Decimal Numbers and Percent Unit - Nova Scotia School for Adult

Decimal Numbers and Percent Unit - Nova Scotia School for Adult
Decimal Numbers and
Percent Unit
NSSAL
(Draft)
C. David Pilmer
2013
(Last Updated: June 2014)
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This resource is the intellectual property of the Adult Education Division of the Nova Scotia
Department of Labour and Advanced Education.
The following are permitted to use and reproduce this resource for classroom purposes.
• Nova Scotia instructors delivering the Nova Scotia Adult Learning Program
• Canadian public school teachers delivering public school curriculum
• Canadian non-profit tuition-free adult basic education programs
The following are not permitted to use or reproduce this resource without the written
authorization of the Adult Education Division of the Nova Scotia Department of Labour and
Advanced Education.
• Upgrading programs at post-secondary institutions (exception NSCC's ACC)
• Core programs at post-secondary institutions
• Public or private schools outside of Canada
• Basic adult education programs outside of Canada
Individuals, not including teachers or instructors, are permitted to use this resource for their own
learning. They are not permitted to make multiple copies of the resource for distribution. Nor
are they permitted to use this resource under the direction of a teacher or instructor at a learning
institution.
Acknowledgments
The Adult Education Division would like to thank Dr. Genevieve Boulet (MSVU) for reviewing
this resource and providing valuable feedback.
The Adult Education Division would also like to thank the following ALP instructors for piloting
this resource and offering suggestions during its development.
Eileen Burchill (IT Campus)
Lynn Cuzner (Marconi Campus)
Carissa Dulong (Truro Campus)
Krys Galvin (Truro Campus)
Barbara Gillis (Burridge Campus)
Nancy Harvey (Akerley Campus)
Barbara Leck (Pictou Campus)
Suzette Lowe (Lunenburg Campus)
Shelly Meisner (IT Campus)
Alice Veenema (Kingstec Campus)
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Table of Contents
Introduction (for Learners) ………………………………………………………………..
Prerequisite Knowledge ……………………………………………………………………
Introduction (for Instructors) ………………………………………………………………
iv
v
vi
Introduction to Decimal Numbers …………………………………………………………
Comparing Decimals ………………………………………………………………………
Rounding Decimals ……………………………………………………………………….
Equivalent Fractions and Decimals ………………………………………………………..
Introduction to Percent …………………………………………………………………….
Comparing Fractions, Decimals and Percentages …………………………………………
Adding and Subtracting Decimal Numbers ……………………………………………….
Multiplying Decimal Numbers …………………………………………………………….
Dividing Decimal Numbers ………………………………………………………………..
Estimation Questions Involving Percentages ………………………..…………………….
Calculator Questions ……………………………………………………………………….
1
6
13
14
18
26
30
37
43
54
58
Appendix ……………………………………………………………………………………
Connect Four Fraction Decimal Equivalency Game …………………………………..
Connect Four Fraction Percent Equivalency Game ……………………………………
Connect Four Percentage Game ……………………………………………………….
Additional Practice: Ordering Decimals ……………………………………………….
Additional Practice: Ordering Decimals and Fractions ………………………………..
Additional Practice: Ordering Decimals, Fractions and Percentages ………………….
Post-Unit Reflection ……………………………………………………………………
Soft Skills Rubric ………………………………………………………………………
Answers ………………………………………………………………………………...
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63
64
65
66
68
69
72
73
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Introduction (for Learners)
Welcome to the Decimal Numbers and Percent Unit. Like the Fractions Unit, we will initially
spending a bit of time understanding what a decimal and percent are and how to order decimals
and percentages from smallest to largest. This understanding is very important before we try to
introduce operations (i.e. addition, subtraction, multiplication, and division) with decimals and
percents.
Prerequisite Knowledge
This unit was written under the assumption that learners understand the concepts covered in the
Level III Whole Number Operations Unit and Level III Fractions Unit. We will be revisiting
many of the concepts addressed in those units.
The expectation for this unit is that learners are comfortable with:
•
The addition, subtraction, and multiplication of multi-digit numbers.
e.g. 198 + 35
•
e.g. 928 − 294
e.g. 251× 58
Divide a multi-digit number by a single digit number.
e.g. 8456 ÷ 7
•
Ordering fractions from smallest to largest without using a calculator.
e.g. Order 2
1 12 3
6
1 1
4
8
, 1 ,
, 2 ,
,
, 2 , and
from smallest to largest.
10
8 16
5
9
7 12 16
Introduction (for Instructors)
This unit is similar to the Fraction Unit in that learners initially spend a significant amount of
time understanding the magnitude of decimals and percentages, before ever completing
operations with decimals and percentages. We actually tap into much of the information and
understanding that the learners acquired in the Fractions Unit; hence, it is a prerequisite for this
unit. Please ensure that learners arrange fractions, decimals and percentages from smallest to
largest before they proceed to the sections involving operations.
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Introduction to Decimal Numbers
We see decimal numbers everywhere.
• Money:
"Yoshi deposited $210.75 into his bank account."
• Measurement: "The distance from my house to work is 14.3 kilometres."
"The container holds 1.57 litres of fluid."
"The package weighs 6.8 kilograms."
"The property is 0.85 acres in size."
"The winning time in the 100 metre dash was 9.72 seconds."
"When we started the experiment, the fluid was at 18.2oC."
• Probability:
"The probability of obtaining a head when flipping a fair coin is 0.5."
• Statistics:
"The mean weight (i.e. average weight) of males in the class is 85.2 kg."
Decimals are just another way of writing fractions, and vice versa.
1
("one-tenth"), which is represented by the
10
area model on the right, can also be written in its decimal form as 0.1
("zero decimal one").
For example, the fraction
1
("one-hundredth"), which is represented
100
by the area model on the right, can also be written in its decimal form as
0.01 ("zero decimal zero one").
For example, the fraction
Fractional Form
7
("seven tenths")
10
81
("eighty-one hundredths')
100
3
("three hundredths")
100
417
("four hundred seventeen thousandths")
1000
79
("seventy-nine thousandths")
1000
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Decimal Form
0.7
("zero decimal seven")
0.81 ("zero decimal eight one")
0.03 ("zero decimal zero three")
0.417 ("zero decimal four one seven")
0.079 ("zero decimal zero seven nine")
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Please note that many math resources and math teachers will say that decimal numbers should be
read in the same manner as their fractional counterparts.
e.g. 0.7 should be read as "seven tenths."
e.g. 0.81 should be read as "eighty-one hundredths."
e.g. 0.417 should be read as "four hundred seventeen thousandths."
The rationale for this approach from people in the education community is that it forces learners
to understand place value, and therefore conveys a deeper level of understanding. However,
mathematicians disagree with this approach stating that decimals like 0.7 should be read as "zero
decimal seven" because it clearly conveys to the listener that we are dealing with a decimal,
rather than its equivalent fraction. In this resource, we are going to follow the practices of the
mathematicians.
Place Value and Decimals
Ten-Thousandths
Thousandths
Hundredths
Tenths
•
Ones
Tens
Ones
Period
Hundreds
Thousands
Ten Thousands
Hundred Thousands
Thousands
Period
Millions
Ten Millions
Hundred Millions
Millions
Period
Some decimals are larger than 1.
28.93
"twenty-eight decimal nine three"
93
Fractional Form: 28
("twenty-eight and ninety-three hundredths")
100
Expanded Form: 20 + 8 + 0.9 + 0.03
or
1
1
( 2 ×10 ) + (8 ×1) +  9 ×  +  3 × 
 10   100 
4319.2
"four thousand, three hundred nine decimal two"
2
Fractional Form: 4309
("four thousand three hundred nine and two tenths")
10
Expanded Form: 4000 + 300 + 9 + 0.2
or
1
( 4 ×1000 ) + ( 3 ×100 ) + ( 9 ×1) +  2 × 
 10 
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7.065
"seven decimal zero six five"
65
Fractional Form: 7
("seven and sixty-five thousandths')
1000
Expanded Form: 7 + 0.06 + 0.005
or
1
1 
( 7 ×1) +  6 ×  +  5 ×

 100   1000 
25.304
"twenty-five decimal three zero four"
304
Fractional Form: 25
("twenty-five and three hundred four thousandths")
1000
Expanded Form: 20 + 5 + 0.3 + 0.004
or
1
1 
( 2 ×10 ) + ( 5 ×1) +  3 ×  +  4 ×

 10   1000 
Questions
1. What decimal numbers are represented by each of these area models?
(a)
(b)
(c)
Answer: ________
(d)
(e)
Answer: ________
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Answer: ________
Answer: ________
(d)
Answer: ________
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2. Circle the adjoining numbers that are equivalent decimals and fractions.
9
10
0.05
5
10
0.5
46
1000
0.46
1
0.07
7
100
0.087
87
100
0.046
1
8
100
1.8
527
1000
0.0527
37
1000
0.037
67
100
0.13
8
10
0.009
9
100
0.09
37
100
6.7
13
1000
1.26
64
1000
2.07
2
7
10
0.27
3
9
100
1.3
7
1000
2.7
9
1000
3.09
56
1000
0.126
0.39
3
9
10
0.056
56
100
2
3
3.64
2
604
1000
3.604
3
3
3
406
1000
1
8
10
26
100
3. Express each fraction in its decimal form.
(a)
256
=
1000
(d) 1
97
=
1000
(b)
6
=
100
(e) 2
(c) 3
7
=
1000
9
=
10
(f) 13
58
=
100
4. Express each decimal in its fractional form. Do not put the fraction in its simplest form.
(a) 0.95 =
(b) 0.4 =
(c) 4.508 =
(d) 1.08 =
(e) 2.003=
(f) 6.059 =
5. Write the decimal equivalent to each of the following.
(a) thirty-five and six tenths
_____________
(b) seven and nine hundredths
_____________
(c) fifty-eight thousandths
_____________
(d) one thousand and fifteen hundredths
_____________
(e) two hundred six and three hundred nine thousandths
_____________
(f) seventy and one tenth
_____________
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(g) five and thirty-seven thousandths
_____________
(h) four hundred and twenty-nine thousandths
_____________
6. Write each decimal as a fraction, using both numerals and words. A completed example has
been provided.
Fraction
Decimal Numerals Words
4
2
two and four hundredths
e.g.
2.04
100
(a)
32.8
(b)
0.472
(c)
13.067
(d)
7.59
(e)
327.09
7. Write each decimal number in both expanded forms.
(a) 42.8
(b) 9.31
(c) 302.429
(d) 18.034
(e) 4209.07
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Comparing Decimals
In this section we will be comparing decimal numbers, and in a few instances comparing decimal
numbers to fractions. You will have to remember the strategies we used in the Fraction Unit that
you completed earlier.
There are two techniques that we would like you to learn for comparing decimals.
1. Benchmarks
When we were working with fractions, we used the benchmarks 0,
1
, and 1 to gauge the
2
size of fractions. We will do the same for decimals.
Examples of Decimals that
are Close to Zero
1

0.1  i.e.

10 

8 

0.08  i.e.

100 

17 

0.017  i.e.

1000 

Examples of Decimals that
are Close to One Half
6

0.6  i.e.

10 

46 

0.46  i.e.

100 

519 

0.519  i.e.

1000 

Examples of Decimals that
are Close to One
9

0.9  i.e.

10 

94 

0.94  i.e.

100 

4 

1.004  i.e. 1

1000 

Example 1
Order the numbers 0.53, 1.1 and 0.091 from smallest to largest.
Answer:
53 

0.53  or
 is close to one half.
 100 
1

1.1  or 1  is close to one.
10 

91 

0.091  or
 is close to zero.
 1000 
Order from Smallest to Largest: 0.091, 0.53, 1.1
Example 2
Order the numbers 0.952, 0.489,
9
9
, 0.07, and
from smallest to largest.
8
16
Answer:
The question involves decimals and fractions, but we can use the benchmark strategy
with all of these numbers.
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952 

0.952  or
 is close to one (slightly less than one).
 1000 
489 

0.489  or
 is close to one half (slightly less than one half).
 1000 
9
is close to one (slightly more than one).
8
7 

