Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges
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When dividing decimal numbers, remember that a whole number will have its decimal
point immediately to the right of the units digit.
To divide one decimal number by another, follow these steps.
54 Part 1 Fundamental Review
EXAMPLE H EXAMPLE I
0.76 0.5 0.380 (3 places) 12.5 1.6 20.00 (2 places)
May be written as 0.38 May be written as 20
a. Multiply 2.36 3.4 b. Multiply 0.236 0.34
2.36 (2 places) 0.236 (3 places)
3.4 (1 place) 0.34 (2 places)
944 9 44
7 08 .......70 8
8.024 (3 places) 0.08024 (5 places; insert 1 zero)
✔
CONCEPT CHECK 3.5
Dividing Decimal Numbers
6
Learning Objective
Divide one decimal number by
another decimal number.
to Divide one Decimal Number by Another
1. Arrange the divisor, dividend, and division bracket ( ) as in whole-
number long division.
2. Move the decimal point in the divisor to the right until the divisor is
a whole number. (You won’t have to move it if the divisor is already a
whole number.)
3. Move the decimal point in the dividend to the right exactly the same
number of decimal places as you did in Step 2. If necessary, attach more
zeros to the right end of the dividend. (See example K.)
4. Write the decimal point in the quotient directly above the new decimal
point in the dividend (in Step 3).
5. Write zeros, if necessary, in the quotient between the decimal point and
the first nonzero digit. (See example L.)
6. Divide as you would for whole numbers.
q
STEPS
EXAMPLE J
$8.40 6.5 $54.600 (3 places)
Should be written as $54.60
Chapter 3 Decimals 55
EXAMPLE K
215
120
2120
0
2.7 4 0.15
is
0.15q2.7 5 0.15.q2.70. 5 15.q270.
.
5 15.q270.
18.
STEP 6STEP 4STEP 3STEP 2STEP 1
EXAMPLE L
8
16
16
0
0.096 4 4
is
4q0.096 5 4.q0.096
.
5 4.q0.096
0.024
STEPS 5 & 6STEP 4STEP 1
Recall from Chapter 1 that, in long division with two whole numbers, you write a
remainder when the division doesn’t come out evenly, for example, 17 8 2 with a
remainder of 1. In division with decimals, you do not write remainders. You simply keep
dividing until you have some required number of decimal places. To get the required
number of decimal places, you may have to keep attaching zeros to the right end of the
dividend. (See example M.)
EXAMPLE M
Calculate 17 8 to three decimal places.
16
10
8
20
16
40
40
0
17 4 8
is
8q17 5 8.q17.
.
5 8.q17.
2.
5 8.q17.000
2.125
STEP 6STEP 4STEP 1
3.7 Remind students that “calculators
do not do remainders”; only humans do.
In example M, 17 4 8 5 2.125. But recall that 17 4 8 can also be written as the fraction
. 2.125 is called the decimal equivalent of . Decimal equivalents can be useful when
you are working with fractions and have a calculator available. Even with simple frac-
tions, and no calculator, it is often simpler to use decimal equivalents because you don’t
need a common denominator.
EXAMPLE N
Compute . This requires that all fractions have a common denominator of 20.
But , , and Therefore, we have
For difficult fractions, use a calculator to convert the fractions to their decimal equiva-
lents. Then use the calculator to perform the required operation. (If possible, you should
use the memory of your calculator to store the intermediate answers.)
EXAMPLE O
Compute [8] [4] [15] [5] gives 0.53333333
[7] [4] [12] [5] gives 0.58333333
[3] [4] [7] [5] gives 1
0.42857143
1.54523809
The preceeding example assumes that your calculator is displaying eight decimal
places. Also, if you use the memory to store the intermediate answers, your
calculator may round off the intermediate answers and give you a final answer of
1.54523810 or 1.5452381. A few calculators make it easier to compute fractions with
an “algebraic operating system” that automatically does multiplication and division
before addition and subtraction. For those calculators, you might use keystrokes
like these:
[8] [4] [15] [1] [7] [4] [12] [1] [3] [4] [7] [5] 1.5452380, or possibly 1.5452381
8
15
1
7
12
1
3
7
.
5 0.85.
1
2
1
3
4
2
2
5
5 0.5 1 0.75 2 0.4
2
5
5 0.4.
3
4
5 0.75
1
2
5 0.5
1
2
1
3
4
2
2
5
17
8
17
8
56 Part 1 Fundamental Review
✔
CONCEPT CHECK 3.6
a. Divide 1.026 by 15. b. Divide 0.009 by 0.4.
