Section 2.3
PRE-ACTIVITY
PREPARATION
Multiplying Decimal Numbers
How much carpeting must you purchase for your 18.3 feet by 14.6 feet
family room?
How far can you travel on 18.5 gallons of gas in your car that gets 20.2
miles per gallon?
What will be the total of your student loan payments at $123.16 per
month for three years?
If the price of electricity is $0.090361 per kilowatt-hour, what is this
month’s bill for 584 kilowatt-hours of electrical use?
In each situation, you must multiply decimal numbers to determine the answer.
LEARNING OBJECTIVE
Master the multiplication process for decimal numbers.
TERMINOLOGY
PREVIOUSLY USED
factor
place
product
multiplier
power of ten
trailing zeros
TECHNIQUE
The first technique to understand is similar to the shortcut used for multiplying whole numbers by the powers
of ten (10, 100, 1000, and so on). Recall that to multiply by 100, for example, you simply added two zeros
to the right of the whole number.
Example: 493 × 100 = 49,300
In other words, to multiply by 100, you moved the understood decimal point in 493• two place values to the
right (493.00 ).
165
Chapter 2 — Decimal Numbers
166
Multiplying a Decimal Number by a Power of Ten
Technique
Move the decimal point to the right as many places as there are zeros in the power of ten.
MODELS
Model 1
A
►
10 × 63.05 = 630.5
Move the decimal point one place to the right.
B
►
100 × 63.05 = 6305. or 6305
C
►
1000 × 63.05 = 1000 × 63.050 = 63,050. or 63,050
Note: Use trailing zeros in the original decimal number, if you find it helpful to do so.
Model 2
A
►
10 × 0.0052 = 0.052
B
►
100 × 0.0052 = 0.52
C
►
1000 × 0.0052 = 5.2
D
►
10,000 × 0.0052 = 52. or 52
Model 3
Write 13.7 million in its standard form.
Note: “13.7 million” is the word form for 13.7 millions or 13.7 × 1,000,000
13.700000 × 1,000,000 = 13,700,000
Section 2.3 — Multiplying Decimal Numbers
167
METHODOLOGY
Multiplying Decimal Numbers
►
►
Example 1: 3.7 × 0.468
Try It!
Example 2: 0.293 × 7.4
Steps in the Methodology
Step 1
Set up the
problem.
Example 1
Right-align the two decimal numbers. Digits
should be neatly aligned in columns, but not
necessarily by place value. That is, it is not
necessary to line up the decimal points by
using trailing zeros!
.468
×3.7
For ease of calculation, use the number with
fewer digits as the multiplier (bottom
number). It is not necessary to include 0
whole numbers in the set-up.
???
Why do you do this?
Special More than two factors to multiply
Case: (see page 171, Model 3)
Step 2
Multiply.
???
Why do you do this?
Step 3
Count
decimal
places.
2 2
Ignore the decimal points and multiply as you
would multiply whole numbers. This will give
you the digits in the answer.
Count the number of decimal places
in each factor and add the results.
4 5
.4 6 8
×3.7
1
32 76
14040
17316
3 decimal places
1 decimal place
.468
×3.7
1
32 76
14040
17316
3 +1 =4
decimal places
Example 2
Chapter 2 — Decimal Numbers
168
Steps in the Methodology
Step 4
Position
the decimal
point.
Example 1
Example 2
1•7316
Starting from the right of the last digit in the
product, count the total number of places you
found in Step 3 and position the decimal point.
4 places from
the right
???
Why do you do this?
Special Not enough decimal places in the answer
Case: to count (see page 170, Model 1)
Step 5
Present the
answer.
Step 6
Validate
your
answer.
Present your answer.
1.7316
Special Dropping trailing zeros in the answer
Case: (see page 170, Model 2)
Validate using division. Divide your answer by
your multiplier.
Tip: Look at the other original factor as a target
result for the quotient. Prove that it is the
quotient.
5
4
2
)
.4 6 8 9
6
3 .7 1. 713 1 6
−1 4 8
4
2 511
−2 2 2
296
−2 9 6
0
???
Why do you do Step 1?
When multiplying, you will ignore the decimal points to do the numeric calculations (in Step 2), making the
alignment by decimal place values and the use of trailing zeros both unnecessary and inefficient.
???
Why do you do Step 2?