0.07  or
 is close to zero.
 100 
9
is close to one half (slightly more than one half).
16
9
9
Order from Smallest to Largest: 0.07, 0.489,
, 0.952,
8
16
2. Comparing Digits
Start on the left of both numbers and compare corresponding digits. If the digit of one
number is larger, then this is the larger decimal number. If the digits are the same, move one
place to the right and repeat the procedure. In some cases, you might want to add additional
zeros to the decimal number for comparison purposes (e.g. 0.54 = 0.540).
Example 3
Which is larger?
(a) 1.6 or 1.4
(b) 0.576 or 0.582
(c) 2.95 or 2.9
Answers:
(a) Step 1: Start on the left and compare the unit digits
Same
1.6
1.4
Step 2: Move one place to the right (to the tenths) and compare the digits
Different
1.6
1.4
Step 3: Since the 6 is bigger than the 4, we can conclude that 1.6 is larger than 1.4
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(b) Step 1: Unit digits are the same.
Same
0.576
0.582
Step 2: Tenths digits are the same.
Same
0.576
0.582
Step 3: Hundredths digits are different.
Different
0.576
0.582
Step 4: Since the 8 is bigger than the 7, we can conclude that 0.582 is larger than
0.576.
(c) Step 1: Add a zero to 2.9 such that both numbers have three digits
Step 2: Unit digits are the same.
Same
2.95
2.90
Step 3: Tenths digits are the same.
Same
2.95
2.90
Step 4: Hundredths digits are different.
Different
2.95
2.90
Step 5: Since the 5 is bigger than the 0, we can conclude that 2.95 is larger than 2.9.
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Questions
Do not use a calculator for any of these questions.
1
, or 1.
2
Decimal Closest
to:
1. For each of the following decimal numbers, indicate whether it is closer to 0,
Decimal
Closest
to:
Decimal
Closest
to:
(a)
0.6
(b)
0.1
(c)
0.9
(d)
0.08
(e)
0.45
(f)
1.01
(g)
0.502
(h)
0.89
(i)
0.12
(j)
0.901
(k)
0.005
(l)
0.486
(m)
0.892
(n)
0.59
(o)
0.092
2. In each case, you are given two numbers. Circle the larger number. You will have to use the
benchmark strategy because every question deals with both decimals and fractions.
(a)
0.89
1
12
(b)
0.56
7
8
(c)
4
8
0.1
(d)
3
32
0.907
(e)
0.451
3
3
(f)
0.879
5
12
(g)
13
12
0.009
(h)
7
16
0.58
3. In each case, you are given two numbers. Circle the larger number.
(a)
0.7
0.3
(b)
0.47
0.52
(c)
0.198
0.192
(d)
1.24
1.04
(e)
2.04
2.4
(f)
0.09
0.078
(g)
3.1
3.098
(h)
0.57
0.507
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(i)
5.61
5.618
(j)
2.09
1.98
(k)
7.029
7.08
(l)
12.899
12.988
(m) 0.4
0.409
(n)
31.29
31.3005
(o)
3.01
2.999
(p)
7.5
7.809
(q)
15.35
15.2
(r)
0.75
0.739
(s)
15
16
0.44
(t)
1
8
0.81
(u)
0.51
1
10
(v)
2.04
2
(w)
4
1
2
4.7
(x)
3
11
12
5
6
3.009
4. Place the following numbers by the appropriate arrow on the number line below.
1.4, 1.9, 0.6, 2.7, 0.1, 3.1, 0.8, 2.2, 1.3, 2.5
0
1
2
3
5. Place the following numbers by the appropriate arrow on the number line below.
0.54, 2.89, 2.4, 0.097, 1.46, 1.039, 2.62, 1.75, 3.05, 1.95
0
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6. Place the following numbers by the appropriate arrow on the number line below.
7
1
4 3
2.78,
, 3.3, 1.57, 2 , 0.07, 3
, , 1.2, 2.44
10
16
100 8
0
1
2
3
7. Order the following numbers from smallest to largest.
(a) 0.9, 1.3, 0.4, 1.6, 0.2
_________, _________, _________, _________, _________
(b) 0.59, 1.23, 0.08, 0.55, 1.14
_________, _________, _________, _________, _________
(c) 0.8, 0.09, 0.52, 1.01, 0.83, 1.1
_________, _________, _________, _________, _________, _________
(d) 0.26, 0.19, 1, 0.98, 0.3, 0.2
_________, _________, _________, _________, _________, _________
(e) 0.2, 0.08, 0.72, 0.006, 0.24, 0.209
_________, _________, _________, _________, _________, _________
(f) 0.64, 0.7, 0.05, 0.78, 0.619, 0.092, 0.4
_________, _________, _________, _________, _________, _________, _________
(g) 0.542, 0.9,
7
1
, 0.85, , 0.862, 0.3
100
2
_________, _________, _________, _________, _________, _________, _________
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(h) 1
8
43
, 0.16, , 0.201, 0.6, 1.3, 0.649
8
1000
_________, _________, _________, _________, _________, _________, _________
(i) 0.48, 1.002,
1
5
, 0.509, 1 , 0.4, 1.1
16
8
_________, _________, _________, _________, _________, _________, _________
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Rounding Decimals
Let's revisit the rounding rules for whole numbers.
• Decide which is the last digit to keep
• Leave that digit the same if the next digit is less than 5 (this is called rounding down)
• However, increase it by 1 if the next digit is 5 or more (this is called rounding up)
e.g.
e.g.
e.g.
e.g.
When 83 is rounded to the nearest ten, we obtain 80.
When 86 is rounded to the nearest ten, we obtain 90.
When 1452 is rounded to the nearest hundred, we obtain 1500.
When 23 805 is rounded to the nearest thousand, we obtain 24 000.
For this course, we will be rounding decimal numbers to:
• the nearest tenth (i.e. first decimal place)
• the nearest hundredth (i.e. second decimal place)
• the nearest thousandth (i.e. third decimal place)
The rules for rounding decimal numbers are the same as those for rounding whole numbers;
round down if the next digit is less than 5 and round up if the next digit is 5 or more.
e.g. 7.36 rounded to the nearest tenth (i.e. first decimal place) is 7.4.
e.g. 7.34 rounded to the nearest tenth is 7.3.
e.g. 0.5293 rounded to the nearest tenth is 0.5.
e.g. 0.5293 rounded to the nearest hundredth (i.e. second decimal place) is 0.53.
e.g. 0.5293 rounded to the nearest thousandth (i.e. third decimal place) is 0.529.
e.g. 4.196 rounded to the nearest hundredth is 4.20 (not 4.2 ← rounded to nearest tenth)
e.g. 2.63 rounded to the nearest whole number is 3.
Questions
1. (a) When 14.547 is rounded to the nearest hundredth, we obtain __________.
(b) When 14.547 is rounded to the nearest whole number, we obtain __________.
(c) When 14.547 is rounded to the nearest tenth, we obtain __________.
2. (a) When 251.93 is rounded to the nearest tenth, we obtain __________.
(b) When 251.93 is rounded to the nearest ten, we obtain __________.
(c) When 251.93 is rounded to the nearest whole number, we obtain __________.
3. (a)
(b)
(c)
(d)
When 7.0648 is rounded to the nearest thousandth, we obtain __________.
When 7.0648 is rounded to the nearest whole number, we obtain __________.
When 7.0648 is rounded to the nearest tenth, we obtain __________.
When 7.0648 is rounded to the nearest hundredth, we obtain __________.
4. (a)
(b)
(c)
(d)
(e)
When 437.3295 is rounded to the nearest whole number, we obtain __________.
When 437.3295 is rounded to the nearest tenth, we obtain __________.
When 437.3295 is rounded to the nearest thousandth, we obtain __________.
When 437.3295 is rounded to the nearest hundred, we obtain __________.
When 437.3295 is rounded to the nearest hundredth, we obtain __________.
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Equivalent Fractions and Decimals
We are aware that decimals are just another way of expressing fractions, and vice versa.
However, there are some common equivalent fractions and decimals that we should all know offthe-top-of-our-heads.
Fractions with a denominator of 2
1
2
= 0.5
=1
2
2
3
1
= 1= 1.5
2
2
4
=2
2
Notice the resulting sequence: 0.5, 1, 1.5, 2,…
1
(As we go up by with the fractions, the corresponding decimals go up by 0.5)
2
Fractions with a denominator of 3
2
1
= 0.666...
= 0.333...
3
3
= 0.6
= 0.3
3
=1
3
4
1
= 1= 1.333...
3
3
= 1.3
Many of these result in repeating decimals. The line above a digit indicates that digit
repeats indefinitely.
Notice the resulting sequence: 0.3 , 0.6 , 1, 1.3 ,…
1
(As we go up by with the fractions, the corresponding decimals go up by 0.333… or
3
0.3 .)
Fractions with a denominator of 4
1
2
= 0.25
= 0.5
4
4
3
= 0.75
4
4
=1
4
Notice the resulting sequence: 0.25, 0.5, 0.75, 1, 1.25,…
1
(As we go up by with the fractions, the corresponding decimals go up by 0.25)
4
Fractions with a denominator of 5
1
2
= 0.4
= 0.2
5
5
3
= 0.6
5
4
= 0.8
5
Notice the resulting sequence: 0.2, 0.4, 0.6, 0.8, 1,…
1
(As we go up by with the fractions, the corresponding decimals go up by 0.2)
5
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Example 1:
Order the numbers
3
2
1
5
, 0.58, , 0.46, 1.3,
, 1 0.962 from smallest to largest.
5
3
20 9
Answer:
It is important not to convert all the fractions to their decimals equivalents. Such conversions
work in some cases, but not in all. Do not forget to use benchmarks.
3
can be converted to 0.6, its decimal equivalent.
5
2
can be converted to 0.666.. or 0.6 , its decimal equivalent.
3
1
is close to the benchmark 0.
20
5
1
1 is close to (and slightly more than) the benchmark 1 .
2
9
1
5
3 2
, 0.46, 0.58, , , 0.962, 1.3, 1
Appropriate Order:
20
9
5 3
Questions
Do not use a calculator to complete any of these questions.
1. For each of the following decimals, state the equivalent fraction or mixed number.
(a) 0.75 =
(b) 3.5 =
(c) 1.8 =
(d) 2.3 =
(e) 7.2 =
(f) 5.6 =
2. For each of the following decimals, state the equivalent fraction or mixed number.
1
2
1
(a) 9 =
(b) =
(c) 6 =
4
3
5
(d) 7
3
=
4
1
(e) 9 =
2
2
(f) 8 =
3
3. In each case, you are given two numbers. Circle the larger number.
(a)
3
5
(c)
1
(e)
4
NSSAL
©2012
0.65
(b)
3
4
1
3
1.09
(d)
2
1
2
2.6
2
3
4.59
(f)
6
4
5
6.78
15
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C. D. Pilmer
(g)
2
1
4
2.304
(h)
7
1
5
7.19
4. Place the following numbers by the appropriate arrow on the number line below.
7 1
3
2
1
1
1 , 2.069, , 0.9, 3 1.44, 2 , , 3.37, 2.539,
8 3
5
25
4
5
0
1
2
3
5. Place the following numbers by the appropriate arrow on the number line below.
95
9
2
2
1
, 0.098,
, 2.2, 3 , 2.43, 1 , 2.71
1 , 1.87, 3.12,
100
16
5
3
4
0
1
2
3
6. Order the following numbers from smallest to largest.
3 1
11 5
93
(a) 2 , 1 , 1.6, 0.09,
, 2.4, 1 ,
5 4
100
12 8
_________, _________, _________, _________, _________, _________, _________, _________
1
9
3
1
41
(b) 1 , 2 , 0.587, 1 , 2 , 1.58, 2.7,
4 16
3 10
100
_________, _________, _________, _________, _________, _________, _________, _________
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1 9
1
19 4
(c) 2 ,
, 2.011, 0.892, 1 , 2.6, 1 ,
4 16
10
20 5
_________, _________, _________, _________, _________, _________, _________, ________
7. Open-ended Questions (i.e. more than one acceptable answer)
Your Answer
(a)
Provide a decimal number that is between 3.4 and 3.5.
(b)
Provide a mixed number that is between 2.5 and 2.8.
(c)
(d)
1
1
and 1
5
3
2
9
Provide a decimal number that is between
and
10
3
Provide a decimal number that is between 1
(e)
Provide a mixed number that is between 3.1 and 3.3.
(f)
Provide a decimal number that is between 2
4
and 3.01
5
(Have your instructor check your answers to question 7.)
8. With a classmate, friend, or family member, play at least two rounds of the Connect Four
Fraction Decimal Equivalency Game found in the appendix of this resource. Record in the
chart below whom you played and who won.
Opponent
Winner
Round #1
Round #2
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Introduction to Percent
We see percentages everywhere.
• The union negotiated a 2% wage increase for this year.
• The dress is marked 30% off.
• Approximately 70% of the class is female.
• Babe Ruth, who played professional baseball from 1914 to 1935, hit a homerun 11.76%
of the time at bat.
• Candice left a tip of 20% for the exceptional service she received at the restaurant.
• The mortgage rate on Lei's condominium is 5.25% per annum.
The word percent comes from the Latin phrase per centum, which means
"per 100." For example, when one says 13%, it means 13 per 100 and
13
, or by the decimal 0.13 . The
can be represented by the fraction
100
area model for this particular percentage is shown on the right; 13 of the
100 equal parts are shaded.
Percentages are just another way of expressing fractions or decimals; they all mean the same
thing but look slightly different.
Percent
7%
23%
89%
109%
16.7%
0.1%
25%
80%
150%
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Fraction
(or Mixed Number)
7
100
23
100
89
100
109
9
=1
100 100
16.7 167
=
100 1000
0.1
1
=
100 1000
25 1
=
100 4
80 4
=
100 5
150
50
1
= 1= 1
100 100
2
Decimal
0.07
0.23
0.89
1.09
0.167
0.001
0.25
0.80
1.50
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C. D. Pilmer
Converting Percentages to Decimals
Simply remove the percent sign and slide the decimal point two places to the left (i.e. divide by
100).
Example 1
Convert the following percentages to decimals.
(a) 68%
(b) 135%
(c) 15.9%
Answers:
Slide decimal point
two places to the left
Remove percent sign
(a) 68%
68.
135.
1.35
Therefore: 135% = 1.35
0.159
Therefore: 15.9% = 0.159
Slide decimal point
two places to the left
Remove percent sign
(c) 15.9%
Therefore: 68% = 0.68
Slide decimal point
two places to the left
Remove percent sign
(b) 135%
0.68
15.9
Converting Decimals to Percentages
Simply slide the decimal point two places to the right (i.e. multiply by 100) and add the percent
sign.
Example 2
Convert the following percentages to decimals.
(a) 0.52
(b) 2.68
(c) 0.743
Answers:
Slide decimal point
two places to the right
(a) 0.52
Add a percent sign
