Dividing Decimals 4.5 OBJECTIVES

4.5
Dividing Decimals
4.5
OBJECTIVES
1.
2.
3.
4.
Divide a decimal by a whole number
Divide a decimal by a decimal
Divide a decimal by a power of ten
Apply division to the solution of an application problem
The division of decimals is very similar to our earlier work with dividing whole numbers.
The only difference is in learning to place the decimal point in the quotient. Let’s start with the
case of dividing a decimal by a whole number. Here, placing the decimal point is easy. You
can apply the following rule.
Step by Step: To Divide a Decimal by a Whole Number
Step 1 Place the decimal point in the quotient directly above the decimal
point of the dividend.
Step 2 Divide as you would with whole numbers.
Example 1
Dividing a Decimal by a Whole Number
Divide 29.21 by 23.
NOTE Do the division just as if
you were dealing with whole
numbers. Just remember to
place the decimal point in the
quotient directly above the one
in the dividend.
1.27
2329.21
23
62
46
1 61
1 61
0
The quotient is 1.27.
CHECK YOURSELF 1
Divide 80.24 by 34.
© 2001 McGraw-Hill Companies
Let’s look at another example of dividing a decimal by a whole number.
Example 2
Dividing a Decimal by a Whole Number
Divide 122.2 by 52.
NOTE Again place the decimal
point of the quotient above
that of the dividend.
2.3
52122.2
104
18 2
15 6
26
355
356
CHAPTER 4
DECIMALS
We normally do not use a remainder when dealing with decimals. Add a 0 to the dividend
and continue.
NOTE Remember that adding
a 0 does not change the value
of the dividend. It simply allows
us to complete the division
process in this case.
2.35
52 122.20
104
18 2
15 6
2 60
2 60
0
Add a 0.
So 122.2 52 2.35. The quotient is 2.35.
CHECK YOURSELF 2
Divide 234.6 by 68.
Often you will be asked to give a quotient to a certain place value. In this case, continue
the division process to one digit past the indicated place value. Then round the result back
to the desired accuracy.
When working with money, for instance, we normally give the quotient to the nearest
hundredth of a dollar (the nearest cent). This means carrying the division out to the thousandths place and then rounding back.
Example 3
Dividing a Decimal by a Whole Number and Rounding the Result
place past the desired place,
and then round the result.
Find the quotient of 25.75 15 to the nearest hundredth.
1.716
1525.750
15
10 7
10 5
25
15
100
90
10
Add a 0 to carry the division to
the thousandths place.
So 25.75 15 1.72 (to the nearest hundredth).
CHECK YOURSELF 3
Find 99.26 35 to the nearest hundredth.
As we mentioned, problems similar to the one in Example 3 often occur when working
with money. Example 4 is one of the many applications of this type of division.
© 2001 McGraw-Hill Companies
NOTE Find the quotient to one
DIVIDING DECIMALS
SECTION 4.5
357
Example 4
An Application Involving the Division of a Decimal by a Whole Number
A carton of 144 items costs $56.10. What is the price per item to the nearest cent?
To find the price per item, divide the total price by 144.
NOTE You might want to
review the rules for rounding
decimals in Section 4.1.
0.389
14456.100
43 2
12 90
11 52
1 380
1 296
84
Carry the division to the thousandths
place and then round back.
The cost per item is rounded to $0.39, or 39¢.
CHECK YOURSELF 4
An office paid $26.55 for 72 pens. What was the cost per pen to the nearest cent?
We want now to look at division by decimals. Here is an example using a fractional form.
Example 5
Rewriting a Problem That Requires Dividing by a Decimal
Divide.
2.57
3.4
Write the division as a fraction.
2.57 10
3.4 10
25.7
34
We multiply the numerator and denominator by 10
so the divisor is a whole number. This does not
change the value of the fraction.
Multiplying by 10, shift the decimal point in the
numerator and denominator one place to the right.
