2.3: Division of Whole Numbers
- be able to divide a whole number by a single or multiple digit divisor
- be able to interpret a calculator statement that a division results in a remainder
Division with a Single Digit Divisor
Our experience with multiplication of whole numbers allows us to perform such divisions as \(75 \div 5\). We perform the division by performing the corresponding multiplication, \(5 \times Q = 75\). Each division we considered in [link] had a one-digit quotient. Now we will consider divisions in which the quotient may consist of two or more digits. For example, \(75 \div 5\).
Let's examine the division \(75 \div 5\). We are asked to determine how many 5's are contained in 75. We'll approach the problem in the following way.
- Make an educated guess based on experience with multiplication.
- Find how close the estimate is by multiplying the estimate by 5.
- If the product obtained in step 2 is less than 75, find out how much less by subtracting it from 75.
- If the product obtained in step 2 is greater than 75, decrease the estimate until the product is less than 75. Decreasing the estimate makes sense because we do not wish to exceed 75.
We can suggest from this discussion that the process of division consists of
The Four Steps in Division
- an educated guess
- a multiplication
- a subtraction
- bringing down the next digit (if necessary)
The educated guess can be made by determining how many times the divisor is contained in the dividend by using only one or two digits of the dividend.
Find \(75 \div 5\).
Solution
\(5\overline{)75}\) Rewrite the problem using a division bracket.
\(\begin{array} {r} {10} \\ {5\overline{)75}} \end{array}\)
Make an educated guess by noting that one 5 is contained in 75 at most 10 times.
Since 7 is the tens digit, we estimate that 5 goes into 75 at most 10 times.
\(\begin{array} {r} {10} \\ {5\overline{)75}} \\ {\underline{-50}} \\ {25} \end{array}\)
Now determine how close the estimate is.
10 fives is \(10 \times 5 = 50\). Subtract 50 from 75.
Estimate the number of 5's in 25.
There are exactly 5 fives in 25.
Check:
Thus, \(75 \div 5 = 15\).
The notation in this division can be shortened by writing.
\(\begin{array} {r} {15} \\ {5\overline{)75}} \\ {-5 \downarrow} \\ {\overline{\ \ 25}} \\ {-25} \\ {\overline{\ \ \ \ 0}} \end{array}\)
\(\begin{cases} \text{Divide:} & \text{5 goes into 7 at most 1 time.} \\ \text{Multiply:} & 1 \times 5 = 5. \text{Write 5 below 7.} \\ \text{Subtract:} & 7 - 5 = 2. \text{Bring down the 5.} \end{cases}\) \(\begin{cases} {\text{Divide:}} & {\text{5 goes into 25 exactly 5 times}} \\ {\text{Multiply:}} & {5 \times 5 = 25. \text{Write 25 below 25.}} \\ {\text{Subtract:}} & {25 - 25 = 0} \end{cases}\)
Find \(4,944 \div 8\).
Solution
\(8\overline{)4944}\)
Rewrite the problem using a division bracket.
\(\begin{array} {r} {600} \\ {8\overline{)4944}} \\ {\underline{-4800}} \\ {144} \end{array}\)
8 goes into 49 at most 6 times, and 9 is in the hundreds column. We'll guess 600.
Then, \(8 \times 600 = 4800\).
\(\begin{array} {r} {10} \\ {600} \\ {8\overline{)4944}} \\ {\underline{-4800}} \\ {144} \\ {\underline{-\ \ 80}} \\ {64} \end{array}\)
8 goes into 14 at most 1 time, and 4 is in the tens column. We'll guess 10.
\(\begin{array} {r} {8} \\ {10} \\ {600} \\ {8\overline{)4944}} \\ {\underline{-4800}} \\ {144} \\ {\underline{-\ \ 80}} \\ {64} \\ {\underline{-64}} \\ {0} \end{array}\)
8 goes into 64 exactly 8 times.
600 eights + 10 eights + 8 eights = 618 eights.