52.
Slide decimal point
two places to the right
(b) 2.68
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©2012
Therefore: 0.52 = 52%
268%
Therefore: 2.68 = 268%
74.3%
Therefore: 0.743 = 74.3%
Add a percent sign
268.
Slide decimal point
two places to the right
(c) 0.743
52%
Add a percent sign
74.3
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C. D. Pilmer
Converting Percentages to Fractions
Simply drop the percent sign, express as fraction with a denominator of 100, and simplify the
fraction if necessary.
Example 3
Convert the following percentages to fractions (or mixed numbers).
(a) 43%
(b) 65%
(c) 108%
(d) 7.9%
(e) 0.6%
(f) 216.4%
Answers:
With questions (d) through (f), we initially have fractions with decimals in them. We do not
leave the number in this form. If we multiply the numerator and denominator by 10, we can
rectify this problem.
43
(a) 43% =
100
(b) 65%
=
65
65 ÷ 5 13
=
=
100 100 ÷ 5 20
108
8
8÷4
2
(c) 108% = = 1= 1 = 1
100 100 100 ÷ 4
25
(d) 7.9%
=
7.9 7.9 ×10
79
=
=
100 100 ×10 1000
(e) 0.6%
=
0.6 0.6 ×10
6
6÷2
3
=
=
=
=
100 100 ×10 1000 1000 ÷ 2 500
216.4
16.4
16.4 ×10
164
164 ÷ 4
41
(f) 216.4%
= = 2= 2 = 2= 2 = 2
100
1000
100 ×10
1000
1000 ÷ 4
250
Equivalent Fractions, Decimals, and Percentages
In the previous section, we examined equivalent fractions and decimals; we are going to expand
on this slightly by also including equivalent percentages.
Fractions with a denominator of 2
1
2
= 1= 100%
= 0.5
= 50%
2
2
3
= 1.5
= 150%
2
4
= 2= 200%
2
Fractions with a denominator of 3
1
2
= 0.3
= 33.3%
= 0.6
= 66.6%
3
3
3
= 1= 100%
3
4
= 1.3
= 133.3%
3
Fractions with a denominator of 4
1
2
= 0.5
= 50%
= 0.25
= 25%
4
4
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3
= 0.75
= 75%
4
20
4
= 1= 100%
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C. D. Pilmer
Fractions with a denominator of 5
1
2
= 0.4
= 40%
= 0.2
= 20%
5
5
3
= 0.6
= 60%
5
4
= 0.8
= 80%
5
Questions
Calculators are not permitted for any of these questions.
1. For each of the area models below, supply the corresponding percent, decimal, and fraction.
(a)
(b)
(c)
Percent:
Percent:
Percent:
Decimal:
Decimal:
Decimal:
Fraction:
Fraction:
Fraction:
2. When you are downloading program or application for your digital device, you will often see
a bar on your screen indicating what portion of that program or application has been
downloaded at that instant. Below, you have been supplied with download bars with shaded
portions. In each case, estimate the percentage of the program or application that has been
downloaded at that time (i.e. There is a range of acceptable answers.).
(a) Percent: _______
(b) Percent: _______
(c) Percent: _______
(d) Percent: _______
(e) Percent: _______
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3. Convert the following percentages to decimals. No work needs to be shown.
(a) 79% = _______
(b) 16% = _______
(c) 9% = _______
(d) 145% = _______
(e) 29.4% = _______
(f) 7% = _______
(g) 208% = _______
(h) 81.7% = _______
(i) 4.5% = _______
(j) 0.8% = _______
4. Convert the following decimals to percentages. No work needs to be shown.
(a) 0.19 = _______
(b) 0.48 = _______
(c) 1.73 = _______
(d) 0.692 = _______
(e) 0.06 = _______
(f) 2.09 = _______
(g) 0.073 = _______
(h) 1.548 = _______
(i) 0.002 = _______
(j) 1.7 = _______
5. Convert the following percentages to fractions (or mixed number). In some cases, the
fraction will have to be simplified.
(a) 39%
(b) 91%
(c) 16%
(d) 129%
(e) 235%
(f) 5.1%
(g) 4.6%
(h) 48.2%
(i) 0.4%
(j) 320.6%
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C. D. Pilmer
6. Convert the following fractions or mixed numbers to percentages. No work needs to be
shown; it all comes down to remembering equivalent fractions and percentages.
3
2
(a) =
(b) 1 =
4
5
1
(c) 3 =
2
(e)
1
(d) 2 =
3
4
=
5
1
(f) 3 =
4
2
(g) 1 =
3
1
(h) 2 =
5
7. Complete the following table of equivalent fractions, decimals and percentages.
Percent
(a)
(c)
(e)
(g)
Fraction
83
100
3
4
5
Decimal
Percent
(b)
Fraction
Decimal
67%
(d)
5%
0.39
1
(f)
0.719
(h)
3
4
216.3%
8. Of the three percentages supplied, which one makes the most sense in the context of the
given situation.
Situation
It was a fantastic sale item. The price of the item had
been reduced by _______.
120%
3%
40%
(b)
When Jacob renewed his mortgage, he was pleased that
the rate of interest had dropped by _______ per annum.
13%
28%
1%
(c)
With a few more men in the course than woman, we
were not surprised when we were told that _______ of
the class was comprised of women.
45%
62%
18%
Maxine's science teacher was very pleased with Maxine's
performance on the test. Her mark was _______, which
was the highest mark in the class.
72%
96%
58%
Montez was satisfied with the service at the restaurant
and therefore left a tip of _______ for the waiter.
15%
45%
2%
(a)
(d)
(e)
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C. D. Pilmer
9. Below you have been supplied with diagrams of cylindrical containers filled with fluid.
Match each of the numbers below with the most appropriate diagram. Place your answers in
the boxes below each diagram. This is an estimation activity; no calculations are required.
Do not assume that equivalent fractions, decimals, and percentages will be going in the same
4
box. For example 82%, 0.81 and might all go in the same box even though they are not
5
equivalent; they are, however, very close to each other.
2.16
140%
0.8
1
4
7
9
0.52
69%
194%
0.72
2.57
218%
1
3
7
0.91
2
1
6
89%
5
8
51%
7
10
1.93
81%
7
8
0.24
260%
19
20
26%
1.43
1
2
2
1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
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10. Circle the two adjoining numbers that are equivalent decimals, fractions, or percentages.
93
100
9.3%
17
1000
1.7%
0.053
93%
0.8
0.17
5.3%
53
100
1
8
100
0.8%
0.008
0.28
0.53
2.75%
13.1%
1.31
131
10
280%
5.03%
503
1000
275%
131
1000
0.131%
2.75
47%
3.3
1
3
34.7%
0.347
0.275%
4.4
4
4.6
25%
47
100
347%
0.3%
3
10
30%
3
100
3
3
2
4
5
2
3
1.387
138.7%
387
100
0.275
2
3
4
11. Use the numbers in the chart below to correctly complete the following statement.
There are _____ people in the Sampson family. Of those, ____ are female. That means that
percentage of females in this family is _____%, which can also be represented by the fraction
_____. The percentage of males in this family is _____%, which can also be represented by
the fraction _____.
75
6
3
4
1
4
8
25
12. With a classmate, friend, or family member, play at least two rounds of the Connect Four
Fraction Percent Equivalency Game found in the appendix of this resource. Record in the
chart below whom you played and who won.
Opponent
Winner
Round #1
Round #2
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Comparing Fractions, Decimals and Percentages
In this section we will be comparing fractions, decimals and percentages for the purpose of
ordering them from smallest to largest. Most learners will tend to leave the decimals as
1
decimals, convert percentages to decimals mentally, and use the benchmarks 0, , and 1 for
2
fractions. This strategy works well in most, but not all, cases.
Example
Order the following from smallest to largest.
8
1
7
3
107%, , 1.546, 9.8%,
,
, 0.51,
9
100 16
4
Answer:
107% = 1.07 (slightly larger than the benchmark 1)
8
is slightly less than the benchmark 1.
9
1
1.546 is slightly larger than 1.5 or 1 .
2
9.8% = 0.098 (close to the benchmark 0)
1
= 0.01 (close to the benchmark 0, and smaller than 9.8% or 0.098)
100
7
1
is slightly less than the benchmark .
16
2
1
0.51 is slightly more than the benchmark .
2
3
1
= 0.75, which is half way between the benchmarks
and 1.
4
2
Proper Order:
1
7
3 8
, 9.8%,
, 0.51, , , 107%, 1.546
100
16
4 9
Questions
Do not use a calculator to complete any of these questions.
1. For each of the following, indicate whether it is closer to 0,
Closest
to:
1
, or 1.
2
Closest
to:
Closest
to:
(a)
98%
(b)
11%
(c)
45%
(d)
0.02
(e)
0.899
(f)
0.6
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Closest
to:
Closest
to:
Closest
to:
(g)
8
9
(h)
11
20
(i)
1
16
(j)
0.3%
(k)
1.05
(l)
102%
(m)
13
24
(n)
56.2%
(o)
16
15
2. In each case, circle the larger number.
(a)
39%
83%
(b)
14.7%
14.2%
(c)
136%
98%
(d)
3.1%
2.99%
(e)
9%
0.48
(f)
83%
0.78
(g)
2.3%
0.005
(h)
1
12
0.65
(i)
0.99
105%
(j)
1.45
89%
(k)
7
8
29%
(l)
47%
9
16
(m) 135%
14
15
(n)
1
10
2%
(o)
0.198
7
16
(p)
93.5%
3
4
(q)
2
3
30%
(r)
1
(s)
0.546
21.5%
4
5
(t)
7.59%
0.48
1
4
(u)
5
8
0.19
81.2%
(v)
8.3%
1
100
0.009
1
3
215%
3. Place the following by the appropriate arrow on the number line below.
5
1
1
1
200%, 0.93, 2 , 54.7%, 1.099, 3 , 125%,
, 0.422, 2 , 180%
20
4
8
16
0
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2
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C. D. Pilmer
4. Place the following by the appropriate arrow on the number line below.
19
3
3
1 , 2.85, 9.7%, 2 , 0.713, 155%, 3.2, , 1.389, 96%, 215%
20
6
4
0
1
2
3
5. Order the following from smallest to largest.
(a) 32%, 124%, 0.7%, 91.2%, 5.8%
_________, _________, _________, _________, _________
(b) 0.82, 14%, 0.1, 64%, 0.745
_________, _________, _________, _________, _________
(c) 123%, 1.45, 8.2%, 0.61, 57.2%
_________, _________, _________, _________, _________
(d)
5
9
, 0.792, 3.8%, 86%,
12
10
_________, _________, _________, _________, _________
(e) 1.96, 68.5%, 0.4,
1
1
, 1 , 20%
32 10
_________, _________, _________, _________, _________, _________
(f) 0.276, 57.6%,
8
31
, 1.1,
, 30.2%
16
32
_________, _________, _________, _________, _________, _________
1
5
(g) 1 , 209%, 0.89, , 0.096, 64.5%
4
8
_________, _________, _________, _________, _________, _________
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(h) 0.956, 91%,
6
1
, 50.3%, 0.08,
14
100
_________, _________, _________, _________, _________, _________
3
7
1
(i) 1 , 28%, 0.9, , 194%, 1.02,
5
7
3
_________, _________, _________, _________, _________, _________, _________
(j) 214%, 0.34,
1
7
7
, 2 , 94.5%, 1 , 1.092
5
8
12
_________, _________, _________, _________, _________, _________, _________
6. Open-ended Questions (i.e. more than one acceptable answer)
Your Answer
(a)
Provide a percent that is between 34% and 35%.
(b)
Provide a percent that is between
(c)
Provide a percent that is between 0.78 and 0.81.
(d)
Provide a decimal number that is between 7% and 9.3%.
(e)
Provide a decimal number that is between 8.2 and 8.3.
(f)
Provide a decimal number that is between 1
(g)
Provide a mixed number that is between 145% and 156%.
(h)
Provide a mixed number that is between 2
(i)
Provide a mixed number that is between 3.01 and 3.25
3
2
and .
5
5
1
2
and 1 .
10
10
3
and 3.
4
(Have your instructor check your answers to question 6.)
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Adding and Subtracting Decimal Numbers
In the Whole Number Operations Bridging Unit, we learned how to add multi-digit whole
numbers. To accomplish this, we start by stacking the numbers vertically such that
corresponding place values line up (e.g. units with units, tens with tens) and add from right to
left. If the sum in any corresponding place value is 10 or greater, we regroup (i.e. carry the
excess to the next larger place value).
e.g. 158 + 265
Answer:
Add the Units
Add the Tens
↓
↓
1
1
Add the Hundreds
↓
1
1
1
1 5 8
+ 2 6 5
1 5 8
+ 2 6 5
1 5 8
+ 2 6 5
3
2 3
4 2 3
8 units plus 5 units is 13
units. Regroup the 13 to 1
ten and 3 units.
1 ten plus 5 tens plus 6 tens 1 hundred plus 1 hundred
is 12 tens. Regroup the 12 plus 2 hundreds is 4
to 1 hundred and 2 tens.
hundreds
Adding Decimals
We follow the same procedure when adding decimal numbers. We start by stacking the numbers
vertically such that the corresponding place values line up (i.e. tenths with tenths, hundredths
with hundredths, etc.). Again we add from right to left. If the sum in any corresponding place
value is 10 or greater, we regroup (i.e. carry the excess to the next larger place value).
e.g. 0.67 + 2.84
Answer:
Add the Hundredths
↓
1
6 7
8 5
0
+ 2
1
0
+ 2
.
.
2
7 hundredths plus 5
hundredths is 12
hundredths. Regroup the
12 to 1 tenth and 2
hundredths.
NSSAL
©2012
Add the Tenths
↓
1
.
.
6 7
8 5
. 5 2
1 tenth plus 6 tenths plus 8
tenths is 15 tenths.
Regroup the 15 to 1 unit
and 5 tenths. Transfer
down the decimal point.
30
Add the Units
↓
1
0
+ 2
.
.
6 7
8 5
3 . 5 2
1 unit plus 2 units is 3
units. The final answer is
3.52.
Draft
C. D. Pilmer
e.g. 0.471 + 4.89 + 0.055
Answer:
Add the Thousandths
0
4
+ 0
Add the Hundredths
↓
4 7 1
8 9
0 5 5
.
.
.
2
0
4
+ 0
.
.
.
4 7 1
8 9
0 5 5
1 6
6
1 thousandth plus 5 thousandths is 6
thousandths.
7 hundredths plus 9 hundredths plus 5
hundredths is 21 hundredths. Regroup
the 21 to 2 tenths and 1 hundredth.
Add the Tenths
↓
1
Add the Units
↓
2
0
4
+ 0
.
.
.
↓
1
4 7 1
8 9
0 5 5
2
0
4
+ 0
.
.
.
4 7 1
8 9
0 5 5
. 4 1 6
5 . 4 1 6
2 tenths plus 4 tenths plus 8 tenths is
1 unit plus 4 units is 5 units. The final
14 tenths. Regroup the 14 to 1 unit and answer is 5.416.
4 tenths. Transfer down the decimal
point.
e.g. 2.95 + 14.86 + 0.7
Answer:
Add the Hundredths
↓
9 5
8 6
7
1
2
1 4
+
0
.
.
.
1
NSSAL
©2012
Add the Tenths
↓
2
2
1 4
+
0
Add the Units
↓
1
.
.
.
2
9 5
8 6
7
. 5 1
31
2
1 4
+
0
1
.
.
.
9 5
