Download Dividing Decimal Numbers

3.4 DIVIDING DECIMAL NUMBERS
Lindy has decided to plant Gerbera daisies along the border of her flower garden. The border
is 22.5 feet long. She has been gardening for many years and, based on her experience, she
knows that Gerberas require 1.25 feet of space per plant (so they have the right amount of
sunlight and room to grow). She knows what to do with them, once she buys them; she’s
just not sure how many to buy. Can you help her?
How many plants should she purchase?
18 plants
___________________
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it on!
• the long division methodology
• the positioning of the decimal point in the quotient
• when and how to use trailing zeros in the dividend
• when to place zeros in the quotient
• how to carry out rounding instructions
• the validation of the answer by multiplication
•
Dividing any two decimal numbers
– neatness of presentation
– rounding to the specified place
– validation of the answer
159
Chapter 3 — Decimal Numbers
While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem.
Example 1: 18.275 ÷ 0.56 Round the answer to the nearest hundredth.
Try It!
Example 2: 16.46 ÷ 4.3 Round the answer to the nearest tenth.
Steps in the Methodology
Step 1
Set up the
problem.
Step 2
Move the
decimal
point in the
divisor.
Set up the division problem, carefully
recognizing which number is the
dividend and which is the divisor. It is
not necessary to include leading zero (0)
whole numbers in the set-up.
Move the decimal point in the divisor to the
right of its last digit.
For ease of computation in Step 5, it is most
efficient to work with a whole number divisor.
Step 2 takes care of this, by multiplying the
divisor by the appropriate power of ten.
Example 1
)
.56 18.275
)
.56 18.275
(Move two places
in the divisor.)
Special The divisor is a whole number
Case: (see Model 2 & Model 5)
Step 3
Move the
decimal
point in the
dividend.
Move the decimal point in the dividend the
same number of places to the right as you
did in the divisor.
Note that this is actually an application of the
Identity Property of Multiplication; you are
creating an equivalent expression that will have
the same answer as the original expression.
Not enough digits in the dividend
Special
to reposition its decimal point
Case:
(see Model 1)
)
.56 18.275
(Move two
places also
in the
dividend.)
Special The dividend is a whole number
Case: (see Model 3 & Model 5)
Step 4
Bring up
the decimal
point.
160
Position the decimal point in the quotient
directly above the new position of the
decimal point in the dividend.
To understand why, think about the multiplication
of decimal numbers: adding the number of
decimal places for both factors gives the number
of decimal places in their product. Because
division is the inverse operation of multiplication,
the number of decimal places in the dividend
(including its trailing zeros) minus the number of
decimal places in the divisor gives the number of
decimal places in the resulting quotient.
)
.56 18.275
Example 2
Activity 3.4 — Dividing Decimal Numbers
Steps in the Methodology
Step 5
Divide.
Example 1
Perform the computation and placement of
the digits in the quotient as you did with
whole numbers.
Carry out the division one more place
beyond the place value specified for
rounding.
1
1
3
1
1
Example 2
3 2.6 3 3
)
7
1
5 6. 1 8 2 7.5 0 0
−1 6 8
14 7
−1 1 2
To do this, add as many trailing zeros in
the dividend as are necessary to do the
required number of divisions.
4 1
35 5
−3 3 6
8
Special
Case:
1 910
−1 6 8
Placeholder zeros after
decimal point in the quotient
(see Model 4)
1 1
1
2 20
−1 6 8
52
(Note the 2 trailing
zeros needed in the
dividend to carry
the division out 3
places.)
Step 6
Present the
answer.
Step 7
Present your answer by rounding the
quotient to the specified place value.
32.633
Answer: 32.63
Validate:
Validate your • Multiply the non-rounded
answer.
quotient (from Step 5) by the
divisor in its original form.
1 3
1 3
3 decimal places
2 decimal places
32 . 6 33
× .56
1
1 1
1957 9 8
1 6 31 6 5 0
• Add the remainder digits of your
final subtraction.
• Position the decimal point in your
answer.
• The result must match the
original dividend.
1 1
1 1
1 1
18 2 7 4 4 8
+
52
5 decimal places
18.275 00
= original
dividend
161
Chapter 3 — Decimal Numbers
Model 1
Special Case: Not Enough Digits in the Dividend to Reposition its Decimal Point
Divide 2.33 by 0.036 and round the answer to the nearest tenth.
Step 1
Set up the division problem.
Step 2
.036 2.33
Step 3
One trailing zero is needed to move
the decimal point in the dividend.
)
)
.036 2.330
Step 4
Step 5
)
.036 2.33
If there are not enough digits in the dividend,
use as many trailing zeros in the dividend as
are necessary to reposition its decimal point.
)
.036 2.330
1
4
2
3
)
To round to tenths, compute the quotient to its
hundredths place.
6 4.7 2
2
1
Two trailing zeros are needed in the new dividend.