Check:
Thus, \(4,944 \div 8 = 618\).
As in the first problem, the notation in this division can be shortened by eliminating the subtraction signs and the zeros in each educated guess.
\(\begin{cases} \text{Divide:} & \text{8 goes into 49 at most 6 times.} \\ \text{Multiply:} & 6 \times 8 = 48. \text{Write 48 below 49.} \\ \text{Subtract:} & 49 - 48 = 1. \text{Bring down the 4.} \end{cases}\) \(\begin{cases} \text{Divide:} & \text{8 goes into 14 at most 1 time.} \\ \text{Multiply:} & 1 \times 8 = 8. \text{Write 8 below 14.} \\ \text{Subtract:} & 14 - 8 = 6. \text{Bring down the 4.} \end{cases}\)
\(\begin{cases} \text{Divide:} & \text{8 goes into 64 exactly 8 times.} \\ \text{Multiply:} & 8 \times 8 = 64. \text{Write 64 below 64.} \\ \text{Subtract:} & 64 - 64 = 0. \end{cases}\)
Not all divisions end in zero. We will examine such divisions in a subsequent subsection.
Practice Set A
Perform the following divisions.
\(126 \div 7\)
- Answer
-
18
Practice Set A
\(324 \div 4\)
- Answer
-
81
Practice Set A
\(2,559 \div 3\)
- Answer
-
853
Practice Set A
\(5,645 \div 5\)
- Answer
-
1,129
Practice Set A
\(757,125 \div 9\)
- Answer
-
84,125
Division with a Multiple Digit Divisor
The process of division also works when the divisor consists of two or more digits. We now make educated guesses using the first digit of the divisor and one or two digits of the dividend.
Find \(2,232 \div 36\).
Solution
\(36 \overline{)2232}\)
Use the first digit of the divisor and the first two digits of the dividend to make the educated guess.
3 goes into 22 at most 7 times.
Try 7: \(7 \times 36 = 252\) which is greater than 223. Reduce the estimate.
Try 6: \(6 \times 36 = 216\) which is less than 223.
\(\begin{array} {ll} {\text{Multiply: }} & {6 \times 36 = 216. \text{Write 216 below 223.}} \\ {\text{Subtract: }} & {223 - 216 = 7. \text{Bring down the 2.}} \end{array}\)
Divide 3 into 7 to estimate the number of times 36 goes into 72. The 3 goes into 7 at most 2 times.
Try 2: \(2 \times 36 = 72\).
Check :
Thus \(2,232 \div 36 = 62\).
Find \(2,417,228 \div 802\).
Solution
\(802 \overline{)2417228}\)
First, the educated guess: \(24 \div 8 = 3\). Then \(3 \times 802 = 2406\), which is less than 2417. Use 3 as the guess. Since \(3 \times 802 = 2406\), and 2406 has four digits, place the 3 above the fourth digit of the dividend.
Subtract: 2417 - 2406 = 11.
Bring down the 2.
The divisor 802 goes into 112 at most 0 times. Use 0.
\(\begin{array} {ll} {\text{Multiply:}} & {0 \times 802 = 0.} \\ {\text{Subtract:}} & {112 - 0 = 112.} \\ {\text{Bring down the 2.}} & {} \end{array}\)
The 8 goes into 11 at most 1 time, and \(1 \times 802 = 802\), which is less than 1122. Try 1.
Subtract 1122 - 802 = 320
Bring down the 8.
8 goes into 32 at most 4 times.
\(4 \times 802 = 3208\).
Use 4.
Check:
Thus, \(2,417,228 \div 802 = 3,014\).
Practice Set B
Perform the following divisions.
\(1,376 \div 32\)
- Answer
-
43
Practice Set B
\(6,160 \div 55\)
- Answer
-
112
Practice Set B
\(18,605 \div 61\)
- Answer
-
305
Practice Set B
\(144,768 \div 48\)
- Answer
-
3,016
Division with a Remainder
We might wonder how many times 4 is contained in 10. Repeated subtraction yields
\(\begin{array} {r} {10} \\ {\underline{-\ \ 4}} \\ {6} \\ {\underline{-4}} \\ {2} \end{array}\)
Since the remainder is less than 4, we stop the subtraction. Thus, 4 goes into 10 two times with 2 remaining. We can write this as a division as follows.