8 6
7
8 . 5 1
Add the Tens
↓
2
2
1 4
+
0
1
.
.
.
9 5
8 6
7
1 8 . 5 1
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C. D. Pilmer
In the Whole Number Operations Bridging Unit, we learned how to subtract multi-digit whole
numbers. To accomplish this, we start by stacking the numbers vertically such that
corresponding place values line up (e.g. units with units, tens with tens) and subtract from right
to left. If the digit being subtracted is larger than the digit from which it is being subtracted,
regroup (i.e. borrow) one from the digit in the next larger place value.
e.g. 392 - 145
Answer:
Subtract the Units
3
− 1
Subtract the Tens
↓
↓
8
12
8
12
9
4
2
5
9
4
2
5
4
7
3
− 1
7
We cannot take 5 units
from 2 units. Therefore we
regroup (i.e. borrow) 1
from the tens.
Subtract the Hundreds
↓
8 tens minus 4 tens is 4
tens.
8
12
9
4
2
5
2 4
7
3
− 1
3 hundreds minus 1
hundred is 2 hundreds.
Subtracting Decimals
We follow the same procedure when subtracting decimal numbers. We start by stacking the
numbers vertically such that the corresponding place values line up (i.e. tenths with tenths,
hundredths with hundredths, etc.). Again we work from right to left. If the digit being
subtracted is larger than the digit from which it is being subtracted, regroup (i.e. borrow) one
from the digit in the next larger place value.
e.g. 5.96 - 3.45
Answer:
Subtract the Hundredths
↓
5
− 3
.
.
9 6
4 5
1
6 hundredths minus 5
hundredths is 1 hundredth.
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Subtract the Tenths
↓
5
− 3
.
.
9 6
4 5
. 5 1
9 tenths minus 4 tenths is 5
tenths. Transfer down the
decimal point.
32
Subtract the Units
↓
5
− 3
.
.
9 6
4 5
2 . 5 1
5 units minus 3 units is 2
units. The final answer is
2.51.
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C. D. Pilmer
e.g. 7.63 - 2.18
Answer:
Subtract the Hundredths
↓
7
− 2
.
.
5
13
6
1
3
8
Subtract the Tenths
↓
7
− 2
5
5
13
.
.
6
1
3
8
.
4
5
Subtract the Units
↓
5
13
.
.
6
1
3
8
5 .
4
5
7
− 2
We cannot take 8
5 tenths minus 1 tenth is 4
hundredths from 3
tenths. Transfer down the
hundredths. Therefore we decimal point.
regroup (i.e. borrow) 1
from the tenths. 13
hundredths minus 8
hundredths is 5 hundredths.
7 units minus 2 units is 5
units. The final answer is
5.45.
e.g. 40.59 - 12.7
Answer:
This question is a little more challenging because in the second step (i.e. subtracting the
tenths), we cannot initially regroup (i.e. borrow) from the units because there are zero units.
That means we have to regroup from the tens to the units, and then the units to the tenths.
Subtract the Hundredths
4 0
− 1 2
.
.
Subtract the Tenths
↓
9
↓
3
10
5 9
7
4
1
0
2
−
15
.
.
9
Subtract the Tens
9
−
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9
. 8 9
Subtract the Units
↓
3
10
4
1
0
2
.
.
7
. 8 9
15
5
7
5
7
9
−
33
↓
9
3
10
4
1
0
2
.
.
5
7
2 7
.
8 9
15
9
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C. D. Pilmer
e.g. 6.35 - 0.728
Answer:
Initially change 6.35 to 6.350; they are equivalent decimals.
Subtract the Thousandths
↓
6
− 0
.
.
3
7
4
10
5
2
0
8
Subtract the Hundredths
↓
6
− 0
.
.
3
7
4
10
5
2
0
8
2
2 2
Subtract the Tenths
↓
Subtract the Units
↓
5
−
6
0
.
.
13
4
10
5
3
7
5
2
0
8
6
0
.
.
5
. 6 2 2
−
. 6 2 2
13
4
10
3
7
5
2
0
8
Questions
Do not use a calculator for any of these questions.
1. Complete the indicated operation. Show all your work.
(a) 12.72 + 34.16
(b) 62.53 + 7.31
(c) 38.6 + 50.27
(d) 6.423 + 0.39
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(e) 6.39 + 35.572
(f) 142.8 + 87.53
(g) 0.265 + 6.81 + 38.7
(h) 7.46 + 0.085 + 0.93
2. Complete the indicated operation. Show all your work.
(a) 64.87 - 21.52
(b) 6.95 - 2.91
(c) 3.547 - 1.819
(d) 13.52 - 7.8
(e) 9.28 - 0.415
(f) 7.042 - 0.36
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(g) 28.049 - 6.27
(h) 7.406 - 0.85
3. A container filled with water weighs 4.56 kilograms. Once the water is removed, the
container weighs 0.89 kilograms. What was the weight of the water that was removed?
4. Jack gained 1.36 kilograms in the first week and 2.06 kilograms in the
second week. How much weight did he gain over that two week period?
5. The odometer on Akira's car initially read 23 467.4 kilometers. After driving 825.7
kilometres, what would be the new odometer reading?
6. Montez's time on the 100 metre dash was 10.54 seconds. Hinto's time was 10.92 seconds.
How many seconds earlier did Montez arrive at the finishing line as compared to Hinto?
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Multiplying Decimal Numbers
In the Level III Whole Number Operations Unit, we learned three techniques for multiplying
whole numbers: traditional algorithm, multiplying using the expanded form, and the lattice
method. You chose the method you preferred; the same will apply here in this section.
e.g. 67 × 49
Traditional
Algorithm
6 7
× 4 9
6 0 3
2 6 8 0
3 2 8 3
Using
Expanded
Form
Lattice Method
60 + 7
× 40 + 9
6
5 4
2 8
2 4 0
carry 1
6
carry
7
3
0
0
0
4
3
9
2
2
2
4
5
8
6
4
8
3 2 8 3
3
3
Multiplying Decimals
1. Multiply the decimals as though they were whole numbers (i.e. initially ignore the decimal
points)
2. The decimal point in the product is placed so that the number of decimal places in the
product is equal to the sum of the number of decimal places in the factors.
Note: When multiplying decimals, you do not need to line up the decimal points, unlike question
involving addition and subtraction of decimal numbers.
Example 1
Complete the indicated operation.
(a) 67 × 4.9
(b) 6.7 × 4.9
(c) 0.67 × 4.9
(d) 0.67 × 0.49
Answers:
These questions were chosen because all of their solutions rely on knowing that
67 × 49 =
3283 , which was calculated above.
(a) 67 - zero decimal places
4.9 - one decimal place
The final answer should have one decimal place (0 + 1 = 1)
Therefore: 67 × 4.9 =
328.3
(b) 6.7 - one decimal place
4.9 - one decimal place
The final answer should have two decimal places (1 + 1 = 2)
Therefore: 6.7 × 4.9 =
32.83
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(c) 0.67 - two decimal places
4.9 - one decimal place
The final answer should have three decimal places (2 + 1 = 3)
Therefore: 0.67 × 4.9 =
3.283
(d) 0.67 - two decimal places
0.49 - two decimal places
The final answer should have four decimal places (2 + 2 = 4)
Therefore: 0.67 × 0.49 =
0.3283
Example 2
Complete the following operation. Show all your work.
49.7 × 0.53
Answer:
Change the question to 497 × 53 ; we will deal with the decimals points in a later step. Again,
you can choose one of the three multiplication techniques that you prefer.
Traditional
Algorithm
4 9 7
×
5 3
1 4 9 1
2 4 8 5 0
2 6 3 4 1
Using Expanded Form
Lattice Method
400 + 90 + 7
×
50 + 3
2
1 2
3
4 5
2 0 0
2
7
0
5
0
0
4
9
7
5
1
0
0
0
0
0
2 6 3 4 1
3
1
2
2
6
5
2
2
3
3
4
0
1
1
5
2
7
4
1
1
We will now consider the decimal points.
49.7 - one decimal place
0.53 - two decimal places
The final answer should have three decimal places (1 + 2 = 3)
Therefore: 49.7 × 0.53 =
26.341
To determine whether the answer is reasonable, round the decimal numbers to numbers that
are more manageable. We could round 49.7 to 50, and round 0.53 to 0.5. Since
50 × 0.5 =
25 , then we can assume that the answer of 26.341 is reasonable.
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Questions
Do not use a calculator for any of these questions.
1. Complete the indicated operation. Show all your work. Please note that we have addition,
subtraction and multiplication questions in here.
(b) 7.43 × 2.6
(a) 3.7 × 6.5
(c) 0.45 × 0.52
(d) 3.4 + 18.92
(e) 1.73 × 4
(f) 0.49 × 9.1
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(g) 83.91 - 7.28
(h) 63 × 0.29
(i) 0.453 × 0.78
(j) 90.36 - 12.5
2. For each question, we have provided three possible solutions. Use your estimation skills to
determine which of three the correct answer is. We do not want you to work these out on
paper or use a calculator. Instead, we want you to round the decimals to numbers that are
more manageable, and estimate the final answer in your head. For example, 9.13 × 4.9 could
be changed to 9 × 5 , which has a product of 45. You would then look for the answer that is
close to 45.
(a) 7.08 × 3.2
10.28
22.656
2.2125
(b) 82.53 − 6.89
75.64
89.42
568.6317
(c) 153.6 + 23.7
129.9
194.3
177.3
(d) 4.3 × 89.5
384.85
512.75
319.65
(e) 452.5 + 351.7
924.2
734.2
804.2
(f) 51.3 × 49.68
2548.584
3264.374
1956.924
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3. Six questions are supplied below. You must use your estimation skills to determine which
arrow on the number line below best represents the solution to each of the six questions.
Question
0.98 × 2.1
5.23 − 4.37
1.1 + 1.97
a
0
Arrow
b
Question
0.326 + 2.21
2.34 − 1.98
0.49 × 2.88
Arrow
d
c
1
e
2
f
3
4. Use your estimation skills to match up each question with each answer.
Questions
(a) 2.93 + 3.208
Answers
32.185
(b)
16.08 - 5.239
311.74
(c)
7.85 × 4.1
21.78
(d)
47.9 + 32.7
6.138
(e)
29.58 - 7.8
126.7
(f)
39.8 × 6.1
80.6
(g)
98.3 + 28.4
1110.9
(h)
409.8 - 98.06
10.841
(i)
52.9 × 21
242.78
5. John's car holds 48.7 litres of gas. If his vehicle can travel 15
kilometres on a litre of gas, how far can it travel on a full tank?
6. Kadeer had $14.15 in his iTunes account. If he purchases a song
for $1.29 from iTunes, how much will be left in his account?
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7. If Meera makes $12.65 per hour, how much will she make, before
deductions, in a 38 hour work week?
8. The shrub is 38.7 cm tall. They expect that it will grow an
additional 3.5 cm over the year. What is the expected height of the
tree in a year's time?
9. If each plastic pellet weighs 0.58 grams, how much does 45 pellets
weigh?
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Dividing Decimal Numbers
In this section, we will show you two ways to solve these types of division question. One
technique uses the traditional algorithm; the other uses the partial quotient method. You chose
the method you prefer.
Dividing a Decimal Number by a Whole Number
With a first few questions we are only going to be looking at questions where we are dividing a
decimal number by a whole number (e.g. 165.2 ÷ 7 , 4.23 ÷ 9 ).
Example 1
Complete the operation 165.2 ÷ 7 .
Answer:
In these explanations, and the ones that follow, we will be using the
terms divisor, quotient, and dividend. These terms have been
described in the diagram on the right. For this particular question,
the dividend is 165.2, the divisor is 7, and the quotient is the final
answer.
Traditional Algorithm
Do the long division as you would
with whole numbers, then place the
decimal point in the quotient directly
above the decimal point in the
dividend.
23.6
7 165.2
-14
25
- 21
42
- 42
0
Therefore: 165.2 ÷ 7 =
23.6
quotient
divisor dividend
Partial Quotient Method
Initially ignore the decimal point and pretend that
you are dividing two whole numbers.
236
7 1652
200
1400
252
30
210
42
6
42
0
Now move the decimal point in the quotient, the
same number of places and in the same direction
as the decimal point in the dividend. In this case,
the dividend should be 165.2, where the decimal
point is one place to the left. Therefore our
quotient should be 23.6; notice that the decimal
point is also one place to the left.
Therefore: 165.2 ÷ 7 =
23.6
This answer looks reasonable because we know that 140 ÷ 7 =
20 , therefore we would expect
that 165.2 ÷ 7 would be a little more than 20.
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Example 2
Complete the operation 4.23 ÷ 9 .
Answer:
Traditional Algorithm
Do the long division as you would
with whole numbers, then place the
decimal point in the quotient directly
above the decimal point in the
dividend.
0.47
9 4.23
- 36
63
- 63
0
Therefore: 4.23 ÷ 9 =
0.47
Partial Quotient Method
Initially ignore the decimal point and pretend that
you are dividing two whole numbers.
47
9 423
40
360
63
7
63
0
In this case, the dividend should be 4.23, where
the decimal point is two places to the left.
Therefore our quotient should be 0.47; notice that
the decimal point is also two places to the left.
Therefore: 4.23 ÷ 9 =
0.47
4.5
1
is equal to
or 0.5.
9
2
Therefore we would expect that 4.23 ÷ 9 is slightly less than 0.5.
This answer looks reasonable because we know that 4.5 ÷ 9 or
Example 3
Match each division question with the appropriate answer. We are not asking you to work these
out using paper-and-pencil or a calculator; rather, we are asking you to use your estimation
skills.
Questions
(a) 2090.2 ÷ 7
(b) 51.84 ÷ 6
(c) 315.84 ÷ 8
(d) 2.13 ÷ 4
(e) 467.37 ÷ 9
Answers
39.48
0.5325
298.6
51.93
8.64
Answers:
• We know that 2100 ÷ 7 =
300 , therefore 2090.2 ÷ 7 is likely equal to 298.6.
• We know that 48 ÷ 6 =
8 , therefore 51.84 ÷ 6 is likely equal to 8.64.
• We know that 320 ÷ 8 =
40 , therefore 315.84 ÷ 8 is likely equal to 39.48.
2
• We know that 2 ÷ 4 or is equal to 0.5, therefore 2.13 ÷ 4 likely equals 0.5325.
4
• We know that 450 ÷ 9 =
50 , therefore 467.37 ÷ 9 is likely equal to 51.93.
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Briefly Revisiting Fractions
We know that fractions are one way of expressing the operation of division.
3
= 3÷4
4
e.g.
e.g.
9
= 9÷5
5
e.g.
27