© 2001 McGraw-Hill Companies
2.57 3.4 NOTE It’s always easier to
rewrite a division problem so
that you’re dividing by a whole
number. Dividing by a whole
number makes it easy to place
the decimal point in the
quotient.
25.7 34
Our division problem is rewritten so that the divisor
is a whole number.
So
2.57 3.4 25.7 34
After we multiply the numerator and denominator
by 10, we see that 2.57 3.4 is the same as 25.7 34.
CHECK YOURSELF 5
Rewrite the division problem so that the divisor is a whole number.
3.42 2.5
358
CHAPTER 4
DECIMALS
NOTE Of course, multiplying
by any whole-number power of
10 greater than 1 is just a
matter of shifting the decimal
point to the right.
Do you see the rule suggested by this example? We multiplied the numerator and the denominator (the dividend and the divisor) by 10. We made the divisor a whole number without altering the actual digits involved. All we did was shift the decimal point in the divisor
and dividend the same number of places. This leads us to the following rule.
Step by Step: To Divide by a Decimal
Step 1 Move the decimal point in the divisor to the right, making the divisor a
whole number.
Step 2 Move the decimal point in the dividend to the right the same number
of places. Add zeros if necessary.
Step 3 Place the decimal point in the quotient directly above the decimal
point of the dividend.
Step 4 Divide as you would with whole numbers.
Let’s look at an example of the use of our division rule.
Example 6
Rounding the Result of Dividing by a Decimal
Divide 1.573 by 0.48 and give the quotient to the nearest tenth.
Write
0.48 1.57
^
^
3
Shift the decimal points two places
to the right to make the divisor a
whole number.
Now divide:
statement is rewritten, place
the decimal point in the
quotient above that in the
dividend.
3.27
48157.30
144
13 3
96
3 70
3 36
34
Note that we add a 0 to carry the
division to the hundredths place. In
this case, we want to find the
quotient to the nearest tenth.
Round 3.27 to 3.3. So
1.573 0.48 3.3 (to the nearest tenth)
CHECK YOURSELF 6
Divide, rounding the quotient to the nearest tenth.
3.4 1.24
Let’s look at some applications of our work in dividing by decimals.
© 2001 McGraw-Hill Companies
NOTE Once the division
DIVIDING DECIMALS
SECTION 4.5
359
Example 7
Solving an Application Involving the Division of Decimals
Andrea worked 41.5 hours in a week and earned $239.87. What was her hourly rate of pay?
To find the hourly rate of pay we must use division. We divide the number of hours
worked into the total pay.
NOTE Notice that we must
add a zero to the dividend to
complete the division process.
5.78
41.5 239.8 70
^
^
207 5
32 3
29 0
33
33
70
7
5
20
20
0
Andrea’s hourly rate of pay was $5.78.
CHECK YOURSELF 7
A developer wants to subdivide a 12.6-acre piece of land into 0.45-acre lots. How
many lots are possible?
Example 8
Solving an Application Involving the Division of Decimals
At the start of a trip the odometer read 34,563. At the end of the trip, it read 36,235. If
86.7 gallons (gal) of gas were used, find the number of miles per gallon (to the nearest tenth).
First, find the number of miles traveled by subtracting the initial reading from the final
reading.
36,235
34,563
1 672
Final reading
Initial reading
Miles covered
Next, divide the miles traveled by the number of gallons used. This will give us the miles
per gallon.
1 9.28
86.7 1672.0 00
^
© 2001 McGraw-Hill Companies
^
867
805 0
780 3
24 7
17 3
73
69
4
0
4
60
36
24
Round 19.28 to 19.3 mi/gal.
CHAPTER 4
DECIMALS
CHECK YOURSELF 8
John starts his trip with an odometer reading of 15,436 and ends with a reading of
16,238. If he used 45.9 gallons (gal) of gas, find the number of miles per gallon
(mi/gal) (to the nearest tenth).