\(\begin{array} {r} {2} \\ {4 \overline{)10}} \\ {\underline{-\ \ 8}} \\ {2} \end{array}\)
\(\begin{array} {ll} {\text{Divide:}} & {\text{4 goes into 10 at most 2 times.}} \\ {\text{Multiply:}} & {2 \times 4 = 8. \text{Write 8 below 0.}} \\ {\text{Subtract:}} & {10 - 8 = 2.} \end{array}\)
\(\begin{array} {r} {\text{2R2}} \\ {4\overline{)\ \ \ 10}} \\ {\underline{-8}} \\ {2} \end{array}\) or \(10 \div 4 = \begin{matrix} \underbrace{\text{2R2}} \\ {\text{2 with remainder 2}} \end{matrix}\)
Find \(85 \div 3\).
Solution
\(\begin{cases} \text{Divide:} & \text{3 goes into 8 at most 2 times.} \\ \text{Multiply:} & 2 \times 3 = 6. \text{ Write 6 below 8.} \\ \text{Subtract:} & 8 - 6 = 2. \text{ Bring down the 5.} \end{cases}\) \(\begin{cases} \text{Divide:} & \text{3 goes into 25 at most 8 times.} \\ \text{Multiply:} & 3 \times 8 = 24. \text{ Write 24 below 25.} \\ \text{Subtract:} & 25 - 24 = 1 \end{cases}\)
Find \(726 \div 23\).
Solution
Check: Multiply 31 by 23, then add 13.
Thus, \(726 \div 23 = 31R13\).
Practice Set C
Perform the following divisions.
\(75 \div 4\)
- Answer
-
18 R3
Practice Set C
\(346 \div 8\)
- Answer
-
43 R2
Practice Set C
\(489 \div 21\)
- Answer
-
23 R6
Practice Set C
\(5,016 \div 82\)
- Answer
-
61 R14
Practice Set C
\(41,196 \div 67\)
- Answer
-
614 R58
Calculators
The calculator can be useful for finding quotients with single and multiple digit divisors. If, however, the division should result in a remainder, the calculator is unable to provide us with the particular value of the remainder. Also, some calculators (most nonscientific) are unable to perform divisions in which one of the numbers has more than eight digits.
Use a calculator to perform each division.
\(328 \div 8\).
Solution
| Type | 328 |
| Press | \(\div\) |
| Type | 8 |
| Press | = |
The display now reads 41.
\(53,136 \div 82\).
Solution
| Type | 53136 |
| Press | \(\div\) |
| Type | 82 |
| Press | = |
The display now reads 648.
\(730,019,001 \div 326\)
Solution
We first try to enter 730,019,001 but find that we can only enter 73001900. If our calculator has only an eight-digit display (as most nonscientific calculators do), we will be unable to use the calculator to perform this division.
\(3727 \div 49\).
Solution
| Type | 3727 |
| Press | \(\div\) |
| Type | 49 |
| Press | = |
The display now reads 76.061224.
This number is an example of a decimal number (see [link] ). When a decimal number results in a calculator division, we can conclude that the division produces a remainder.
Practice Set D
Use a calculator to perform each division.
\(3,330 \div 74\)
- Answer
-
45
Practice Set D
\(63,365 \div 115\)
- Answer
-
551
Practice Set D
\(21,996,385,287 \div 53\)
- Answer
-
Since the dividend has more than eight digits, this division cannot be performed on most nonscientific calculators. On others, the answer is 415,026,137.4
Practice Set D
\(4,558 \div 67\)
- Answer
-
This division results in 68.02985075, a decimal number, and therefore, we cannot, at this time, find the value of the remainder. Later, we will discuss decimal numbers.
