3.3 MULTIPLYING DECIMAL NUMBERS
Darren makes $6.74 per hour delivering pizzas on the weekends.
He is looking forward to his next payday and trying to calculate
how much money he has earned. He worked 31.5 hours this last
pay period.
What will his gross pay be
(before taxes or any other
deductions are taken out)?
$
If he is reimbursed at $.05 per
mile and has logged 42.3 miles
on his scooter, how much will
he be paid for mileage?
212.31
$
Assess your readiness to complete this activity. Rate how well you understand:
2.12
Not
ready
Almost
ready
Bring
it on!
• how to set up a multiplication problem with decimal numbers
• the placement of the decimal point in the product of decimal numbers
• the validation of multiplication by division
•
Multiplying decimal numbers
– neatness of presentation
– correct placement of the decimal point in the product
– validation of the answer
147
Chapter 3 — Decimal Numbers
Technique
Move the decimal point to the right as many places as there are zeros in the power of ten.
Model 1
10 × 63.05 = 630.5
Move the decimal point one place to the right.
100 × 63.05 = 6305. or 6305
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
10 × 0.0052 = 0.052
100 × 0.0052 = 0.52
1000 × 0.0052 = 5.2
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
148
Activity 3.3 — Multiplying Decimal Numbers
While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem.
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
Example 2
.293
×7.4
For ease of calculation, use the number with
fewer digits as the multiplier (bottom number).
If a decimal number includes 0 (zero) as its
whole number (such as in 0.468), it is not
necessary to include the 0 when setting up the
problem.
Special More than two factors to
Case: multiply (see Model 3)
Step 2
Multiply.
Ignore the decimal points and multiply as you
would multiply whole numbers. This will give
you the digits in the answer.
2 2
4 5
.4 6 8
×3.7
Note that ignoring the decimal points in the factors
until step 4 eliminates the complication of trying to
place decimal points in the partial products.
1
32 76
14040
.293
×7.4
1172
20510
21682
17316
Step 3
Count
decimal
places.
Count the number of decimal places
in each factor and add the results.
3 decimal places
1 decimal place
.468
×3.7
1
32 76
14040
17316
.293
7.4
3
1
4
4 decimal
places
3 +1 =4
decimal places
149
Chapter 3 — Decimal Numbers
Steps in the Methodology
Step 4
Position
the decimal
point.
Example 1
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.
2.1682
4 places from
the right
To understand the placement of the decimal point
in the answer, consider the two factors and their
product in fraction notation.
Example 2
4 places from
the right
Notice that the numerator of the answer is the
product of the whole number numerators of the
fractions and the denominator of the answer is the
product of the denominators of the fractions.
For Example 1, 0.468 × 3.7 means
468 × 3 7
1000
10
= 468 × 37 = 17316
1000
10
10000
= 1 7 3 1 6 , which is 1.7316 in decimal form.
1000
Special Not enough decimal places in the
Case: answer to count (see Model 1)
Step 5
Present the
answer.
Step 6
Present your answer.
Special Dropping trailing zeros in
Case: the answer (see Model 2)
Validate using division. Divide your answer by
Validate your your multiplier.
answer.
Tip: Look at the other original factor as a target
result for the quotient. Prove that it is the
quotient.
150
Answer:
2.1682
1.7316
5
4
2
)
.4 6 8
6
1
3 .7 1. 7 3 1 6
−1 4 8
.293
7.4 2.1682
)
−148
688
4
2 511
−2 2 2
−666
222
296
−2 9 6
−222
0
0
Activity 3.3 — Multiplying Decimal Numbers
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
2 052
51300
307800
Step 3
4 decimal places
+
3 decimal places
= 7 decimal places
1
361152
Step 4
.0361152
Step 6
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 values.
3 1
1
Validate:
.1 0 2 6
)
5
9 15
−7 0 4
7 decimal places in the answer
Step 5
1
. 352 .03 6 115 2
−3 5 2
2 112
−2 112
Answer: .0361152 or 0.0361152
0
Model 2
Solve:
Step 1
Special Case: Dropping Trailing Zeros in the Answer
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
3 decimal places
+ 2 decimal places =
5 decimal places
19.25950
5 decimal places
in the answer
1925950
1
Step 5
Answer: 19.2595
Step 6
Validate:
3
1
4
2
5.9 2 6
3 . 25 19.25 95 0
−16 25
2
1
)
2 91 1
It is customary to drop
the trailing zero(s) when
presenting the answer.