Recall that you can multiply decimals by powers of 10 by simply shifting the decimal point
to the right. A similar approach will work for division by powers of 10.
Example 9
Dividing a Decimal by a Power of 10
(a) Divide.
3.53
10 35.30
30
53
50
30
30
0
The dividend is 35.3. The quotient is 3.53. The
decimal point has been shifted one place to the left.
Note also that the divisor, 10, has one zero.
(b) Divide.
3.785
100 378.500
300
78 5
70 0
8 50
8 00
500
500
0
Here the dividend is 378.5, whereas the quotient is
3.785. The decimal point is now shifted two places
to the left. In this case the divisor, 100, has two
zeros.
CHECK YOURSELF 9
Perform each of the following divisions.
(a) 52.6 10
(b) 267.9 100
Example 9 suggests the following rule.
Rules and Properties:
To Divide a Decimal by a Power of 10
Move the decimal point to the left the same number of places as there are
zeros in the power of 10.
© 2001 McGraw-Hill Companies
360
DIVIDING DECIMALS
SECTION 4.5
361
Example 10
Dividing a Decimal by a Power of 10
Divide.
27.3 10 2
(a)
^
7.3
Shift one place to the left.
2.73
(b)
57.53 100 0
^
57.53
Shift two places to the left.
0.5753
NOTE As you can see, we may
have to add zeros to correctly
place the decimal point.
(c) 39.75 1000 0
^
039.75
Shift three places to the left.
0.03975
85 1000 0
(d)
^
085.
The decimal after the 85 is implied.
0.085
REMEMBER: 104 is a 1
(e) 235.72 104 0
followed by four zeros.
^
0235.72
Shift four places to the left.
0.023572
CHECK YOURSELF 10
Divide.
(a) 3.84 10
(b) 27.3 1000
Let’s look at an application of our work in dividing by powers of 10.
Example 11
Solving an Application Involving a Power of 10
© 2001 McGraw-Hill Companies
To convert from millimeters (mm) to meters (m), we divide by 1000. How many meters
does 3450 mm equal?
3450 mm 3
^
450. m
3.450 m
Shift three places to the left to divide by 1000.
CHAPTER 4
DECIMALS
CHECK YOURSELF 11
A shipment of 1000 notebooks cost a stationery store $658. What was the cost per
notebook to the nearest cent?
Recall that the order of operations is always used to simplify a mathematical expression
with several operations. You should recall the order of operations as the following.
Rules and Properties:
1.
2.
3.
4.
The Order of Operations
Perform any operations enclosed in parentheses.
Apply any exponents.
Do any multiplication and division, moving from left to right
Do any addition and subtraction, moving from left to right.
Example 12
Applying the Order of Operations
Simplify each expression.
(a) 4.6 (0.5 4.4)2 3.93
4.6 (2.2)2 3.93
parentheses
4.6 4.84 3.93
exponent
9.44 3.93
add (left of the subtraction)
5.51
subtract
(b) 16.5 (2.8 0.2)2 4.1 2
16.5 (3)2 4.1 2
parentheses
16.5 9 4.1 2
exponent
16.5 9 8.2
multiply
7.5 8.2
subtraction (left of the addition)
15.7
add
CHECK YOURSELF 12
Simplify each expression.
(a) 6.35 (0.2 8.5)2 3.7
(b) 2.52 (3.57 2.14) 3.2 1.5
CHECK YOURSELF ANSWERS
1. 2.36
2. 3.45
3. 2.84
4. $0.37, or 37¢
5. 34.2 25
6. 2.7
7. 28 lots
8. 17.5 mi/gal
9. (a) 5.26; (b) 2.679
10. (a) 0.384;
(b) 0.0273
11. 66¢
12. (a) 5.54; (b) 9.62
© 2001 McGraw-Hill Companies
362
Name
4.5
Exercises
Section
Date
Divide.