Exercises
For the following problems, perform the divisions.
The first 38 problems can be checked with a calculator by multiplying the divisor and quotient then adding the remainder.
Exercise \(\PageIndex{1}\)
\(52 \div 4\)
- Answer
-
13
Exercise \(\PageIndex{2}\)
\(776 \div 8\)
Exercise \(\PageIndex{3}\)
\(603 \div 9\)
- Answer
-
67
Exercise \(\PageIndex{4}\)
\(240 \div 8\)
Exercise \(\PageIndex{5}\)
\(208 \div 4\)
- Answer
-
52
Exercise \(\PageIndex{6}\)
\(576 \div 6\)
Exercise \(\PageIndex{7}\)
\(21 \div 7\)
- Answer
-
3
Exercise \(\PageIndex{8}\)
\(0 \div 0\)
Exercise \(\PageIndex{9}\)
\(140 \div 2\)
- Answer
-
70
Exercise \(\PageIndex{10}\)
\(528 \div 8\)
Exercise \(\PageIndex{11}\)
\(244 \div 4\)
- Answer
-
61
Exercise \(\PageIndex{12}\)
\(0 \div 7\)
Exercise \(\PageIndex{13}\)
\(177 \div 3\)
- Answer
-
59
Exercise \(\PageIndex{14}\)
\(96 \div 8\)
Exercise \(\PageIndex{15}\)
\(67 \div 1\)
- Answer
-
67
Exercise \(\PageIndex{16}\)
\(896 \div 56\)
Exercise \(\PageIndex{17}\)
\(1,044 \div 12\)
- Answer
-
87
Exercise \(\PageIndex{18}\)
\(988 \div 19\)
Exercise \(\PageIndex{19}\)
\(5,238 \div 97\)
- Answer
-
54
Exercise \(\PageIndex{20}\)
\(2530 \div 55\)
Exercise \(\PageIndex{21}\)
\(4,264 \div 82\)
- Answer
-
52
Exercise \(\PageIndex{22}\)
\(637 \div 13\)
Exercise \(\PageIndex{23}\)
\(3,420 \div 90\)
- Answer
-
38
Exercise \(\PageIndex{24}\)
\(5,655 \div 87\)
Exercise \(\PageIndex{25}\)
\(2,115 \div 47\)
- Answer
-
45
Exercise \(\PageIndex{26}\)
\(9,328 \div 22\)
Exercise \(\PageIndex{27}\)
\(55,167 \div 71\)
- Answer
-
777
Exercise \(\PageIndex{28}\)
\(68,356 \div 92\)
Exercise \(\PageIndex{29}\)
\(27,702 \div 81\)
- Answer
-
342
Exercise \(\PageIndex{30}\)
\(6,510 \div 31\)
Exercise \(\PageIndex{31}\)
\(60,536 \div 94\)
- Answer
-
644
Exercise \(\PageIndex{32}\)
\(31,844 \div 38\)
Exercise \(\PageIndex{33}\)
\(23,985 \div 45\)
- Answer
-
533
Exercise \(\PageIndex{34}\)
\(60,606 \div 74\)
Exercise \(\PageIndex{35}\)
\(2,975,400 \div 285\)
- Answer
-
10,440
Exercise \(\PageIndex{36}\)
\(1,389,660 \div 795\)
Exercise \(\PageIndex{37}\)
\(7,162,060 \div 879\)
- Answer
-
8,147 remainder 847
Exercise \(\PageIndex{38}\)
\(7,561,060 \div 909\)
Exercise \(\PageIndex{39}\)
\(38 \div 9\)
- Answer
-
4 remainder 2
Exercise \(\PageIndex{40}\)
\(97 \div 4\)
Exercise \(\PageIndex{41}\)
\(199 \div 3\)
- Answer
-
66 remainder 1
Exercise \(\PageIndex{42}\)
\(573 \div 6\)
Exercise \(\PageIndex{43}\)
\(10,701 \div 13\)
- Answer
-
823 remainder 2
Exercise \(\PageIndex{44}\)
\(13,521 \div 53\)
Exercise \(\PageIndex{45}\)
\(3,628 \div 90\)
- Answer
-
40 remainder 28
Exercise \(\PageIndex{46}\)
\(10,592 \div 43\)
Exercise \(\PageIndex{47}\)
\(19,965 \div 30\)
- Answer
-
665 remainder 15
Exercise \(\PageIndex{48}\)
\(8,320 \div 21\)
Exercise \(\PageIndex{49}\)
\(61,282 \div 64\)
- Answer
-
957 remainder 34