90 8
126 10
120 8
60 20
60 20
00
0.4.q0.0.09
.
5 4q0.0900
0.0225
15q1.026
.
5 15q1.0260
0.0684
3.8 There are also relatively inexpen-
sive ($35) calculators that allow the
student to change between “chain”and
“algebraic”ordering of arithmetic op-
erations.(The Texas Instruments BA II
Plus is one example.)
Learning
Objective
In Chapter 1, we showed simple multiplication and division shortcuts when the multi-
plier or the divisor is a whole number ending in zeros (e.g., 30, 200, or 1,000). The same
shortcuts may be used with decimal numbers. We just “manage the decimal point.”
If the multiplier is 10, 100, 1,000, etc., there is just one step.
Step 1 Move the decimal point in the multiplicand to the right the same num-
ber of places as the number of zeros in the multiplier. (See example P.) If
necessary, attach zeros to the right end of the multiplicand before mul-
tiplying. (See example Q.)
EXAMPLE P EXAMPLE Q
0.56 10 0.5.6 5.6 4.73 1,000 4.730 4,730
(1 place) (3 places)
If the multiplier ends in zeros but has a first digit that is not 1 (for example, 300 or
2,000), there are two steps.
Step 1 Multiply the multiplicand by the nonzero part of the multiplier.
Step 2 Move the decimal point in the product from Step 1 to the right the
same number of places as the number of zeros in the multiplier.
EXAMPLE R
Multiply 3.431 by 2,000
Multiply by 2: 3.431 2 6.862
Move the decimal point three places to the right: 6.862. 6,862.
If the divisor is 10, 100, 1,000, etc., there is just one step.
Step 1 Move the decimal point in the dividend to the left the same number
of places as the number of zeros in the divisor. (See example S.) If
necessary, attach zeros to the left end of the dividend. (See example T.)
EXAMPLE S EXAMPLE T
735.1 100 9.64 1,000
735.1 100 7.35.1 7.351 9.64 1,000 .009.64 0.00964
(2 places) (3 places)
Other calculators that do not have an “algebraic operating system” may have parentheses,
permitting this type of calculation:
[(] [8] [4] [15] [)] [1] [(] [7] [4] [12] [)] [1] [(] [3] [4] [7] [)] [5] 1.5452380, or possi-
bly 1.5452381
Chapter 3 Decimals 57
Using Multipliers and Divisors that End
with Zeros
Multiply and divide by decimal num-
bers that end with zeros.
7
Learning Objective
✔
CONCEPT CHECK 3.7
If the divisor ends in zeros but has a first digit that is not 1 (for example, 300 or
2,000), there are two steps.
Step 1 Divide the dividend by the nonzero part of the divisor.
Step 2 Move the decimal point in the quotient from Step 1 to the left the same
number of places as the number of zeros in the divisor.
EXAMPLE U
Divide 615.24 by 300
Divide by 3: 615.24 3 205.08
Move the decimal point two places to the left: 2.05.08 2.0508
58 Part 1 Fundamental Review
a. Multiply 0.413 by 300 b. Divide 4.375 by 10
0.413 3 1.239
Move the decimal point Move the decimal point
two places to the right: one place to the left:
1.23.9 123.9 4.375 10 .4.375 0.4375
8
Learning Objective
Approximate products and
quotients.
Business people today almost always use calculators or computers to do important com-
putations. But calculators are perfect only if every single key is pressed correctly. Often,
you can discover a calculator error by doing some simple mental approximations. The
objective is to determine whether the answer is approximately the right size—that is,
whether the decimal point is in the correct position. To do so, we round each decimal
number to only one nonzero digit and all the rest to zeros. Follow these steps.
Approximating Products and Quotients
to Approximate a Multiplication Problem
1. Round the first nonzero digit from the left end in each factor. (How does
the digit to its right compare to 5?)
2. Change all the digits to the right of the first nonzero digit to zero.
3. Multiply the two new factors.
4. Place the decimal point in the product.
STEPS
EXAMPLE V EXAMPLE W
Approximate 3.764 7.4 Approximate 0.089 61.18
3.764 4.000 4 0.089 0.090 0.09
7.4 7.0 7 61.18 60.00 60
28 5.40
The actual answers are 27.8536 and 5.44502.
In division, the mental approximation will be easier if you change the decimal num-
bers so that the division will end evenly after one step. To do this, first round the divisor
to one nonzero digit and then round the dividend to two nonzero digits, evenly divisible
by the new divisor.