.0 3 6 2. 3 3 0 00
−2
1 6
6
1
170
−1 4 4
5
2 610
−2 5 2
7
1
80
−7 2
8
Step 6
Step 7
Round to the tenths place. 64.72
1 2
2 4
Validate:
1
6 4 .7 2
×.036
1
Answer: 64.7
quotient to 2 decimal places (in Step 5)
original divisor, 3 decimal places
3 8 8 32
1
+ 19 4 1 6 0
1
Add the remainder digits of the final subtraction.
2.33 000
Properly position the decimal point (5 places).
= 2.33
162
1 1
2 32 9 92
8
+
Activity 3.4 — Dividing Decimal Numbers
Model 2
Special Case: Divisor is a Whole Number
Solve 87.28 ÷ 31 and round to the nearest hundredth.
)
Step 1
Set up the division problem. 31 87.28
.
31. 87.28
)
Steps 2, 3 & 4
2.81 5
Step 5 31 8 7.28 0
−6 2
)
4
When the divisor is a whole number, the decimal point is already
understood to be after its right-most digit, so it does not move.
Therefore, the decimal point in the dividend does not move.
Step 6
2.815
Step 7
2
Answer: 2.82
1
252
−2 4 8
Validate:
1
3 decimal places
2 .815
× 31
0 decimal places
1
2 815
+84450
48
−3 1
1
8 7 265
+
15
6
1 710
−1 5 5
87.280
15
3 decimal places
= 87.28
Model 3
Special Case: Dividend is a Whole Number
Divide 4.7 into 62 and round to the nearest hundredth.
)
Step 1
4.7 62
Steps 2, 3 & 4
.
4.7 62.0
Step 5
6
2
)
5
)
When the dividend is a whole number, place a decimal point to the
right of its ones digit and use trailing zero(s) to move the decimal
point the required number of decimal places.
1 3.191
1
4 .7 6 2 .0 00 0
−4 7
4
1
15 0
−1 4 1
13.191
Answer: 13.19
8
910
−4 7
2
Step 6
1
430
−4 2 3
6
710
−4 7
23
Step 7
Validate:
1
2 1
3
6
13 .191
× 4.7
92337
1
527640
1 1 1 1
619 9 7 7
+
23
62.0000 = 62
163
Chapter 3 — Decimal Numbers
Model 4
Special Case: Placeholder Zeros After the Decimal Point in the Quotient
Divide: 0.38 ÷ 52.1 Round to the nearest thousandth.
.
52.1 .3 8
)
Steps 1, 2, 3 & 4
.0 0 7 2
)
1
9
1 1
7
52.1 .3 8 0 0 0
−3 6 4 7
Step 5
If the first partial product that the divisor will divide into
extends beyond the tenths place of the quotient, you must use
zeros as placeholders after the decimal point in the quotient.
1
4
1 5 2310
−1 0 4 2
488
Step 7
521 does not divide into 38.
Hold the (tenths) place with a zero.
.0 0 7 2
Validate: × 52.1
521 does not divide into 380.
Hold the (hundredths) place with a
zero.
521 does divide into 3800—7 times.
.0072
Step 6
3 1
1
Answer: 0.007
1
0 0 72
0 1440
03 6 000
1 1 1
037512
+
488
0.3 8000 = 0.38
Model 5
Solve: 8 ÷ 135 Round to the nearest thousandth.
Note: Even though 8 < 135, in 8 ÷ 135,
8 is the dividend and 135 is the divisor.
)
Step 1 135 8
.
135 8.
)
Steps 2, 3, & 4
The decimal point is understood to be to the right of the whole number
divisor 135, so it does not move. Therefore, the decimal point in the
dividend 8 does not move and remains to the right of 8.
135 does not divide into 80. Hold the (tenths) place in the quotient with a zero.
135 divides into 800 5 times.
3
Step 5
1
4
1 2
)
. 0 5 92
7
9
1
135. 8.10 0 0 0
−6 7 5
4
1
12 50
−1 2 1 5
2
1
3 50
−2 7 0
80
164
Step 6
.0592
Answer: 0.059
Activity 3.4 — Dividing Decimal Numbers
Step 7
Validate:
1 2
2 4 1
0. 05 9 2
× 135
1 1
2 96 0
1
1 7 7 60
05 9 2 00
1 1 1
79920
+
80
8.0 0 00 = 8
The technique that follows is a shortcut for dividing decimal numbers by the powers of ten (10, 100, 1000, and
so on). It is the reverse of the technique used for multiplying decimal numbers by the powers of ten.
Technique
Move the decimal point to the left as many places as there are zeros in the power of ten.
Model 1
7.3 ÷ 10 = .73
Move the decimal point one place to the left.
07.3 ÷ 100 = .073
Model 2
480 ÷ 10 = 480. ÷ 10 = 48.0 or 48
480 ÷ 100 = 4.80 or 4.8
480 ÷ 1000 = .480 or .48
requires a zero placeholder
480 ÷ 10,000 =
007.3 ÷ 1000 = .0073
requires two zero placeholders
.0480 ÷ 10,000 = .0480 or .048
requires a zero placeholder
165
Chapter 3 — 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 5
Step 2
Step 6
Step 3
Step 7
Step 4
166
Activity 3.4 — Dividing Decimal Numbers
1. What are three additional situations (other than the context problems for this activity) in which you would
need to divide decimal numbers?