1. 16.68 6
2. 43.92 8
ANSWERS
1.
3. 1.92 4
4. 5.52 6
2.
3.
4.
5. 5.48 8
6. 2.76 8
5.
6.
7. 13.89 6
8. 21.92 5
7.
8.
9. 185.6 32
10. 165.6 36
9.
10.
11. 79.9 34
12. 179.3 55
11.
12.
13.
13. 5213.78
14. 7626.22
14.
15.
15. 0.611.07
16. 0.810.84
16.
17.
17. 3.87.22
18. 2.913.34
18.
19.
19. 5.211.622
20. 6.43.616
20.
© 2001 McGraw-Hill Companies
21.
21. 0.271.8495
22. 0.0380.8132
22.
23.
23. 0.0461.587
24. 0.523.2318
24.
25.
25. 0.658 2.8
26. 0.882 0.36
26.
363
ANSWERS
27.
Divide by moving the decimal point.
28.
27. 5.8 10
28. 5.1 10
30.
29. 4.568 100
30. 3.817 100
31.
31. 24.39 1000
32. 8.41 100
32.
33. 6.9 1000
34. 7.2 1000
35. 7.8 102
36. 3.6 103
37. 45.2 105
38. 57.3 104
29.
33.
34.
35.
36.
Divide and round the quotient to the indicated decimal place.
37.
39. 23.8 9
38.
tenth
40. 5.27 8
hundredth
41. 38.48 46
hundredth
42. 3.36 36
43. 125.4 52
tenth
44. 2.563 54
thousandth
39.
thousandth
40.
41.
45. 0.7 1.642
hundredth
46. 0.6 7.695
42.
47. 4.5 8.415
tenth
48. 5.8 16
49. 3.12 4.75
hundredth
50. 64.2 16.3
43.
tenth
hundredth
thousandth
44.
45.
Solve the following applications.
46.
51. Cost of CDs. Marv paid $40.41 for three CDs on sale. What was the cost per CD?
47.
52. Contributions. Seven employees of an office donated $172.06 during a charity
drive. What was the average donation?
48.
53. Book purchases. A shipment of 72 paperback books cost a store $190.25. What was
the average cost per book to the nearest cent?
© 2001 McGraw-Hill Companies
49.
50.
51.
52.
53.
364
ANSWERS
54. Cost. A restaurant bought 50 glasses at a cost of $39.90. What was the cost per glass
to the nearest cent?
55. Cost. The cost of a case of 48 items is $28.20. What is the cost of an individual item
54.
55.
56.
to the nearest cent?
57.
56. Office supplies. An office bought 18 hand-held calculators for $284. What was the
cost per calculator to the nearest cent?
57. Monthly payments. Al purchased a new refrigerator that cost $736.12 with interest
included. He paid $100 as a down payment and agreed to pay the remainder in
18 monthly payments. What amount will he be paying per month?
58. Monthly payments. The cost of a television set with interest is $490.64. If you
make a down payment of $50 and agree to pay the balance in 12 monthly payments,
what will be the amount of each monthly payment?
58.
59.
60.
61.
62.
63.
64.
65.
66.
59. Mileage. In five readings, Lucia’s gas mileage was 32.3, 31.6, 29.5, 27.3, and
33.4 miles per gallon (mi/gal). What was her average gas mileage to the nearest tenth
of a mile per gallon?
60. Pollution. Pollution index readings were 53.3, 47.8, 41.9, 55.8, 43.7, 41.7, and 52.3
for a 7-day period. What was the average reading (to the nearest tenth) for the 7 days?
61. Label making. We have 91.25 inches (in.) of plastic labeling tape and wish to make
labels that are 1.25 in. long. How many labels can be made?
62. Wages. Alberto worked 32.5 hours (h), earning $306.15. How much did he make per
hour?
63. Cost per pound. A roast weighing 5.3 pounds (lb) sold for $14.89. Find the cost per
© 2001 McGraw-Hill Companies
pound to the nearest cent.