STEPS 2 & 3STEP 1STEPS 2 & 3STEP 1
Chapter 3 Decimals 59
to Approximate a Division Problem
1. Round the divisor to a single nonzero digit at the left, followed by all zeros.
2. Round the dividend to a two-digit number at the left, followed by all
zeros.Select the two-digit number so that it is evenly divisible by the
new divisor.
3. Divide the new dividend by the new divisor.
4. Place the decimal point correctly in the quotient.
STEPS
EXAMPLE X
Approximate 4.764 8.1
0.6
2
4.8
0
EXAMPLE Y
Approximate 61.18 0.089
700.
63
0
The actual answers are 0.5881 and 687.4157 (to four decimal places).
9.q6300.0.09q63.000.090q61.180.089q61.18
STEPS 3 & 4STEP 2STEP 1
8.q4.88.q4.8008.0q4.7648.1q4.764
STEPS 3 & 4STEP 2STEP 1
3.9 In example Y,we say that 0.089 has
two significant digits,0.090 has one
significant digit,and 63.00 has two
significant digits.Significant digits are
nonzero digits that are either preceded
or followed by all zeros.
60 Part 1 Fundamental Review
✔
CONCEPT CHECK 3.8
a. Approximate 6.891 0.614 b. Approximate 0.0738 92.65
Remember to round off the divisor first.
6.891 7.000 92.65 90.00
0.614 0.600 0.0738 0.0720
0.6 (1 place)
7 (0 places)
4.2 (1 place) 720
0
Compare with 6.891 0.614 4.231074 Compare with 0.0738 92.65 0.000796546
COMPLETE ASSIGNMENTS 3.2 and 3.3
90q0.0720
.0008
90q0.072
.
decimal equivalent
decimal places
decimal point
mixed decimal
pure decimal
rounding off
Chapter Terms for Review
Try Microsoft
®
Excel
Try working the problems using the Microsoft Excel templates found on your student
CD. Solutions for the problems are also shown on the CD.
1. Set up and complete the following tables using the appropriate Excel formulas.
Date Auto Sales Part Sales Total Sales
6/4/10 $ 36,628.14 $ 1,782.28
6/5/10 $ 42,789.40 $ 2,047.33
6/6/10 $ 58,334.98 $ 1,132.48
6/7/10 $ 96,782.04 $ 3,006.04
6/8/10 $ 29,765.55 $ 2,333.33
Total
Date Total Receipts Total Cash Cash Short
7/15/10 $ 974.58 $ 969.30
7/16/10 $ 888.07 $ 888.02
7/17/10 $ 1,384.17 $ 1,350.23
Total
Date Units Sold Price Per Unit Total Sales
5/24/10 47 $ 107.16
5/25/10 63 $ 107.16
5/26/10 72 $ 107.16
5/27/10 39 $ 107.16
Total
Date Total Sale Price Per Unit Units Sold
5/24/10 $ 5,036.52 $ 107.16
5/25/10 $ 6,751.08 $ 107.16
5/26/10 $ 7,715.52 $ 107.16
5/27/10 $ 4,179.24 $ 107.16
Total
THE BOTTOM LINE
Chapter 3 Decimals 61
Summary of chapter learning objectives:
Learning Objective
3.1
Read decimal numbers.
Example
1. Write 8.427 using words.
2. Write forty-one and eleven ten-thousandths using digits.
Answers: 1. eight and four hundred twenty-seven thousandths 2. 41.0011 3. 0.506 4. 13.27 5. 99.862
6. 4.7423 7. 16.38 8. 0.021 9. 0.0256 10. 69.782 11. 0.09462 12. 1.246 13. 0.282 14. 25 15. 0.4
3.2
Round decimal numbers.
3. Round 0.506489 to the nearest thousandth (that is, to
three decimal places).
4. Round up 13.26012 to the next hundredth (that is, to
two decimal places).
3.3
Add two or more decimal numbers.
5. Add 82.9, 14.872, and 2.09.
3.4
Subtract one decimal number from another.
6. Subtract 14.5977 from 19.34.
3.5
Multiply two decimal numbers.
7. Multiply: 4.68 3.5
3.6
Divide one decimal number by another decimal number.
8. Divide: 0.084 4
9. Divide: 0.064 2.5
3.7
Multiply and divide by decimals that end with zeros.
10. Multiply: 0.069782 1000
11. Divide: 9.462 by 100
12. Multiply: 0.0623 20
13. Divide: 84.6 by 300
3.8
Approximate products and quotients.