= 27 ÷ 100
100
We also know that equivalent fractions can be created by multiplying or dividing the numerator
and denominator of a fraction by the same number.
12 12 ÷ 3 4
e.g.= =
15 15 ÷ 3 5
35 35 ÷ 5 7
e.g.= =
20 20 ÷ 5 4
20 20 ÷ 10 2
e.g.= =
30 30 ÷ 10 3
7 7 × 2 14
e.g.= =
8 8 × 2 16
2 2×3 6
e.g. = =
5 5 × 3 15
9 9 ×10 90
e.g.= =
4 4 × 10 40
We will use both of these pieces of knowledge to help us understand the first step in dividing a
decimal number by another decimal number.
Consider the question 9 ÷ 0.6 .
9
.
0.6
• We could create an equivalent fraction by multiplying the numerator and denominator
by 10.
9
9 ×10
90
= =
0.6 0.6 ×10 6
90
• The
can be expressed as 90 ÷ 6 .
6
• We have shown that the answer (i.e. quotient) to 9 ÷ 0.6 (or 0.6 9 ) is equal to answer
•
The question 9 ÷ 0.6 can be expressed as
to 90 ÷ 6 (or 6 90 ).
Consider the question 8.1 ÷ 0.03 .
8.1
.
0.03
• We could create an equivalent fraction by multiplying the numerator and denominator
by 100.
8.1
8.1×100 810
= =
0.03 0.03 ×100
3
810
• The
can be expressed as 810 ÷ 3 .
3
• We have shown that the answer (i.e. quotient) to 8.1 ÷ 0.03 (or 0.03 8.1 ) is equal to
•
The question 8.1 ÷ 0.03 can be expressed as
answer to 810 ÷ 3 (or 3 810 ).
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Consider the question 7.675 ÷ 0.5 .
7.675
.
0.5
• We could create an equivalent fraction by multiplying the numerator and denominator
by 10.
7.675 7.675 ×10 76.75
= =
0.5
0.5 ×10
5
76.75
can be expressed as 76.75 ÷ 5 .
• The
5
• We have shown that the answer (i.e. quotient) to 7.675 ÷ 0.5 (or 0.5 7.675 ) is equal
•
The question 7.675 ÷ 0.5 can be expressed as
to answer to 76.75 ÷ 5 (or 5 76.75 ).
We can take this and apply it to a variety of division questions.
e.g. 0.7 3.01 = 3.01 ÷ 0.7 =
3.01 3.01×10 30.1
=
=
= 7 30.1
0.7
0.7 ×10
7
5
5 ×10
50
e.g. 0.8 5 =÷
5 0.8 = =
= =
8 50
0.8 0.8 ×10 8
e.g. 0.02 9.28 = 9.28 ÷ 0.02 =
9.28 9.28 ×100 928
=
=
= 2 928
0.02 0.02 ×100
2
e.g. 0.09 0.0387 = 0.0387 ÷ 0.09 =
0.0387 0.0387 ×100 3.87
=
=
= 9 3.87
0.09
0.09 ×100
9
e.g. 0.004 0.0224 =0.0224 ÷ 0.004 =
0.0224 0.0224 ×1000 22.4
=
=
=4 22.4
0.004
0.004 ×1000
4
Let's look at all the division questions with equivalent quotients that we have discussed in the
last two pages.
0.6 9 = 6 90
0.03 8.1 = 3 810
0.5 7.675 = 5 76.75
0.7 3.01 = 7 30.1
0.8 5 = 8 50
0.02 9.28 = 2 928
0.09 0.0387 = 9 3.87
0.004 0.0224 = 4 22.4
Notice that every case we started with a question where we were dividing by a decimal number,
but in the end we had changed the question to one where we were dividing by a whole number.
We should be able to solve new question as we have already learned how to divide a decimal by
a whole number.
So how could we describe the process of changing a division question from one where we are
dividing by a decimal number to one where we are dividing by a whole number? We obviously
do not want to go through the lengthy process of converting the division question to fraction
question, creating an equivalent fraction, and then converting from a fraction question to a
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division question. Instead, we use the following shortcut. Start by
quotient
moving the decimal point to the right in the divisor until the divisor is a
divisor dividend
whole number. Then move the decimal point to the right in the dividend
the same number of places as was moved for the divisor. If you move
both one place to the right, it is equivalent to multiplying the numerator
and denominator of a fraction by 10. If you move both two places to the right, it is equivalent to
multiplying the numerator and denominator of a fraction by 100.
Dividing a Whole Number or Decimal Number by a Decimal Number
Step 1: Move the decimal point to the right in the divisor until the divisor is a whole number.
Step 2: Move the decimal point to the right in the dividend the same number of places as was
done in Step 1.
Step 3: Divide through using the procedure that you prefer for dividing a decimal number by a
whole number (i.e. what we did in Examples 1 and 2)
Example 4
Complete the operation 21.87 ÷ 0.9 .
Answer:
Regardless of whether you prefer the traditional algorithm or the partial quotient method, you
must start by changing the divisor (0.9) to a whole number. This is accomplished by moving
the decimal point one place to the right in the divisor. We must then move the decimal point
one place to the right in the dividend. This means that the question changes from 21.87 ÷ 0.9
to 218.7 ÷ 9
Traditional Algorithm
Partial Quotient Method
243
24.3
9 218.7
-18
38
- 36
27
- 27
0
If 218.7 ÷ 9 =
24.3 ,
then 21.87 ÷ 0.9 =
24.3
9 2187
200
1800
387
40
360
27
3
27
0
Now move the decimal point in the quotient, the
same number of places and in the same direction
as the decimal point in the dividend.
If 218.7 ÷ 9 =
24.3 ,
then 21.87 ÷ 0.9 =
24.3
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Example 5
Complete the operation 0.2922 ÷ 0.06 .
Answer:
Start by changing the divisor (0.06) to a whole number. This is accomplished by moving the
decimal point two places to the right in the divisor. We must then move the decimal point
two places to the right in the dividend. This means that the question changes from
0.2922 ÷ 0.06 to 29.22 ÷ 6 .
Traditional Algorithm
Partial Quotient Method
487
4.87
6 29.22
- 24
52
- 48
42
- 42
0
If 29.22 ÷ 6 =
4.87 ,
then 0.2922 ÷ 0.06 =
4.87
6 2922
400
2400
522
80
480
42
7
42
0
Now move the decimal point in the quotient, the
same number of places and in the same direction
as the decimal point in the dividend.
If 29.22 ÷ 6 =
4.87 ,
then 0.2922 ÷ 0.06 =
4.87
Example 6
Match each division question with the appropriate answer. In many cases, you may wish to
move the decimal points in both divisor and dividend to make the question more manageable.
We are not asking you to work these out using paper-and-pencil or a calculator; rather, we are
asking you to use your estimation skills.
Questions
(a) 58.08 ÷ 0.8
(b) 0.3474 ÷ 0.06
(c) 83.72 ÷ 0.4
(d) 880.2 ÷ 9
(e) 0.0336 ÷ 0.07
Answers
209.3
0.48
5.79
72.6
97.8
Answers:
• Change 58.08 ÷ 0.8 to 580.8 ÷ 8 . We know that 560 ÷ 8 =
70 , therefore 580.8 ÷ 8 (or
58.08 ÷ 0.8 ) is likely equal to 72.6.
• Change 0.3474 ÷ 0.06 to 34.74 ÷ 6 . We know that 36 ÷ 6 =
6 , therefore 34.74 ÷ 6 (or
0.3474 ÷ 0.06 ) is likely equal to 5.79.
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•
•
•
Change 83.72 ÷ 0.4 to 837.2 ÷ 4 . We know that 800 ÷ 4 =
200 , therefore 837.2 ÷ 4 (or
83.72 ÷ 0.4 ) is likely equal to 209.3
We know that 900 ÷ 9 =
100 , therefore 880.2 ÷ 9 is likely equal to 97.8.
3.5
Change 0.0336 ÷ 0.07 to 3.36 ÷ 7 . We know that 3.5 ÷ 7 or
equals 0.5. Therefore
7
it is likely that 3.36 ÷ 7 (or 0.0336 ÷ 0.07 ) is equal to 0.48.
Questions
Do not use a calculator to complete any of these questions.
1. Complete each of the operations. Show all your work.
(a) 32.04 ÷ 6
(b) 240.3 ÷ 9
(c) 2.415 ÷ 5
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2. Match each division question with the appropriate answer. We are not asking you to work
these out using paper-and-pencil or a calculator; rather, we are asking you to use your
estimation skills.
Questions
(a) 389.6 ÷ 8
(b) 49.02 ÷ 6
(c) 2.024 ÷ 4
(d) 257.8 ÷ 9
(e) 32.83 ÷ 7
Answers
0.506
48.7
4.69
8.17
28.62
3. In each case, four division questions have been provided. From the last three division
questions, circle the one which has the same quotient (i.e. generates the same answer) as the
first division question. You do not want you to work any of these out using paper-and-pencil
or a calculator.
(a) 0.6 45.36
6 4.536
6 453.6
6 4536
(b) 8 7.36
0.8 73.6
0.8 0.736
0.8 736
(c) 0.5 385.6
5 3856
5 38.56
5 3.856
(d) 8 27.345
0.8 273.45
0.8 2734.5
0.8 2.7345
(e) 0.5 49
5 4.9
5 490
5 0.49
(f) 0.06 8.2
6 820
6 82
6 0.082
(g) 3 182.6
0.03 18260
0.03 18.26
0.03 1.826
(h) 0.07 58.2
7 582
7 5820
7 0.582
4. Complete the following operations. Show all your work. Please note that we have also
included a few addition, subtraction, and multiplication questions.
(a) 4.48 ÷ 0.8
(b) 2.58 ÷ 0.03
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(c) 7.59 + 12.8
(d) 66.01 ÷ 0.7
(e) 4.7 × 6.8
(f) 3.4 ÷ 0.05
(g) 0.2616 ÷ 0.3
(h) 183.2 − 59.16
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5. Match each question with the appropriate answer. Note that we have included addition,
subtraction, multiplication, and division questions. With some of the division questions, you
may wish to move the decimal points in both divisor and dividend to make the question more
manageable. We are not asking you to work these out using paper-and-pencil or a calculator;
rather, we are asking you to use your estimation skills.
Questions
(a) 639.1 ÷ 7
(b) 289.4 + 315.7
(c) 23.76 ÷ 0.6
(d) 20.5 × 61.8
(e) 0.24 ÷ 0.5
(f)
453.6 − 198.8
(g) 0.496 ÷ 0.08
(h) 0.9 ×135.6
(i)
32.68 ÷ 0.4
Answers
39.6
122.04
0.48
91.3
254.8
6.2
817
1266.9
605.1
6. Six questions are supplied below. You must use your estimation skills to determine which
arrow on the number line below best represents the solution to each of the six questions.
Question
6.65 ÷ 7
1.49 + 1.04
0.145 ÷ 0.05
a
0
Arrow
b
Question
3.05 − 2.97
0.31× 5.2
1.278 ÷ 0.6
Arrow
c
1
d
2
e
f
3
7. The prize money of $169.80 has to be shared equally by 6 people.
How much money does each person get?
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8. Hamid uses 0.2 kg of ground beef when making a single hamburger
patty. How many patties can he make using 3.64 kg of ground beef?
9. Ryan makes $13.50 per hour. How much will he make, before deductions, if he works 6.5
hours?
10. Tylena paid $5.79 for 0.6 kg of meat. How much would one kilogram
of the meat cost?
11. Jessie cycled 85.3 km on day one, 93.6 km on day two, and 78.8 km on
day three. How far did she cycle in that three day period?
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Estimation Questions Involving Percentages
We use percentages every day when we work out the price of an item after taxes, determine the
sale price of an item, and calculate the tip for your waiter or waitress. In many cases, we use our
estimation skills when addressing these real-life situations.
•
In Nova Scotia, when you purchase most items, you have to pay a 15% harmonized sales
tax on those goods. For example, a bedroom suite advertised at $1395 will be subject to
sales tax. It is important that you be able to estimate the tax on that purchase and the total
cost of the purchase.
•
A discount is a reduction in a price. When a discount on an item is offered, the rate of
discount is often advertised as a percent of the regular price. For example a sofa,
regularly priced at $799, may be advertised as 25% off during a particular sale. It is
important to be able to estimate the cost of the sofa after the discount so that you are not
overcharged for that item.
•
When you go out to a restaurant for a meal, you are expected to tip the waiter or waitress
for good service. Typically people tip between 15% (good service) and 20% (exceptional
service). It is important that you be able to mentally calculate these tips so that the waiter
or waiter receives appropriate amount for their level of service.
Below we solve a variety of estimation questions involving percentages. As with any estimation
question, there are a variety of ways of obtaining a reasonable estimate. In our solutions, we
have only provided one reasonable estimate. We have tried to use the most common approach in
each case, but we recognize that there are other perfectly acceptable techniques.
Example 1
Your bill at a local restaurant is $68.95. The waitress offered exceptional service and you decide
to give a tip of approximately 20%. How much money should she receive?
Answer:
• Round $68.95 to $70.
• We know that 10% of $70 is $7.
• Therefore 20% of $70 is $14.
• The tip for the waitress should be approximately $14.
Example 2
Jorell needs to purchase a new mattress for his bed. It costs $795 before taxes. He has to pay
15% tax. Approximately how much will he pay in taxes, and what will be the approximate cost
of this purchase?
Answer:
• Round $795 to $800
• We know that 10% of $800 is $80, and that 20% of $800 is $160.
• Therefore 15% of $800 would have to be half way between $80 and $160. That means
the tax on the mattress would be approximately $120.
• That means that the total cost who be slightly less than $920 ($800 + $120).
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Example 3
Lei is going to purchase a sofa that is regularly priced at $1195.
Today the sofa is marked down by 30%. What is the approximate
sales price of this item?
Answer:
• Round $1195 to $1200.
• If the price is reduced by 30%, then 70% of the price is retained.
• If 10% of $1200 is $120, then 70% of $1200 is found by multiplying $120 by 7.
• Since 120 × 7 =
840 , then we can conclude that the sale price of the sofa is approximately
$840.
Example 4
The new jeans, Nasrin is interested in, regularly cost $79. Today they are marked down by 25%.
How much will she pay approximately for these jeans including the 15% sales tax?
Answer:
• Round $79 to $80.
• 25% off is the same as one-quarter off.
1