Ignoring the decimal points in the factors (for the time being) eliminates the complication of trying to place
decimal points in the partial products. Multiply as though the numbers were whole numbers, aligning and
adding partial products according to their common whole number place values. (Review Steps 2-6 in the
Methodology for Multiplying Whole Numbers in Section 1.4.)
This simplifies the computation, yet results in the correct digits for the answer. (In Step 4, you will correctly
position the decimal point in the answer.)
Section 2.3 — Multiplying Decimal Numbers
169
???
Why do you do Step 4?
In Step 2, ignoring the decimal points in the factors essentially transforms both decimal numbers into whole
numbers by mentally moving their decimal points to the right the appropriate number of places.
In Example 1 (from the methodology),
.468 changes to 468 by moving the decimal point 3 places to the right (multiplying by 1000)
×3.7 changes to × 37 by moving the decimal point 1 place to the right (multiplying by 10)
3276
14 040
17 316
Note that you changed the original problem so that the resulting product is a whole number. To compensate
for the changes you made to the original factors, you must make the opposite changes to the whole number
answer. That is, in the product you must move the decimal point to the left the same total number of places as
you mentally moved them to the right in the original factors.
For Example 1,
Moving the decimal point 3 places + 1 place (= 4 places) to the left (dividing by 1000 and dividing by 10) in
17316 (17136) yields 1.7316 as the product of the two original decimal numbers.
The process is:
First, .468 × 1000 × 3.7 × 10 = 17316
Then, 17316 ÷ 1000 ÷ 10 = 1.7316
Or, putting it all together,
.468 × 3.7 × 1000 × 10 ÷ 1000 ÷ 10 = 1.7316
Think about the mathematical properties you apply in this process:
.468 × 3.7 × 1000 × 10 ÷ 1000 ÷ 10 = 1.7316
for ease of
multiplication
.468 × 3.7 ×
10,000
for placement of
the decimal point
÷
10,000
= 1.7316
Special Property of Division:
Any number divided by itself equals one (1).
.468 × 3.7 ×
1
= 1.7316
Identity Property of Multiplication:
Any number times one (1) equals that same number.
.468
× 3.7
= 1.7316
Chapter 2 — Decimal Numbers
170
MODELS
Model 1
Special Case: Not Enough Decimal Places in the Answer to Count
Multiply:
Steps 1 & 2
0.352 × 0.1026
1
1 3
1
.10 2 6
×.352
1
Step 3
4 decimal places
+
3 decimal places
2 052
51300
307800
= 7 decimal places
1
If there are not enough places to count, you must insert the
appropriate number of zeros to the left of the leading digit
in the answer to hold the necessary place value.
361152
Step 4
.0361152
7 decimal places in the answer
Step 5
Answer: .0361152 or 0.0361152
Step 6
Validate:
Model 2
Solve:
Step 1
5
1
. 352 .03 6 115 2
−3 5 2
9 15
−7 0 4
2 112
−2 112
Special Case: Dropping Trailing Zeros in the Answer
0
3.25 × 5.926
5.926
×3.25
Step 2
2
1
4
1
1
1 3
5.9 2 6
×3.25
Step 3
2 1
2 9 630
1
118520
1777800
Step 4
Answer: 19.2595
It is customary to drop the
last trailing zero(s) when
presenting the answer.
3 decimal places
+ 2 decimal places =
5 decimal places
19.25950
5 decimal places
in the answer
1925950
Step 5
.1 0 2 6 9
)
3 1
1
Step 6
Validate:
1
3
1
4
2
5.9 2 6 9
3 . 25 19.25 95 0
−16 25
2
1
)
2 91 1
3 0 09
−2 9 25
7 1
845
−6 5 0
1 95 0
−195 0
0
Section 2.3 — Multiplying Decimal Numbers
Model 3
171
Special Case: More than Two Factors to Multiply
Multiply:
0.03 × 0.522 × 1.4
The Methodology for Multiplying Decimal Numbers is for two factors. Apply the
Commutative and Associative Properties of Multiplication and choose any two factors
to multiply first. Then multiply the first product, with its decimal point correctly
positioned, by the next factor. Continue until all factors have been used.
Steps 1-4
2
.522
×1.4
.73 0 8
× .03
3 decimal places
1 decimal place
1
2 0 88
5220
021924
.7308
6 places
4 decimal places
2 decimal places
Note: It was necessary
to use a zero placeholder
to the left of the 2 in
order to have 6 decimal
places in the final answer.