14. Approximate 48.79 0.47
15. Approximate 0.2688 0.713
1 Write “one hundred sixteen and fourteen ten-thousandths” as a number
2 Write 6,431.719, using words
SELF-CHECK
Review Problems for Chapter 3
62 Part 1 Fundamental Review
3 Round 3.475 feet to the nearest tenth 4 Round $12.667 to the nearest cent
5 Add 3.79475 and 739.85 6 Add 12.42, 0.087, and 8.3
7 Subtract 8.693 from 11.41 8 Subtract 287.963 from 410.4511
9 Multiply 3.722 by 0.483 (do not
round off)
10 Multiply $19.75 by 22.45 (round off to the
nearest cent)
In problems 11 and 12, divide to three places and round to the nearest hundredth.
11 Divide 45.88 by 14.2 12 Divide $6.25 by 8.41
In problems 13 and 14, use shortcuts to solve each problem and round to the nearest hundredth.
13 Multiply 86.493 by 100 14 Divide $2,762.35 by 1,000
In problems 15 and 16, pick the best approximate answers from the possible answers.
15 Multiply 48.98 by 11.2 (a) 0.5 (b) 5 (c) 50 (d) 500 (e) 5,000
16 Divide $6.65 by 8.21 (a) $0.008 (b) $0.08 (c) $0.80 (d) $8.00 (e) $80.0
17 DeLois McBryde owns a chain of very large, upscale bookstores. She decides to start selling coffee drinks such
as espresso and cappuccino at one of her stores. During the first day, the store has total sales of $4,188.25. Of the
total, $362.50 was from coffee drinks. How much of the total was from books and other items?
18 Gary Gehlert operates tennis and golf shops at a desert resort. Last year, he started selling on the Internet as well.
He had the following profits last year: Tennis (shop), $52,418.12; Golf (shop), $168,078.51; Tennis (Internet),
$28,570.12; and Golf (Internet), $32,904.82. What were the total profits from these sources?
19 Dean Treggas, a landscape contractor, needed to plant 226 one-gallon plants and 164 five-gallon plants. Dean uses
about 0.8 cubic foot of planting soil for each one-gallon plant and 2.5 cubic feet of planting soil for each five-gallon
plant. How many cubic feet of planting soil will Dean need for all these plants?
20 Planting soil is sold by the cubic yard. To two decimals, how many cubic yards of planting soil will Dean Treggas
need to do his planting in question 19? (Hint: 1 cubic yard equals 27 cubic feet.)
Answers to the Self-Check can be found in Appendix B at the back of the text.
Chapter 3 Decimals 63
Assignment 3.1: Addition and Subtraction of Decimal Numbers
Name
Date Score
A (13 points) Use digits to write each number that is expressed in words. Use words to write each number
that is expressed in digits. (1 point for each correct answer)
1. Six hundred thirteen ten-thousandths
2. Nineteen thousandths
3. Sixty-four hundredths
4. Seventy-six and seventy-one ten-thousandths
5. Eight hundred sixty and ninety-eight hundred-thousandths
6. Eighteen and six thousandths
7. 308.97
8. 0.0004
9. 492.3
10. 0.081
11. 42.0481
12. 6.018
13. 1,007.4
Score for A (13)
B (24 points) Round as indicated. (1 point for each correct answer)
Nearest Tenth Nearest Cent
14. 6.3517 qt 20. $17.375
15. 48.97 mi 21. $0.098
16. 3.824 gal 22. $942.3449
17. 374.29 lb 23. $8.1047
18. 7.35 ft 24. $0.0449
19. 6.375 oz 25. $52.996
Nearest Thousandth UP to the Next Cent
26. 5.37575 pt 32. $9.681
27. 0.00549 gal 33. $0.159
28. 14.6445 oz 34. $72.535
29. 8.1855 in. 35. $2.0917
30. 8.9989 mi 36. $11.4405
31. 0.200499 lb 37. $0.6545
Score for B (24)
$0.660.200 lb
$11.458.999 mi
$2.108.186 in.
$72.5414.645 oz
$0.160.005 gal
$9.695.376 pt
$53.006.4 oz
$0.047.4 ft
$8.10374.3 lb
$942.343.8 gal
$0.1049.0 mi
$17.386.4 qt
one thousand seven and four tenths
six and eighteen thousandths
forty-two and four hundred eighty-one ten-thousandths
eighty-one thousandths
four hundred ninety-two and three tenths
four ten-thousandths
three hundred eight and ninety-seven hundredths
18.006
860.00098
76.0071
0.64
0.019
0.0613
1
Learning Objectives
2 3 4