(many possible answers-these are just a few)
You would use division:
—if you needed to know how many gallons of gas you bought $23.56/$1.399
—when figuring the number of 0.75 pound veal steaks cut from a 12.35 pound rack
—when figuring how long it will take you to drive 535.5 miles at 65.7 miles per hour
2. How do you write a whole number in decimal form?
The decimal is implied in all whole numbers and follows the ones digit.
3. What is the shortcut for dividing a number by 10, 100, 1000, and so on?
The shortcut to dividing by a power of ten is to move the decimal point to the left as many places as there are zeros in
the power of ten that you are dividing by. In other words, when dividing by 100, move the decimal point two places
to the left.
4. How is dividing decimal numbers different from dividing whole numbers?
The difference in the process of dividing decimal numbers and whole numbers is the placement of the decimal point.
You also have to determine if the answer is to be rounded at which place value you will need to end the division
process.
5. How are the Methodologies for Dividing Decimal Numbers and Dividing Whole Numbers similar?
The division of decimals methodology uses the whole number division methodology to perform the actual division.
The placement of the decimal is the part that is added to the whole number methodology. Once the decimal placement
is fixed, the division is the same as for whole numbers.
6. How do you determine where the decimal point belongs in the quotient when dividing decimal numbers?
The long division methodology is for whole number division; therefore, the divisor must be a whole number. One
can move the decimal point in a number by multiplying the number by 10, 100, 1000, and so on. Move the decimal
point to the right until the divisor is a whole number; move the decimal in the dividend the same number of spaces;
that is, multiply the dividend by the same number multiplied by the divisor. If the decimal point is moved the same
number of spaces in both the dividend and divisor, then the result has not been changed. The decimal point in the
answer is directly above the decimal point in the dividend. (You can do this because of the Identity Property of
Multiplication).
7. How do you know when to stop the long division process in a decimal number division problem?
The directions will determine when to stop the division process. You need to perform the division to one more place
than the directions state. Then you can round to the indicated place value. For example if you are asked to round to
hundredths, carry out your division to thousandths place, then round.
167
Chapter 3 — Decimal Numbers
8. How do you validate division of decimal numbers when you have to round your answer?
Rounding the answer requires carrying the division one more place than required. Validate by multiplying the
quotient before rounding times the divisor, and then add the remainder. Validate the rounding as usual method or
by using the number line and midpoint.
9. What aspect of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
1) Fill in the following chart with the correct quotients.
Number
÷ 10
÷ 100
÷ 1000
3.72
0.372
0.0372
0.00372
0.005
0.0005
0.00005
0.000005
416.803
41.6803
4.16803
0.416803
Perform each division and validate your answers.
Problem
2) 0.072 ÷ 2.4
168
Worked Solution
Validation
Activity 3.4 — Dividing Decimal Numbers
Problem
Worked Solution
Validation
3) 2.597 ÷ 3.3
Round your answer to
the nearest hundredth.
4) Divide 27.226 by 90.
Round your answer to
the nearest thousandth.
5) 2.1 divides into 526 how
many times?
Round your answer to
the nearest hundredth.
169
Chapter 3 — Decimal Numbers
Problem
Worked Solution
Validation
6) 0.083 ÷ 32.4
Round your answer to
the nearest thousandth.
7) 17 ÷ 103
Round your answer to
the nearest thousandth.
Perform each division, rounding to the decimal place value specified. Validate each answer.
1. 0.00252 ÷ 0.63
0.004
4. 19 ÷ 0.26
(nearest hundredths) 73.08
2. 654.2 ÷ 40.4
(nearest tenths) 16.2
5. 24.73 ÷ 0.21
(nearest hundredths) 117.76
3. 76.58 ÷ 2.51
(nearest hundredths) 30.51
6. 12 ÷ 43
(nearest thousandths) 0.279
7. 9.685 ÷ 30
(nearest thousandths) 0.323
170
Activity 3.4 — Dividing Decimal Numbers
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) 14.703 ÷ 14.1
Round the answer
to the thousandths
place.
Identify Errors
or Validate
You have to carry out
the division to the tenthousandths place in
the quotient to round
it to its thousandths
place.
(Is the tenthousandths digit 5,
<5, or >5?)
Correct Process
Validation
x
−
−
−
−
+
14.703
1.0427
Answer: 1.043
2) 0.154 ÷ 3.2
Round the answer
to the hundredths
place.
32 does not divide
into 15. There
needs to be a zero
(0) above the 5..
171
Chapter 3 — Decimal Numbers
Worked Solution
What is Wrong Here?
3) 17 ÷ 0.23
Round the answer
to the hundredths
place.
4) 13 ÷ 78.1
Round the answer
to the hundredths
place.
Identify Errors
or Validate
The decimal point
in a whole number
is to the right of
the number.
The problem was
set up incorrectly.
The problem
should be 78.1)13
172
Correct Process
Validation