- Division with a Single Digit Divisor
- Division with a Multiple Digit Divisor
- Division with a Remainder
- Calculators
Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to divide whole numbers. By the end of the module students should be able to be able to divide a whole number by a single or multiple digit divisor and interpret a calculator statement that a division results in a remainder.
Our experience with multiplication of whole numbers allows us to perform such divisions as
Let's examine the division
We can suggest from this discussion that the process of division consists of
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
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.
Now determine how close the estimate is.
10 fives is
Estimate the number of 5's in 25.
There are exactly 5 fives in 25.
Check:
Thus,
The notation in this division can be shortened by writing.
Find
Rewrite the problem using a division bracket.
8 goes into 49 at most 6 times, and 9 is in the hundreds column. We'll guess 600.
Then,
8 goes into 14 at most 1 time, and 4 is in the tens column. We'll guess 10.
8 goes into 64 exactly 8 times.
600 eights + 10 eights + 8 eights = 618 eights.
Check:
Thus,
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.
Perform the following divisions.
18
81
853
1,129
84,125
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
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:
Try 6:
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:
Check:
Thus,
Find
First, the educated guess:
Subtract: 2417 - 2406 = 11.
Bring down the 2.
The divisor 802 goes into 112 at most 0 times. Use 0.
The 8 goes into 11 at most 1 time, and
Subtract
Bring down the 8.
8 goes into 32 at most 4 times.
Use 4.
Check:
Thus,
Perform the following divisions.
43
112
305
3,016
We might wonder how many times 4 is contained in 10. Repeated subtraction yields
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.
Since 4 does not divide into 2 (the remainder is less than the divisor) and there are no digits to bring down to continue the process, we are done. We write
Find
There are no more digits to bring down to continue the process. We are done. One is the remainder.
Check: Multiply 28 and 3, then add 1.
Thus,
Find
Check: Multiply 31 by 23, then add 13.
Thus,
Perform the following divisions.
18 R3
43 R2
23 R6
61 R14
614 R58
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.
Type | 328 |
Press | ÷ |
Type | 8 |
Press | = |
The display now reads 41.
Type | 53136 |
Press | ÷ |
Type | 82 |
Press | = |
The display now reads 648.
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.
Type | 3727 |
Press | ÷ |
Type | 49 |
Press | = |
The display now reads 76.061224.
This number is an example of a decimal number (see (Reference)). When a decimal number results in a calculator division, we can conclude that the division produces a remainder.
Use a calculator to perform each division.
45
551
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
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.
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.
13
67
52
3
70
61
59
67
87
54
52
38
45
777
342
644
533
10,440
8,147 remainder 847
4 remainder 2
66 remainder 1
823 remainder 2
40 remainder 28
665 remainder 15
957 remainder 34
665 remainder 4
458 remainder 13
996 remainder 23
For the following 5 problems, use a calculator to find the quotients.
42
8,216
A mathematics instructor at a high school is paid $17,775 for 9 months. How much money does this instructor make each month?
$1,975 per month
A couple pays $4,380 a year for a one-bedroom apartment. How much does this couple pay each month for this apartment?
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?
$485 each person invested
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?
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?
14 cubes per hour
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?
A computer program consists of 68,112 bits. 68,112 bits equals 8,514 bytes. How many bits in one byte?
8 bits in each byte
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?
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?
38
((Reference)) What is the value of 4 in the number 124,621?
((Reference)) Round 604,092 to the nearest hundred thousand.
600,000
((Reference)) What whole number is the additive identity?
((Reference)) Find the product.
625,600
((Reference)) Find the quotient.
"Used as supplemental materials for developmental math courses."