3 0 09
−2 9 25
7 1
845
−6 5 0
1 95 0
−195 0
0
151
Chapter 3 — Decimal Numbers
Model 3
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
152
.5 2 2
2
1.4 .7 3 0 8
−7 0
)
2
1
30
−2 8
28
−2 8
0
Activity 3.3 — Multiplying Decimal Numbers
Make Your Own Model
Either individually or as a team exercise, create a model demonstrating
how to solve the most difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
153
Chapter 3 — Decimal Numbers
1. What are three additional situations (other than the context problems for this activity) in which you would
need to multiply decimal numbers?
Answers will vary:
—calculating volume (8.5ft × 12.3ft × 17.8ft)
—amount of money owed for a job (pay rate of $9.75 times 32.5 hours)
—amount of meat for serving at a banquet (0.3 pounds times 325 people)
—gas money ($1.399 times 14.3 gallons), etc.
2. How do you determine where the decimal point is placed in the product when multiplying two decimal
numbers?
Count up the number of decimal places in each number; then count over that number of places, from right to left, in
the answer.
3. When you multiply a decimal number less than one (1) by a second number, will the product be greater or
less than the second number? Explain.
It will be less than the second number. A decimal part of any number is smaller than the original number.
4. What is the shortcut for multiplying a decimal number by 10, 100, 1000, and so on?
The short cut is to move the decimal point one place to the right for every zero in the factor.
5. What is the best strategy for multiplying three decimal numbers? What mathematical property or properties
allow you to use this strategy?
Multiply the first two numbers following the methodology and then multiply the third number. Validate by using two
divisions. Keep careful track of the number of decimal places.
Or you could multiply the number with the most digits by the one with the next largest number of digit and take
that answer and multiply by the one with the smallest number of digits. This can be done by using the Associative
Property for Multiplication and/or the Commutative property for Multiplication.
6. How do you validate your answer to a decimal number multiplication problem that contains three factors?
Validate by dividing the answer by the third factor then divide that answer by the second factor
154
Activity 3.3 — Multiplying Decimal Numbers
7. What aspect of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
Multiply as indicated. Validate your answers.
Problem
1)
3.26 × 0.35
2)
0.265 × 18.3
3)
0.005 × 0.26
Worked Solution
Validation
155
Chapter 3 — Decimal Numbers
Worked Solution
Problem
4)
35.2 × 967
5)
5.02 × 0.5 × 8.4
Validation
6. Fill in the following chart with the correct products.
Decimal number
156
× 10
× 100
× 1000
× 10,000
2.61
26.1
261
2610
26100
0.009
0.09
0.9
9
90
345.9178
3459.178
34591.78
345917.8
3459178
Activity 3.3 — Multiplying Decimal Numbers
7. Write each of the following numbers in its standard form.
Number
Standard form
a)
17.5 million
17,500,000
b)
5.3 billion
5,300,000,000
Perform the indicated operations and validate your answers.
1. 45 × 0.02
0.9
4. 4.5 × 0.15 × 1.23
0.83025
2. 7.02 × 5.27
36.9954
5. 72.9 × 0.0301
2.19429
3. 0.23 × 0.009
0.00207
6. 0.347 × 0.026
0.009022
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.
Correct Process
×
.
Should have counted 4
places from the rightmost digit.
)
.
−
−
−
Answer: .4680
or .468
2) 5.36 × 1000
Validation
0
Move the decimal
point one place to
the right for every
zero (0).
Move three places.
157
Chapter 3 — Decimal Numbers
Worked Solution
What is Wrong Here?
Identify Errors
or Validate
3) Multiply:
0.12 × 0.67
The zero (0) should
have been placed to
the left of the 8.
4) 62 × 4.4
Move from the right
to the left place
when placing the
decimal point.
272.8
5) Solve:
9.2 × 4.6
158
Move the decimal
point so that there
are two places to the
right of the decimal
point.
4232
Correct Process
Validation