• One-quarter  i.e.  of $80 is $20.
4

• If the price is reduced by $20, then the approximate sale price of the jeans is $60.
• Now we need to determine the 15% sales tax. We know that 10% of $60 is $6, and that
20% of $60 is $12. Therefore 15% of $60 will be halfway between $6 and $12. The
sales tax will be approximately $15 on this item.
• The total cost of the jeans will be approximately $75 ($60 + $15).
Questions
Do not use a calculator on any of these questions.
1. Solve each of the following. No work needs to be shown (i.e. Do it in your head.).
(a) What is 10% of 40?
_______
(b) What is 10% of 120?
_______
(c) What is 10% of 500?
_______
(d) What is 10% of 1400?
_______
(e) What is 20% of 40?
_______
(f) What is 20% of 120?
_______
(g) What is 20% of 500?
_______
(h) What is 20% of 1400?
_______
(i) What is 15% of 40?
_______
(j) What is 15% of 120?
_______
(k) What is 15% of 500?
_______
(l) What is 15% of 1400?
_______
(m) What is 30% of 40?
_______
(n) What is 30% of 120?
_______
(o) What is 25% of 400?
_______
(p) What is 25% of 1200?
_______
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2. Manish's bill at a local restaurant is $49.45. The waiter offered
exceptional service so Manish decides to give a tip of approximately
20%. How much money should the waiter receive?
3. Krys is purchasing a fall jacket for her son. It costs $39.95. Approximately how much will
she have to pay after taxes (15%) for this item?
4. Ryan is purchasing a DVD boxed set of Season 13 of The Simpsons. It normally sells for
$29.95 but today it is marked down by 30%. What is the approximate sale price of this item?
5. Alice received satisfactory service at the restaurant and therefore felt it was reasonable to
leave a 15% tip on her $81.35 bill. Approximately how much should she leave?
6. All spring stock was marked down by 40% in a local clothing store.
Approximately how much would one pay, after taxes (15%), for a
spring dress regularly costing $59.95?
7. The $160 electronic device was marked down by 25% because a newer model of the same
device was now on the market. Approximately how much will it cost for this discounted
device after the paying sales tax (15%)?
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8. The demand for a particular running shoe was much higher than the
manufacturer expected. They were originally going to sell the shoes
for $89.95 a pair. The manufacturer decides to increase the price by
20%. If they do this, what would be approximate new cost of the
shoes before taxes?
9. The regular price of a season's pass for skiing is $295. If you purchase the pass early, you
can save 30%. What is the approximate total cost, after taxes (15%), if you purchase this
early-bird season's pass?
10. Eight questions are supplied below. You must use your estimation skills to determine which
arrow on the number line below best represents the solution to each of the eight questions.
Question
25% of 11.90
0.784 ÷ 0.4
1.43 − 1.316
30% of 4.90
a
0
Arrow
c
b
Question
0.52 + 0.496
10% of 21.50
20% of 2.99
2.1×1.513
d
Arrow
e
1
f
2
g
h
3
11. With a classmate, friend, or family member, play at least two rounds of the Connect Four
Percent Game found in the appendix of this resource. Record in the chart below whom you
played and who won.
Opponent
Winner
Round #1
Round #2
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Calculator Questions
This is one of the few times in this course where we will allow you to use a
calculator to solve problems. Our rationale is that you should know all the
fundamentals concerning decimals and percentages at this point in time, and that
we now want to expose you to multi-step problems with "messier" numbers that
are better handled with a calculator, as opposed to using paper-and-pencil techniques.
Example 1
John was born on July 8, 1955. In an attempt to get his twelve grandchildren to remember his
birthday, John gives each child $78.55 cash at Christmas. How much money should he take out
of his account to cover his grandchildren's gifts?
Answer:
• Simply multiply 12 by 78.55 on the calculator.
• John needs to take out $942.60 to cover the gifts.
Example 2
Suzzette has a 53.5 litre container of water. She wants to know how many 2.45 litre containers
she can completely fill using the larger container of water.
Answer:
• Using a calculator: 53.5 ÷ 2.45 =
21.8 (rounded to one decimal place)
• Normally we would round 21.8 up to 22, but this question is asking us how many
"containers she can completely fill." For this reason, we will round down and say that
she can completely fill 21 containers.
Example 3
In week one, Carissa's expenses were $496.65, and her earnings were $757.50. The money not
spent went into her savings. In week two she hopes to save twice as much money as week one.
If she is able to do this, how much money will go into her savings in week two?
Answer:
• Her Savings on Week One: 757.50 - 496.65 = $260.85
• Her Desired Savings on Week Two: 2 × 260.85 = $521.70
Example 4
Barb is purchasing at shirt priced at $18.95, a pair of jeans at $46.95, and a
knapsack at $39.95. What is the total cost after tax (15%)?
Answer:
• Find the total before tax.
18.95 + 46.95 + 39.95 = 105.85
• Determine the tax.
15% of 105.85 = 0.15 ×105.85 =
15.88 (rounded to the second decimal point)
• Find the total after tax.
105.85 + 15.88 = $121.73
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Example 5
At one particular store, customers can receive a 30% discount if their
purchases before the discount total $150 or more. Shelly plans on
purchasing a $89.45 set of bath towels and facecloths, a $49.95 set of
blinds for her bedroom window, and a $17.45 toaster. What is her
total cost after tax (15%)?
Answer:
• Find the total before taxes.
89.95 + 49.95 + 17.45 = 157.35
• Since their total purchase exceeds $150, they are able to receive the 30% discount.
• If the price is reduced by 30%, then 70% of the price is retained. Take 70% of $157.35 to
find the new total (before taxes).
70% of 157.35 = 0.70 ×157.35 =
110.15 (rounded off)
• Determine the tax.
15% of 110.15 = 0.15 ×110.15 =
16.52 (rounded off)
• Find the total after tax.
110.15 + 16.52 = $126.67
Example 6
Meera works 47 hours this week. She gets $15.60 per hour for the first 40
hours. She gets "time-and-a-half" for any hours after the 40 hours; this is
considered overtime. How much will she earn, before deductions, for this
work week?
Answer:
• Earnings for the First 40 Hours of Work: 40 ×15.60 = $624
• Hourly Earnings at Time-and-a-Half: 1.5 ×15.60 = $23.40 per hour
• Earnings for the 7 Hours of Overtime: 7 × 23.40 = $163.80
• Total Earnings: 624 + 163.80 = $787.80
Example 7
Nashi makes $13.20 per hour plus a 2.5% commission on all her sales. If she works 36 hours
and makes $6490 worth of sales, what will be her earnings before deductions?
Answer:
• Earnings from Hourly Wage: 36 ×13.20 =
$475.20
• Commission Earnings: 2.5% of $6490 = 0.025 × 6490 =
$162.25
• Total Earnings: 475.20 + 162.25 = $637.45
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Questions
Calculators are permitted with these questions. Show how you solved the each question.
1. If a long-distance phone provider offers a rate of $0.12 per minute, how long can you talk for
$2.76?
2. Suppose it costs $27.50 per day plus $0.11 per kilometer for a rental car. What is the total
bill if you have the car for three days and travel 657 kilometres?
3. Masato purchases a loft of bread ($2.65), a can of beans ($2.29) and hot dogs ($3.89). If he
pays with a $10 bill, how much change will he receive? Please note that sales tax is not
applied to food.
4. The garden center marked all plants down by 60% for their end-of-season sale. How much
would you have to pay after taxes (15%) for a plant that normally cost $129.99?
5. Rana purchases a sweat top ($18.99), jeans ($34.99), and running shoes ($37.99) for each of
her twin boys. What will be the total cost after paying the sales tax (15%)?
6. Last week Meera was making $13.40 per hour and working 36 works. This week her hours
increased to 42 hours and her hourly wage increased to $14.70 per hour. How much will she
make, before deductions, over this two week period?
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7. Kendrick makes $12.50 per hour plus a 3% commission on all his sales. If he worked 38
hours and sold $5840 worth of merchandise this week, what would be his earnings before
deductions?
8. Harris, a pipefitter working in oil project in northern Alberta, has to work a 12 hour shift on a
statutory holiday. For doing so, his employer will pay him time-and-a-half for the first 8
hours and double-time for the remaining 4 hours. If his normal hourly rate is $32.60, how
much will he make, before deductions, for this 12 hour shift?
9. The Boxing Day sale at a local clothing boutique advertised 40% off all purchases. Kimi
wanted to purchase a blouse, regularly priced at $29.95, and a sweater, regularly priced at
$37.59. If she purchases both during the sale, what is the total cost including sales tax
(15%)?
10. Kevin travelled to friend's cottage using his car. The car's odometer initially read 33 407.2
kilometres. Upon arriving at the cottage, the odometer read 33 598.9 kilometres. If he used
13.5 litres of fuel during the trip, how many kilometres per litre did his car achieve on this
trip?
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Appendix
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Connect Four Fraction Decimal Equivalency Game
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. The
square with a specified decimal is captured by creating the equivalent fraction using the
numerator and denominator strips at the bottom of the page. One paper clip is placed on
each strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the
3
denominator, then they can capture one square labeled 0.75 ( is equivalent to 0.75). They
4
either mark the square with an X or place a colored counter on the square. There may be
other squares with that same difference but only one square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with the equivalent decimal using an O or a different
colored marker. If a player cannot move a single paperclip to capture a square, a paperclip
must still be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
0.4
1
0.2
0.4
1
0.5
0.6
0.2
0.4 0.25 0.3
0.4
0.25 0.1 0.75 0.8
0.3
0.6
0.75 0.2
0.3
0.5
0.2
1
0.2
0.8 0.25 0.1
0.8
0.4
0.1
0.5
0.3 0.75
0.6
1
Numerator (Top) Strip:
1
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Denominator (Bottom) Strip:
4
4
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Connect Four Fraction Percent Equivalency Game
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. The
square with a specified percent is captured by creating the equivalent fraction using the