64. Weight. One nail weighs 0.025 ounce (oz). How many nails are there in 1 lb? (1 lb is
16 oz.)
65. Mileage. A family drove 1390 miles (mi), stopping for gas three times. If they
purchased 15.5, 16.2, and 10.8 gallons (gal) of gas, find the number of miles per
gallon (the mileage) to the nearest tenth.
66. Mileage. On a trip an odometer changed from 36,213 to 38,319. If 136 gal of gas
were used, find the number of miles per gallon (to the nearest tenth).
365
ANSWERS
67.
67. Conversion. To convert from millimeters (mm) to inches, we can divide by 25.4. If
film is 35 mm wide, find the width to the nearest hundredth of an inch.
68.
69.
68. Conversion. To convert from centimeters (cm) to inches, we can divide by 2.54. The
rainfall in Paris was 11.8 cm during 1 week. What was that rainfall to the nearest
hundredth of an inch?
70.
71.
69. Construction. A road-paving project will cost $23,500. If the cost is to be shared by
72.
100 families, how much will each family pay?
70. Conversion. To convert from milligrams (mg) to grams (g), we divide by 1000. A
tablet is 250 mg. What is its weight in grams?
73.
71. Conversion. To convert from milliliters (mL) to liters (L), we divide by 1000. If a
bottle of wine holds 750 mL, what is its volume in liters?
74.
72. Unit cost. A shipment of 100 calculators cost a store $593.88. Find the cost per
calculator (to the nearest cent).
73. The blood alcohol content (BAC) of a person who has been drinking is determined by
the formula
BAC oz of alcohol % of alcohol 0.075 of body wt.
(hours of drinking 0.015)
A 125-lb person is driving and is stopped by a policewoman on suspicion of driving
under the influence (DUI). The driver claims that in the past 2 hours he consumed
only six 12-oz bottles of 3.9% beer. If he undergoes a breathalyzer test, what will his
BAC be? Will this amount be under the legal limit for your state?
74. Four brands of soap are available in a local store.
Ounces
Total Price
Squeaky Clean
Smell Fresh
Feel Nice
Look Bright
5.5
7.5
4.5
6.5
$0.36
0.41
0.31
0.44
Compute the unit price, and decide which brand is the best buy.
366
Unit Price
© 2001 McGraw-Hill Companies
Brand
ANSWERS
75. Sophie is a quality control expert. She inspects boxes of #2 pencils. Each pencil
weighs 4.4 grams (g). The contents of a box of pencils weigh 66.6 g. If a box is
labeled CONTENTS: 16 PENCILS, should Sophie approve the box as meeting
specifications? Explain your answer.
75.
76.
76. Write a plan to determine the number of miles per gallon (mpg) your car (or your
family car) gets. Use this plan to determine your car’s actual mpg.
77.
77. Express the width and length of a $1 bill in centimeters (cm). Then express this same
length in millimeters (mm).
78.
79.
78. If the perimeter of a square is 19.2 cm, how long is each side?
80.
81.
82.
P 19.2 cm
83.
84.
79. If the perimeter of an equilateral triangle (all sides have equal length) is 16.8 cm, how
long is each side?
P 16.8 cm
80. If the perimeter of a regular (all sides have equal length) pentagon is 23.5 in., how
long is each side?
© 2001 McGraw-Hill Companies
P 23.5 in.
Simplify each expression.