Exercise \(\PageIndex{50}\)
\(1,030 \div 28\)
Exercise \(\PageIndex{51}\)
\(7,319 \div 11\)
- Answer
-
665 remainder 4
Exercise \(\PageIndex{52}\)
\(3,628 \div 90\)
Exercise \(\PageIndex{53}\)
\(35,279 \div 77\)
- Answer
-
458 remainder 13
Exercise \(\PageIndex{54}\)
\(52,196 \div 68\)
Exercise \(\PageIndex{55}\)
\(67,751 \div 68\)
- Answer
-
996 remainder 23
For the following 5 problems, use a calculator to find the quotients.
Exercise \(\PageIndex{56}\)
\(4,346 \div 53\)
Exercise \(\PageIndex{57}\)
\(3,234 \div 77\)
- Answer
-
42
Exercise \(\PageIndex{58}\)
\(6,771 \div 37\)
Exercise \(\PageIndex{59}\)
\(4,272,320 \div 520\)
- Answer
-
8,216
Exercise \(\PageIndex{60}\)
\(7,558,110 \div 651\)
Exercise \(\PageIndex{61}\)
A mathematics instructor at a high school is paid $17,775 for 9 months. How much money does this instructor make each month?
- Answer
-
$1,975 per month
Exercise \(\PageIndex{62}\)
A couple pays $4,380 a year for a one-bedroom apartment. How much does this couple pay each month for this apartment?
Exercise \(\PageIndex{63}\)
Thirty-six people invest a total of $17,460 in a particular stock. If they each invested the same amount, how much did each person invest?
- Answer
-
$485 each person invested
Exercise \(\PageIndex{64}\)
Each of the 28 students in a mathematics class buys a textbook. If the bookstore sells $644 worth of books, what is the price of each book?
Exercise \(\PageIndex{65}\)
A certain brand of refrigerator has an automatic ice cube maker that makes 336 ice cubes in one day. If the ice machine makes ice cubes at a constant rate, how many ice cubes does it make each hour?
- Answer
-
14 cubes per hour
Exercise \(\PageIndex{66}\)
A beer manufacturer bottles 52,380 ounces of beer each hour. If each bottle contains the same number of ounces of beer, and the manufacturer fills 4,365 bottles per hour, how many ounces of beer does each bottle contain?
Exercise \(\PageIndex{67}\)
A computer program consists of 68,112 bits. 68,112 bits equals 8,514 bytes. How many bits in one byte?
- Answer
-
8 bits in each byte
Exercise \(\PageIndex{68}\)
A 26-story building in San Francisco has a total of 416 offices. If each floor has the same number of offices, how many floors does this building have?
Exercise \(\PageIndex{69}\)
A college has 67 classrooms and a total of 2,546 desks. How many desks are in each classroom if each classroom has the same number of desks?
- Answer
-
38
Exercises for Review
Exercise \(\PageIndex{70}\)
What is the value of 4 in the number 124,621?
Exercise \(\PageIndex{71}\)
Round 604,092 to the nearest hundred thousand.
- Answer
-
600,000
Exercise \(\PageIndex{72}\)
What whole number is the additive identity?
Exercise \(\PageIndex{73}\)
Find the product. \(6,256 \times 100\).
- Answer
-
625,600
Exercise \(\PageIndex{74}\)
Find the quotient. \(0 \div 11\)