numerator and denominator strips at the bottom of the page. One paper clip is placed on
each strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the
3
denominator, then they can capture one square labeled 75% ( is equivalent to 75%). They
4
either mark the square with an X or place a colored counter on the square. There may be
other squares with that same difference but only one square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with the equivalent decimal using an O or a different
colored marker. If a player cannot move a single paperclip to capture a square, a paperclip
must still be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one player clip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
40%
10%
20% 100% 40%
50%
25% 100% 25%
80%
60%
20%
30%
60%
40%
50%
30%
40%
75%
20%
30%
25%
80% 100%
20%
80%
75%
10%
20%
40%
10%
50% 100% 60%
30%
75%
Numerator (Top) Strip:
1
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Denominator (Bottom) Strip:
4
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Connect Four Percentage Game
Number of Players: Two
Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically
or diagonally.
Instructions:
1. Roll a die to see which player will go first.
2. The first player looks at the board and decides which square he/she wishes to capture. They
place two paperclips on two strips below; one on the "Percentage" strip and one on the "Of"
strip. Take the percentage of that number and capture the appropriate square (e.g. 20% of 40
allows one to capture an "8" square). They either mark the square with an X or place a
colored counter on the square. There may be other squares with that same value but only one
square can be captured at a time.
3. Now the second player is ready to capture a square but he/she can only move one of the
paperclips. They then mark the square with that value using an O or a different colored
marker. If a player cannot move a single paperclip to capture a square, a paperclip must still
be moved in order to ensure that the game can continue.
4. Play alternates until one player connects four squares. Remember that only one paperclip is
moved at a time. If none of the players is able to connect four, then the winner is the
individual who has captured the most squares.
Game Board:
10
16
10
12
8
20
30
8
3
24
15
10
2
5
18
4
25
30
25
20
10
6
16
8
6
4
12
2
3
5
18
24
15
20
12
4
Percentage:
Of:
10% 15% 20% 25%
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Additional Practice; Ordering Decimals
This is an optional section of material.
1
, or 1.
2
Decimal Closest
to:
1. For each of the following decimal numbers, indicate whether it is closer to 0,
Decimal
Closest
to:
Decimal
Closest
to:
(a)
0.56
(b)
0.93
(c)
0.07
(d)
1.008
(e)
0.4
(f)
0.19
(g)
0.897
(h)
0.054
(i)
0.089
(j)
0.106
(k)
0.61
(l)
0.403
2. In each case, you are given two numbers. Circle the larger number.
(a)
0.2
0.8
(b)
0.38
0.92
(c)
0.867
0.263
(d)
3.14
3.64
(e)
2.7
2.5
(f)
4.07
4.12
(g)
3.459
3.624
(h)
1.908
1.98
(i)
3.001
2.85
(j)
6.13
6.112
(k)
8.3
8.408
(l)
1.89
1.982
(m) 0.52
0.509
(n)
3.59
3.505
(o)
4.01
4.099
(p)
9.056
9.06
(q)
17.45
17.2
(r)
5.856
5.9
(s)
3.298
3.41
(t)
4.61
4.3
(u)
8.5
8.601
(v)
3.04
3.009
3. In each case, you are given three numbers. Circle the largest number.
(a)
0.408
0.4
0.431
(b)
2.6
2.58
2.098
(c)
7.4
7.59
7.329
(d)
4.02
4.2
4.002
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4. Place the following numbers by the appropriate arrow on the number line below.
2.16, 1.36, 1.63, 0.45, 3.08, 0.72, 2.82, 1.91, 0.14 2.67, 0.87
0
1
2
3
5. Place the following numbers by the appropriate arrow on the number line below.
1.183, 0.91, 2.462, 1.45, 0.076, 0.33, 3.278, 0.6, 2.8, 3.07, 1.782, 2.2
0
1
2
3
6. Order the following numbers from smallest to largest.
(a) 1.5, 1.3, 0.8, 2.1, 0.6, 1.9
_________, _________, _________, _________, _________, _________
(b) 2.06, 1.45, 1.39, 0.73, 2.41, 0.09
_________, _________, _________, _________, _________, _________
(c) 0.907, 1.6, 0.34, 1.241, 0.7, 0.93, 1.78
_________, _________, _________, _________, _________, _________, _________
(d) 1.041, 0.4, 1.9, 0.28, 1.372, 1.83, 0.563
_________, _________, _________, _________, _________, _________, _________
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Additional Practice; Ordering Decimals and Fractions
This is an optional section of material.
1
, or 1.
2
Decimal Closest
to:
1. For each of the following decimal numbers, indicate whether it is closer to 0,
Decimal
Closest
to:
Decimal
Closest
to:
(a)
0.04
(b)
19
20
(c)
0.913
(d)
7
12
(e)
0.521
(f)
1
30
(g)
0.8
(h)
15
32
(i)
0.48
(j)
1.003
(k)
0.1
(l)
33
32
2. In each case, you are given two numbers. Circle the larger number.
(a)
1
16
0.45
(b)
7
8
5
11
(c)
0.178
0.26
(d)
8
14
0.9
(e)
0.56
7
16
(f)
0.189
15
16
(g)
3
1
8
3.6
(h)
1
3
5
1.17
(i)
2.592
2
1
6
(j)
5
3
7
5.625
(k)
4
4.08
(l)
6.4
6
1
32
(m)
17
16
0.98
(n)
1.48
1
11
20
9
16
3. In each case, you are given three numbers. Circle the largest number.
(a)
0.56
11
12
(c)
5.628
5
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4
9
0.095
(b)
2.83
2
1
8
2
9
20
5.91
(d)
4.06
4.306
4
6
10
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4. Place the following numbers by the appropriate arrow on the number line below.
1
15
10
1
9
, 3.052, 1 , 2.8, 1.183, 3 , 0.1, 2
2 , 0.68, 1.4,
5
16
20
6
16
0
1
2
3
5. Order the following numbers from smallest to largest.
9 3
5
, 0.068,
(a) 0.92, ,
8 10
8
_________, _________, _________, _________, _________
3 5
(b) 1.1, 0.427, 1.83, 1 ,
7 9
_________, _________, _________, _________, _________
(c) 1
5
3
13
, 0.9,
, 1 , 1.042, 0.78
12
50 16
_________, _________, _________, _________, _________, _________
(d) 0.17, 1
3
1
8
,
, 1.7,
, 0.905
25 100
15
_________, _________, _________, _________, _________, _________
(e) 0.8, 1.908, 1
7
1
, 0.91, 0.064, 1.15,
4
12
_________, _________, _________, _________, _________, _________, _________
(f) 3.171, 2
29
9
39
, 3 , 2.5, 3.26, 2
, 3.8
20
32
1000
_________, _________, _________, _________, _________, _________, _________
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Additional Practice; Ordering Decimals, Fractions and Percentages
This is an optional section of material.
1
, or 1.
2
Decimal Closest
to:
1
(c)
18
1. For each of the following decimal numbers, indicate whether it is closer to 0,
Decimal
(a)
Closest
to:
Decimal
16%
9
17
7
6
(d)
(g)
Closest
to:
(b)
0.842
(e)
98%
(f)
47%
(h)
8.3%
(i)
0.05
(j)
52.6%
(k)
27
1000
(l)
106%
(m)
12.5%
(n)
0.91
(o)
23
50
2. In each case, you are given two numbers. Circle the larger number.
(a)
49.2%
53%
(b)
13%
3
4
(c)
7
14
0.95
(d)
0.6
38%
(e)
58.3%
1
8
(f)
0.09
13
16
(g)
11%
9
20
(h)
0.939
102%
(i)
25%
1
16
(j)
2
6
7
2.57
(k)
78.9%
2.8
(l)
1
1
8
145%
(m)
23
21
98.5%
(n)
236%
2.1
3. In each case, you are given three numbers. Circle the largest number.
(a)
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57%
1
12
0.357
(b)
70
1.3
1
7
9
119%
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4. Place the following numbers by the appropriate arrow on the number line below.
91
1
3
11
28%, 2
, 95% , 1 , 28%, , 1.7, 140%, 0.08, 2.683, 1 , 216.5%
100
10
6
12
0
1
2
3
5. Order the following numbers from smallest to largest.
5
(a) 0.9, , 4%, 87.5%, 0.38
8
_________, _________, _________, _________, _________
(b) 0.529,
7
17
, 16%, 101%,
100
20
_________, _________, _________, _________, _________
(c) 61%, 0.003,
1
9
, 28.5%, , 0.96
10
6
_________, _________, _________, _________, _________, _________
(d) 153%, 1
1
, 0.72, 1.25, 209.5%, , 1.8
32
_________, _________, _________, _________, _________, _________
4 47
3
, 1.1 , 2 , 250%, 63%, 187%
(e) 1 ,
10
7 100
_________, _________, _________, _________, _________, _________, _________
(f) 109%, 173%, 86.9%, 1.6, 1
31 13
,
, 0.378
100 25
_________, _________, _________, _________, _________, _________, _________
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Post-Unit Reflections
What is the most valuable or important
thing you learned in this unit?
What part did you find most interesting or
enjoyable?
What was the most challenging part, and
how did you respond to this challenge?
How did you feel about this math topic
when you started this unit?
How do you feel about this math topic
now?
Of the skills you used in this unit, which
is your strongest skill?
What skill(s) do you feel you need to
improve, and how will you improve them?
How does what you learned in this unit fit
with your personal goals?
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Soft Skills Rubric
Look back over the module you have just completed and assess yourself using the following
rubric. Use pencil or pen and put a checkmark in the column that you think best describes your
competency for each description. I will look at how accurately you have done this and will
discuss with you any areas for improvement.
You will be better prepared for your next step, whether it is work or further education, if you are
competent in these areas by the end of the course. Keep all of these rubrics in one place and
check for improvement as you progress through the course.
Date:
Competent
demonstrates the concept
fully and consistently
Throughout this module, I…
Approaching
Competency
Developing
Competency
demonstrates the concept
most of the time
demonstrates the concept
some of the time
• Attended every class
• Let my instructor know if not
able to attend class
• Arrived on time for class
• Took necessary materials to class
• Used appropriate language for
class
• Used class time effectively
• Sustained commitment
throughout the module
• Persevered with tasks despite
difficulties
• Asked for help when needed
• Offered support/help to others
• Helped to maintain a positive
classroom environment
• Completed the module according
to negotiated timeline
• Worked effectively without close
supervision
Comments:
(Created by Alice Veenema, Kingstec Campus)
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Answers
Introduction to Decimals (pages 1 to 5)
1. (a) 0.5
(d) 0.41
(b) 0.3
(e) 0.7
(c) 0.13
(f) 0.29
2.
5
10
0.07
0.5
7
100
37
1000
9
100
604
1000
3. (a) 0.256
(d) 1.097
45
100
8
(d) 1
100
4. (a)
5. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
0.046
1.8
0.037
1.26
7
10
3
2.7
3
1
0.09
2
8
10
46
1000
9
100
3.09
3.604
1
26
100
56
1000
0.056
(b) 0.06
(e) 2.007
(b)
(c) 3.9
(f) 13.58
4
10
(e) 2
508
1000
59
(f) 6
1000
(c) 4
3
1000
35.6
7.09
0.58
1000.15
206.309
70.1
5.037
400.029
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6.
Fraction
Decimal Numerals Words
8
32
32.8
thirty-two and eight tenths
10
(a)
(b)
0.472
472
1000
(c)
13.067
13
(d)
7.59
7
(e)
327.09
327
7. (a) 40 + 2 + 0.8
(b) 9 + 0.3 + 0.01
four hundred seventy-two thousandths
67
1000
59
100
seven and fifty-nine hundredths
9
100
and
and
thirteen and sixty-seven thousandths
three hundred twenty-seven and nine hundredths
( 4 ×10 ) + ( 2 ×1) +  8 ×
1