4 places
Step 5
Answer: .021924 or 0.021924
Step 6
Validate by successive divisions, using the multipliers as the divisors,in reverse order.
.7308
.03 .021924
−21
09
−9
)
02
−2
24
−24
0
.5 2 2 9
1.4 .7 3 0 8
−7 0
2
)
2
1
30
−2 8
28
−2 8
0
Chapter 2 — Decimal Numbers
172
How Estimation Can Help
When working with decimal numbers, a very important benefit of estimation is predicting the placement of
the decimal point in the final answer.
An effective way to quickly estimate the product of two decimal numbers is to first round each factor to its
largest non-zero place value and mentally multiply, approximating the digits in the answer.
However, as a second step, you must also mentally position the decimal point according to your rounded
factors.
Example:
0.693 × 5.14
0.7 ×
ROUND
5
7 × 5 = 35 The digits in the estimate are 35 and one decimal place is required.
THINK
Estimated answer:
3.5
Actual answer:
3.56202
Example:
0.439 × 0.0742
.4
ROUND
THINK
×
.07
The digits in the estimate are 28 and three decimal places are required.
Estimated answer: .028
Actual answer:
Example:
293.45 × 0.06
300
ROUND
THINK
.0325738
× .06
300 × 6 or 1800 The digits in the estimate are 1800 and two decimal places are required.
Estimated answer: 18.00
Actual answer:
17.607
Go back and estimate the answer to Example 2 in the Methodology for Multiplying Decimal Numbers.
Was your answer reasonably close to your estimate?
Section 2.3 — Multiplying Decimal Numbers
173
ADDRESSING COMMON ERRORS
Incorrect
Process
Issue
Incorrectly
accounting for
the number of
decimal places
in a whole
number
Correct
Process
Resolution
16 × 1.23
1
The decimal point in
a whole number is
understood to be to
2 places
the immediate right
2 places of the ones digit in
the number.
1
1. 2 3
× 16
738
1230
1968
A whole number
4 places factor has no decimal
places to account for.
Answer: 0.1968
Validation
16 × 1.23
1
1
1. 2 3
× 16
738
1230
1968
2 decimal places
0 decimal places
1.239
16 19.68
−16
1
)
36
−3 2
2+0=2 decimal
places in the
final answer
48
−48
Count two decimal
places from the right.
0
Answer: 19.68
Not counting
the right-most
zero(s) in the
product when
positioning the
decimal point in
the final answer
7.25 × 0.18
7.25
×.18
1
800
58
0
7250
50
0
13050
4 decimal
places
Answer: 0.1305
Incorrectly
inserting zero
placeholders in
the final answer
0.121 × 0.32
.121
×.32
242
24
3630
36 0
.3872 0
5 decimal
places
The right-most zero
digits in the product
are important
placeholders and
must be counted
when positioning the
decimal point. They
may be considered
trailing zeros and
dropped only after
the decimal point
has been properly
positioned for the
final answer.
7.25 × 0.18
2
4
7 . 25
× .18
2 decimal places
2 decimal places
1
5 800
7250
1.3050
4
1
5
)
7.259
2 1
.18 1.3 0 5 0
−1 2 6
3
1
45
−3 6
2+2=4 decimal
places in the
final answer
90
−9 0
0
Count from the right
four places.
Answer: 1.3050 or
1.305
Count decimal places
0.121 × 0.32
in the final answer
.121 3 decimal places
.1219
from the right of
the last digit in the
×.32 2 decimal places .32 .03872
product. If there are
−32
242
not enough places to
67
3630 3+2=5 decimal
count, insert zeros to
the left of the leading .03872 places in the
−67
digit in the answer to
final answer
32
hold the necessary
Count five decimal
−32
place values.
places from the right.
)
0
Answer: 0.03872
Chapter 2 — Decimal Numbers
174
Issue
Incorrectly
positioning the
decimal point
in the answer
if and when
aligning the
decimal points
of the factors
Incorrect
Process
0.12 × 5.6
5.60
× .12
1120
5600
Most importantly, it is
not necessary to align
the decimal points (as
it is for adding and
subtracting).
