aki bought a ounce bag of potting soil. How many ounce pots can he fill from the bag?

    445.

    Hiking Bill hiked miles on the first day of his backpacking trip, miles the second day, miles the third day, and miles the fourth day. What is the total number of miles Bill hiked?

    446.

    Reading Last night Emily read pages in her Business textbook, pages in her History text, pages in her Psychology text, and pages in her math text. What is the total number of pages Emily read?

    447.

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

    448.

    Scouts There are boys in Dave’s scout troop. At summer camp, each boy earned merit badges. What was the total number of merit badges earned by Dave’s scout troop at summer camp?

    Writing Exercises

    449.

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

    450.

    Cat food One bag of cat food feeds Lara’s cat for days. How many bags of cat food does Lara need for days?

    Everyday Math

    451.

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

    452.

    Oswaldo divided by and said his answer was with a remainder of How can you check to make sure he is correct?

    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?

    This page titled 1.6: Divide Whole Numbers was last modified on Sun, 28 May 2023 07:58:00 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.