 10 
1
1
( 9 ×1) +  3 ×  + 1× 
 10   100 
1 
1  
1 
 +  2×
 + 9×

 10   100   1000 
1  
1 

(d) 10 + 8 + 0.03 + 0.004 or (1×10 ) + ( 8 ×1) +  3 ×
 +  4×

 100   1000 
1 

(e) 4000 + 200 + 9 + 0.07 or ( 4 ×1000 ) + ( 2 ×100 ) + ( 9 ×1) +  7 ×

 100 
(c) 300 + 2 + 0.4 + 0.02 + 0.009 and
( 3 ×100 ) + ( 2 ×1) +  4 ×
Comparing Decimals (pages 6 to 12)
1. (a)
1
2
(b) 0
(c) 1
1
2
(f) 1
1
2
(h) 1
(i) 0
(j) 1
(k) 0
(l)
(m) 1
(n)
(d) 0
(g)
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(e)
1
2
1
2
(o) 0
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2. (a)
(c)
(e)
(g)
3. (a)
(c)
(e)
(g)
(i)
(k)
(m)
(o)
(q)
(s)
(u)
(b)
7
8
(d)
0.907
(f)
0.879
(h)
0.58
0.7
0.198
2.4
3.1
5.618
7.08
0.409
3.01
15.35
15
16
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
0.52
1.24
0.09
0.57
2.09
12.988
31.3005
7.809
0.75
(t)
0.81
0.51
(v)
0.89
4
8
3
3
13
12
(w) 4.7
(x)
5
6
11
3
12
2
4. The numbers should be placed along the number line in the following order.
0.1, 0.6, 0.8, 1.3, 1.4, 1.9, 2.2, 2.5, 2.7, 3.1
5. The numbers should be placed along the number line in the following order.
0.097, 0.54, 1.039, 1.46, 1.75, 1.95, 2.4, 2.62, 2.89, 3.05
6. The numbers should be placed along the number line in the following order.
3 7
1
4
0.07, ,
, 1.2, 1.57, 2 , 2.44 2.78, 3
, 3.3
8 10
16
100
7. (a) 0.2, 0.4, 0.9, 1.3, 1.6
(b) 0.08, 0.55, 0.59, 1.14, 1.23
(c) 0.09, 0.52, 0.8, 0.83, 1.01, 1.1
(d) 0.19, 0.2, 0.26, 0.3, 0.98, 1
(e) 0.006, 0.08, 0.2, 0.209, 0.24, 0.72
(f) 0.05, 0.092, 0.4, 0.619, 0.64, 0.7, 0.78
7
1
(g)
, 0.3, , 0.542, 0.85, 0.862, 0.9
100
2
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8
43
,1
, 1.3
8 1000
5
1
, 0.4, 0.48, 0.509, 1.002, 1.1, 1
(i)
8
16
(h) 0.16, 0.201, 0.6, 0.649,
Rounding Decimals (page 13)
1. (a) 14.55
(b) 15
(c) 14.5
2. (a) 251.9
(b) 250
(c) 252
3. (a)
(b)
(c)
(d)
7.065
7
7.1
7.06
4. (a)
(b)
(c)
(d)
(e)
437
437.3
437.330
440
437.33
Equivalent Fractions and Decimals (pages 14 to 17)
1. (a)
3
4
(d) 2
1
3
(b) 3
1
2
(c) 1
4
5
(e) 7
1
5
(f) 5
2
3
2. (a) 9.25
(b) 0.4
(c) 6.3
(d) 7.75
(e) 9.5
(f) 8.6
3. (a)
(c)
(e)
NSSAL
©2012
0.65
1
1
3
2
4
3
(b)
0.875
(d)
2.6
(f)
6
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(g)
2.304
(h)
7
1
5
4. The numbers should appear on the number line in the following order.
1 1 2
3
7 1
, , , 0.9, 1.44, 1 , 2.069, 2.539, 2 , 3 , 3.37
25 3 5
4
8
5
5. The numbers should appear on the number line in the following order.
9 95
2 2
1
0.098,
,
, 1 , 1 , 1.87, 2.2, 2.43, 2.71, 3.12, 3
16 100 5 3
4
5 93
3
1
11
,
, 1 , 1.6, 1 , 2.4, 2
8 100 4
5
12
1
41
3
1
9
(b)
, 0.587, 1 , 1.58, 1 , 2 , 2.7, 2
3
100
4 16
10
1
1
9 4
19
(c)
, , 0.892, 1 , 1 , 2.011, 2 , 2.6
10 20
16 5
4
6. (a) 0.09,
7. In each case we have supplied three possible answers; there are many more than the ones we
have listed.
(a) 3.41, 3.459, 3.499
3
3
2
(b) 2 , 2 , 2
5
3
4
(c) 1.3, 1.25, 1.239
(d) 0.7, 0.74, 0.859
1
1
2
(e) 3 , 3 , 3
5 10
4
(f) 2.9, 2.83, 2.999, 3.005
Introduction to Percent (pages 18 to 25)
1. (a) 17%, 0.17,
17
100
(b) 43%, 0.43,
43
100
(c) 29%, 0.29,
29
100
2. There are a range of acceptable answers.
(a) 40% to 47%
(b) 11% to 19%
(c) 85% to 94%
(d) 63% to 73%
(e) 4% to 7%
3. (a) 0.79
(c) 0.09
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(b) 0.16
(d) 1.45
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(e) 0.294
(g) 2.08
(i) 0.045
4. (a)
(c)
(e)
(g)
(i)
19%
173%
6%
7.3%
0.2%
(b)
(d)
(f)
(h)
(j)
39
100
4
25
7
2
20
23
500
1
250
5. (a)
(c)
(e)
(g)
(i)
6. (a)
(c)
(e)
(g)
(f) 0.07
(h) 0.817
(j) 0.008
(b)
(d)
(f)
(h)
(j)
75%
350%
80%
166.6%
7.
Percent
(a)
83%
(c)
460%
(e)
5%
(g)
71.9%
8. (a)
(b)
(c)
(d)
(e)
(b)
(d)
(f)
(h)
Fraction
83
100
3
4
5
1
20
719
1000
48%
69.2%
209%
154.8%
170%
91
100
29
1
100
51
1000
241
500
103
3
500
140%
233.3%
325%
220%
Decimal
Percent
0.83
(b)
67%
4.6
(d)
139%
0.05
(f)
175%
0.719
(h)
216.3%
Fraction
67
100
39
1
100
3
1
4
163
2
1000
Decimal
0.67
0.39
1.75
2.163
40%
1%
45%
96%
15%
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9. Remember we are not grouping equivalent decimal, fractions, and percentages in the same
boxes; rather, we are completing an estimation activity where we match the numbers to the
most appropriate diagram.
7
1
1
(a) 51%, 0.52,
(b) 26%, 0.24,
(c) 69%, 0.72,
10
2
4
5
3
7
(e) 81%, 0.8,
(f) 260%, 2.57, 2
(d) 140%, 1.43, 1
8
9
7
1
7
19
(g) 218%, 2.16, 2
(h) 89%, 0.91,
(i) 194%, 1.93, 1
6
8
20
10.
93
100
17
1000
1.7%
1.387
53
100
93%
0.8%
13.1%
0.008
2
347%
3
4
275%
4
5
4
47
100
2
0.53
280%
131
1000
3
138.7%
3
10
2
3
4.6
30%
11. There are 8 people in the Sampson family. Of those, 6 are female. That means that
percentage of females in this family is 75%, which can also be represented by the fraction
3
.
4
The percentage of males in this family is 25%, which can also be represented by the fraction
1
.
4
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Comparing Fractions, Decimals, and Percentages (pages 26 to 29)
1.
Closest
to:
Closest
to:
Closest
to:
1
2
1
2
(a)
98%
1
(b)
11%
0
(c)
45%
(d)
0.02
0
(e)
0.899
1
(f)
0.6
(g)
8
9
1
(h)
11
20
1
2
(i)
1
16
0
(j)
0.3%
0
(k)
1.05
1
(l)
102%
1
(m)
13
24
1
2
(n)
56.2%
1
2
(o)
16
15
1
2. (a)
(c)
(e)
(g)
(i)
83%
136%
0.48
2.3%
105%
7
8
(b)
(d)
(f)
(h)
(j)
(m) 135%
(n)
(k)
(o)
(q)
(s)
(u)
7
16
2
3
4
5
81.2%
(l)
14.7%
3.1%
83%
0.65
1.45
9
16
1
10
(p)
93.5%
(r)
215%
(t)
0.48
(v)
8.3%
3. The numbers should occur in this order along the number line (from left to right).
1
1
5
1
, 0.422, 54.7%, 0.93, 1.099, 125%, 180%, 200%, 2 , 2 , 3
20
4
8 16
4. The numbers should occur in this order along the number line (from left to right).
3
3
19
9.7%, , 0.713, 96%, 1.389, 155%, 1 , 215%, 2.85, 2 , 2.85, 3.2
6
4
20
5. (a) 0.7%, 5.8%, 32%, 91.2%, 124%
(b) 0.1, 14%, 64%, 0.745, 0.82
(c) 8.2%, 57.2%, 0.61, 123%, 1.45
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5
9
, 0.792, 86%,
12
10
1
1
, 20%, 0.4, 68.5%, 1 , 1.96
32
10
8
31
0.276, 30.2%,
, 57.6%,
, 1.1
32
16
1
5
0.096, , 64.5%, 0.89, 1 , 209%
4
8
1
6
, 0.08,
, 50.3%, 91%, 0.956,
100
14
3
7
1
28%, , 0.9, , 1.02, 1 , 194%
5
7
3
1
7
7
, 0.34, 94.5%, 1.092, 1 , 214%, 2
5
12
8
(d) 3.8%,
(e)
(f)
(g)
(h)
(i)
(j)
Adding and Subtracting Decimal Numbers (pages 30 to 36)
1. (a)
(c)
(e)
(g)
46.88
88.87
41.962
45.775
(b)
(d)
(f)
(h)
69.84
6.813
230.33
8.475
2. (a)
(c)
(e)
(g)
43.35
1.728
8.865
21.779
(b)
(d)
(f)
(h)
4.04
5.72
6.682
6.556
3. 3.67 kg
4. 3.42 kg
5. 24 293.1 km
6. 0.38 seconds
Multiplying Decimal Numbers (pages 37 to 42)
1. (a)
(c)
(e)
(g)
(i)
24.05
0.234
6.92
76.63
0.35334
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©2012
(b)
(d)
(f)
(h)
(j)
82
19.318
22.32
4.459
18.27
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2. (a)
(b)
(c)
(d)
(e)
(f)
22.656
75.64
177.3
384.85
804.2
2548.584
3.
Question
0.98 × 2.1
5.23 − 4.37
1.1 + 1.97
Arrow
d
b
f
Question
0.326 + 2.21
2.34 − 1.98
0.49 × 2.88
Arrow
e
a
c
4.
Question
(a) 2.93 + 3.208
Answers
(c) 32.185
(b)
16.08 - 5.239
(h)
311.74
(c)
7.85 × 4.1
(e)
21.78
(d)
47.9 + 32.7
(a)
6.138
(e)
29.58 - 7.8
(g)
126.7
(f)
39.8 × 6.1
(d)
80.6
(g)
98.3 + 28.4
(i)
1110.9
(h)
409.8 - 98.06
(b)
10.841
(i)
52.9 × 21
(f)
242.78
5. 730.5 km
6. $12.86
7. $480.70
8. 42.2 cm
9. 26.1 grams
Dividing Decimal Numbers (pages 43 to 53)
1. (a) 32.04
(c) 2.415
NSSAL
©2012
(b) 26.7
(d) 0.093
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C. D. Pilmer
2.
Questions
(a) 389.6 ÷ 8
(b) 49.02 ÷ 6
(c) 2.024 ÷ 4
(d) 257.8 ÷ 9
(e) 32.83 ÷ 7
Answers
(c) 0.506
(a) 48.7
(e) 4.69
(b) 8.17
(d) 28.62
3. (a) 6 453.6
(b) 0.8 0.736
(c) 5 3856
(d) 0.8 2.7345
(e) 5 490
(f) 6 820
(g) 0.03 1.826
(h) 7 5820
4. (a)
(c)
(e)
(g)
5.6
20.39
31.96
0.872
(b)
(d)
(f)
(h)
639.1 ÷ 7
289.4 + 315.7
23.76 ÷ 0.6
20.5 × 61.8
0.24 ÷ 0.5
453.6 − 198.8
0.496 ÷ 0.08
0.9 ×135.6
32.68 ÷ 0.4
5.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
6.
Question
6.65 ÷ 7
1.49 + 1.04
0.145 ÷ 0.05
86
94.3
68
124.04
(c)
(h)
(e)
(a)
(f)
(g)
(i)
(d)
(b)
Arrow
b
e
f
Question
3.05 − 2.97
0.31× 5.2
1.278 ÷ 0.6
39.6
122.04
0.48
91.3
254.8
6.2
817
1266.9
605.1
Arrow
a
c
d
7. $28.30
8. 18.2, but we round down such that the final answer is 18 hamburgers.
9. $87.75
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C. D. Pilmer
10. $9.65
11. 257.7 km
Estimation Questions Involving Percentages (pages 54 to 57)
1. (a) 4
(c) 50
(e) 8
(g) 100
(i) 6
(k) 75
(m) 12
(o) 100
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
12
140
24
280
18
210
36
300
Answers are likely to vary slightly from learner to learner on questions 2 through 9. This is to be
expected with estimation questions. As long as your answer is close to our answer, assume that
you estimation technique was perfectly valid.
2. Approximately $10
3. Approximately $46
4. Approximately $21
5. Approximately $12
6. Approximately $42 ($36 + $6)
7. Approximately $138 ($120 + $18)
8. Approximately $108 ($90 + $18)
9. Approximately $230 ($200 + $30)
10. Question
25% of 11.90
0.784 ÷ 0.4
1.43 − 1.316
30% of 4.90
NSSAL
©2012
Arrow
g
e
a
d
Question
0.52 + 0.496
10% of 21.50
20% of 2.99
2.1×1.513
85
Arrow
c
f
b
h
Draft
C. D. Pilmer
Calculator Questions (pages 58 to 61)
1. 23 minutes
2. $154.77
3. $1.17
4. $59.80
5. Hint: Remember that we are purchasing each of these items for her two boys.
Answer: $211.54
6. $1099.80
7. $650.20
8. $652.00
9. $46.60
10. 14.2 kilometres per litre
Additional Practice; Ordering Decimals (pages 66 and 67)
1.
Decimal
Closest
to:
1
2
Decimal
Closest
to:
(b)
0.93
1
(a)
0.56
(d)
1.008
1
(e)
0.4
(g)
0.897
1
(h)
0.054
(j)
0.106
0
(k)
0.61
(m)
12.5%
0
(n)
0.91
2. (a)
(c)
(e)
(g)
(i)
(k)
NSSAL
©2012
0.8
0.867
2.7
3.624
3.001
8.408
(b)
(d)
(f)
(h)
(j)
(l)
86
1
2
1
2
1
2
1
Decimal
Closest
to:
(c)
0.07
0
(f)
0.19
0
(i)
0.089
0
(l)
0.403
(o)
23
50
1
2
1
2
0.92
3.64
4.12
1.98
6.13
1.982
Draft
C. D. Pilmer
(m)
(o)
(q)
(s)
(u)
3. (a)
(c)
0.52
4.099
17.45
3.41
8.601
(n)
(p)
(r)
(t)
(v)
3.59
9.06
5.9
4.61
3.04
0.431
7.59
(b)
(d)
2.6
4.2
4. 0.14, 0.45, 0.72, 0.87, 1.36, 1.63, 1.91, 2.16, 2.67, 2.82, 3.08
5. 0.076, 0.33, 0.6, 0.91, 1.183, 1.45, 1.782, 2.2, 2.462, 2.8, 3.07, 3.278
6. (a)
(b)
(c)
(d)
0.6, 0.8, 1.3, 1.5, 1.9, 2.1
0.09, 0.73, 1.39, 1.45, 2.06, 2.41
0.34, 0.7, 0.907, 0.93, 1.241, 1.6, 1.78
0.28, 0.4, 0.563, 1.041, 1.372, 1.83, 1.9
Additional Practice; Ordering Decimals and Fractions (pages 68 and 69)
1.
Decimal
Closest
to:
Decimal
(a)
0.04
0
(b)
19
20
(d)
7
12
1
2
(e)
0.521
(g)
0.8
1
(h)
15
32
1
2
1
2
(j)
1.003
1
(k)
0.1
0
(j)
(l)
6.4
(n)
1
0.45
(b)
(c)
0.26
(d)
(e)
0.56
(f)
(g)
3.6
(h)
2.592
9
(k) 4
16
17
(m)
16
NSSAL
©2012
1
7
8
0.9
15
16
3
1
5
5.625
2. (a)
(i)
Closest
to:
87
Decimal
Closest
to:
(c)
0.913
1
(f)
1
30
0
(i)
0.48
1
2
(l)
33
32
1
11
20
Draft
C. D. Pilmer
3. (a)
(c)
11
12
(b)
2.83
5.91
(d)
4
4. 0.1, 0.68,
6
10
1
9
1
15
10
, 1.183, 1.4, 1 , 2 , 2 , 2.8 , 3.052, 3
5
16
6
16
20
9
3 5
, , 0.92,
8
10 8
5
3
0.427, , 1.1, 1 , 1.83
9
7
13
3
5
, 0.78, 0.9, 1.042, 1 , 1
50
12 16
1
8
3
, 0.17,
, 0.905, 1 , 1.7
100
15
25
7
1
0.064, , 0.8, 0.91, 1.15, 1 , 1.908
12
4
29
39
9
, 2.5, 2 , 3.171, 3.26, 3 , 3.8
2
20
32
1000
5. (a) 0.068,
(b)
(c)
(d)
(e)
(f)
Additional Practice; Ordering Decimals, Fractions and Percentages (pages 70 and 71)
1.
(a)
(d)
(g)
(j)
Decimal
Closest
to:
16%
0
9
17
7
6
52.6%
Decimal
Closest
to:
(b)
0.842
1
(c)
1
18
0
1
2
(e)
98%
1
(f)
47%
1
2
1
(h)
8.3%
0
(i)
0.05
0
1
2
(k)
27
1000
0
(l)
106%
1
2. (a)
53%
(b)
(c)
0.95
(d)
(e)
58.3%
(f)
NSSAL
©2012
88
Decimal
Closest
to:
3
4
0.6
13
16
Draft
C. D. Pilmer
(g)
9
20
(h)
(i)
25%
(j)
(k)
2.8
23
(m)
21
3. (a)
57%
4. 0.08, 28%,
102%
(l)
6
7
145%
(n)
236%
(b)
1
2
7
9
3
1
11
91
, 95% , 1 , 140%, 1.7, 1 , 216.5%, 2.683, 2
6
10
100
12
5
, 87.5%, 0.9
8
7
17
, 16%, 0.529,
, 101%
100
20
1
9
0.003, , 28.5%, 61%,
, 0.96
10
6
1
0.72, 1 , 1.25, 153%, 1.8, 209.5%
32
47
4
3
, 63% , 1.1, 1 , 187%, 2 , 250%
100
7
10
13
31
0.378,
, 86.9%, 109%, 1
, 1.6, 173%
25
100
5. (a) 4%, 0.38,
(b)
(c)
(d)
(e)
(f)
NSSAL
©2012
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C. D. Pilmer
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