6720
Thinking of
the original
numbers,
.12 × 5
0.12
5.6
m ans
s3
means
dec
cim
mal
decimal
places in
the answer
Answer: 6.720
or 6.72
Correct
Process
Resolution
0.12 × 5.6
5.6
×.12
112
560
1 decimal place
.672
3 decimal places
2 decimal places
Example:
5.60
× .12
1120
5600
2 decimal places
.6720
4 decimal places
2 decimal places
Answer: 0.6720 or
0.672
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the best strategy for multiplying decimal numbers
the placement of the decimal point in the answer
the validation of multiplication by division
5.6 9
.12 .6 7 2
−6 0
1
1
)
72
−7 2
0
Answer: 0.672
While it is true
that trailing zeros
do not change the
value of a factor, if
you do attempt to
use them to set up
your problem, you
must count them as
decimal places for
your answer. (See
example at right.)
Validation
Section 2.3
ACTIVITY
Multiplying Decimal Numbers
PERFORMANCE CRITERIA
• Multiplying decimal numbers
– neatness of presentation
– correct placement of the decimal point in the product
– validation of the answer
CRITICAL THINKING QUESTIONS
1. What are three additional situations (other than those mentioned in the introduction) in which you would
need to multiply decimal numbers?
•
•
•
2. How do you determine where the decimal point is placed in the product when multiplying two decimal
numbers?
3. How is the process of multiplying decimal numbers different from multiplying whole numbers?
175
176
Chapter 2 — Decimal Numbers
4. What is the short cut for multiplying a decimal number by 10, 100, 1000, and so on?
5. What is the best strategy for multiplying three decimal numbers? What mathematical property or properties
allow you to use this strategy?
6. How do you validate your answer to a decimal number multiplication problem that contains three factors?
Section 2.3 — Multiplying Decimal Numbers
TIPS
FOR
177
SUCCESS
• Be neat. Keep digits vertically aligned. Clearly indicate the decimal points.
• To minimize the number of calculations needed, set up the multiplication problem with the factor having
fewest digits as the multiplier (bottom number).
• When validating by division, carefully track the position of the decimal point in the quotient. Do not just
validate that the digits of the quotient match the digits of the remaining factor.
• Estimation can be useful to further assure the proper placement of the decimal point.
DEMONSTRATE YOUR UNDERSTANDING
Multiply as indicated. Validate your answers.
Problem
1)
3.26 × 0.35
2)
0.265 × 18.3
Worked Solution
Validation
Chapter 2 — Decimal Numbers
178
Problem
3)
35.2 × 967
4)
12.062 × 5.02
5)
0.005 × 0.26
Worked Solution
Validation
Section 2.3 — Multiplying Decimal Numbers
Problem
6)
179
Worked Solution
Validation
0.53 × 5.02 × 8.41
7. Fill in the following chart with the correct products.
Decimal number
× 10
× 100
× 1000
2.61
0.009
345.9178
8. Write each of the following numbers in its standard form.
Number
a)
17.5 million
b)
5.3 billion
Standard form
× 10,000
Chapter 2 — Decimal Numbers
180
IDENTIFY
AND
CORRECT
THE
ERRORS
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
Worked Solution
What is Wrong Here?
1) 1.5 × 0.312
Identify Errors
or Validate
The decimal point
is in the wrong
position in the
answer.
Should have counted
4 places from the
right-most digit.
Correct Process
1
0.312
× 1.5
1560
3120
4680
Answer: .4680
or .468
2) Multiply:
0.12 × 0.67
3) 62 × 4.4
Validation
.312 9
1.5 .4680
−45
18
−15
30
)
−30
0
Section 2.3 — Multiplying Decimal Numbers
Worked Solution
What is Wrong Here?
4) 0.003 × 5.1
5) 5.36 × 1000
6) Solve:
9.2 × 4.6
Identify Errors
or Validate
181
Correct Process
Validation
Chapter 2 — Decimal Numbers
182
ADDITIONAL EXERCISES
Perform the indicated operations and validate your answers.
1. 45 × 0.02
2. 7.02 × 5.27
3. 0.23 × 0.009
4. 4.5 × 0.15 × 1.23
5. 72.9 × 0.0301
6. 0.347 × 0.026
7. Fill in the following chart with the correct products.
Decimal number
× 10
1.2
0.0052
8. Write 6.25 million in its standard form.
× 100
× 1000
× 10,000