81. 4.2 3.1 1.5 (3.1 0.4)2
82. 150 4.1 1.5 (2.5 1.6)3 2.4
83. 17.9 1.1 (2.3 1.1)2 (13.4 2.1 4.6)
84. 6.892 3.14 2.5 (4.1 3.2 1.6)2
367
Answers
1. 2.78
3. 0.48
5. 0.685
7. 2.315
9. 5.8
11. 2.35
13. 0.265
15. 18.45
17. 1.9
19. 2.235
21. 6.85
23. 34.5
25. 0.235
27. 0.58
29. 0.04568
31. 0.02439
33. 0.0069
35. 0.078
37. 0.000452
39. 2.6
41. 0.84
43. 2.4
45. 2.35
47. 1.9
49. 1.52
51. $13.47
53. $2.64
55. $0.59, or 59¢
57. $35.34
59. 30.8 mi/gal
61. 73 labels
63. $2.81
65. 32.7 mi/gal
67. 1.38 in.
69. $235
71. 0.75 L
73.
75.
79. 5.6 cm
81. 11.8
83. 17.0291
© 2001 McGraw-Hill Companies
77.
368
Using Your Calculator
to Divide Decimals
It would be most surprising if you had reached this point without using your calculator to
divide decimals. It is a good way to check your work, and a reasonable way to solve
applications. Let’s first use the calculator for a straightforward problem.
Example 1
Dividing Decimals
Use your calculator to find the quotient
211.56 82
Enter the problem in the calculator to find that the answer is 25.8.
CHECK YOURSELF 1
Use your calculator to find the quotient
304.32 9.6
Now that you’re convinced that it is easy to divide decimals on your calculator, let’s
introduce a twist.
Example 2
An Application Involving the Division of Decimals
© 2001 McGraw-Hill Companies
Omar drove 256.3 miles on a tank of gas. When he filled up the tank, it took 9.1 gallons.
What was his gas mileage?
Here’s where students get into trouble when they use a calculator. Entering these values,
you may be tempted to answer “28.16483516 miles per gallon.” The difficulty is that there
is no way you can compute gas mileage to the nearest hundred-millionth mile. How do you
decide where to round off the answer that the calculator gives you? A good rule of thumb
is to never report more digits than the least number of digits in any of the numbers that you
are given in the problem. In this case, you were given a number with four digits and another
with two digits.Your answer should not have more than two digits. Instead of 28.16483516,
the answer could be 28 miles per gallon. Think about the question. If you were asked for
gas mileage, how precise an answer would you give? The best answer to this question
would be to give the nearest whole number of miles per gallon: 28 miles per gallon.
CHECK YOURSELF 2
Emmet gained a total of 857 yards (yd) in 209 times that he carried the football.
How many yards did he average for each time he carried the ball?
CHECK YOURSELF ANSWERS
1. 31.7
2. 4.10 yd
369
Name
Section
Date
Calculator Exercises
Divide and check.
ANSWERS
1. 8.901 2.58
2. 16.848 0.288
3. 99.705 34.5
4. 171.25 2.74
5. 0.01372 0.056
6. 0.200754 0.00855
1.
2.
3.
4.
5.
Divide and round to the indicated place.
6.
7. 2.546 1.38
hundredth
8. 45.8 9.4
tenth
7.
9. 0.5782 1.236
8.
thousandth
10. 1.25 0.785
hundredth
9.
11. 1.34 2.63
10.
two decimal places
12. 12.364 4.361
three decimal places
11.
Solve the following applications.
12.
13. Salary. In 1 week, Tom earned $178.30 by working 36.25 hours (h). What was his
hourly rate of pay to the nearest cent?
13.
14.
14. Area. An 80.5-acre piece of land is being subdivided into 0.35-acre lots. How many
15.
lots are possible in the subdivision?
16.
18.
15. 3.925 cm
16. 8.3838 in.
17. 1.7584 cm
18. 13.0624 mm
Answers
1. 3.45
13. $4.92
370
3. 2.89
5. 0.245
7. 1.84
15. 1.25 cm
17. 0.56 cm
9. 0.468
11. 0.51
© 2001 McGraw-Hill Companies
If the circumference of a circle is known, the diameter can be determined by dividing the
circumference by p. In each of the following, determine the diameter for the given
circumference. Use 3.14 for p.
17.