Math in the Real World Unit
(Level IV Graduate Math)
Draft
(NSSAL)
C. David Pilmer
©2009
(Last Updated: Dec, 2011)
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
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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
• 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 also like to thank the following NSCC instructors for
piloting this resource and offering suggestions during its development.
Eileen Burchill (IT Campus)
Elliott Churchill (Waterfront Campus)
Barbara Leck (Pictou Campus)
Suzette Lowe (Lunenburg Campus)
Nancy Harvey (Akerley Campus)
Floyd Porter (Strait Area Campus)
Brian Rhodenizer (Kingstec Campus)
Joan Ross (Annapolis Valley Campus)
Tanya Tuttle-Comeau (Cumberland Campus)
Jeff Vroom (Truro Campus)
Table of Contents
Introduction…………………………………………………………………………...
Tracking Your Progress……………………………………………………………….
Negotiated Completion Date…………………………………………………………
The Big Picture………………………………………………………………………..
Course Timelines……………………………………………………………………...
ii
iii
iii
iv
v
The Basics
Order of Operations…………………………………………………………………...
Fractions……………………………………………………………………………….
Adding Fractions…………………………………………………………………..
Subtracting Fractions……………………………………………………………...
Multiplying and Dividing Fractions………………………………………………
Decimals………………………………………………………………………………
Percents……………………………………………………………………………….
Ratio and Proportion……………………………………………………..…………...
Signed Numbers………………………………………………………………………
All Together Now, Part I……………………………………………………………...
All Together Now, Part II……………………………………………………………..
All Together Now, Part III…………………………………………………………….
Reflect Upon Your Learning………………………………………………………….
1
3
3
7
10
12
18
22
25
29
30
31
32
Applying Math to the Real World
Introduction to Application Questions………………………………………………...
Applications - Fractions……………………………………………………………….
Applications - Decimals……………………………………………………………….
Applications - Percents………………………………………………………………..
Applications - Ratio and Proportion…………………………………………………..
Applications - Signed Numbers……………………………………………………….
Reflect Upon Your Learning………………………………………………………….
Putting It Together: Part I……………………………………………………………..
Putting It Together: Part II……………………………………………………………
Putting It Together: Part III…………………………………………………………...
Putting It Together: Part IV…………………………………………………………..
Is It Reasonable?...........................................................................................................
Reflect Upon Your Learning………………………………………………………….
33
35
39
43
48
54
58
59
64
69
74
79
83
Real World Math - Careers and Math
Part I: Math in the Real World and Daily Life………………………………………..
Part II: Careers and Math……………………………………………………………..
84
85
Post-Unit Reflections…………………………………………………………………. 87
Answers………………………………………………………………………………. 88
Online Support………………………………………………………………………... 98
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Introduction
This unit is designed to review fundamental mathematical concepts (e.g. operations with
fractions, operations with signed numbers, order of operations,…), both in and out of context,
that are necessary to be sucessful in this course and in post-secondary studies that do not require
an academic math credit. There are some requirements for this unit.
1. The unit will not be cut into sections such that learners will be assessed on only one
mathematical concept at a time. That means you will not have a separate test on fractions, a
separate test on decimals, and separate tests for the other concepts covered in this unit; tests
will cover multiple concepts.
2. Most of the questions will be completed using paper-and-pencil techniques. Calculators will
only be used on questions identified with the calculator icon ().
3. Learners who have completed Level III math are not required to complete all of this resource.
They are only required to complete the following sections.
• All Together Now, Parts I through III
• Putting It Together, Parts I though IV
• Is It Reasonable?
• Real World Math - Careers and Math
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Tracking Your Progress
This page allows you to keep track of your progress through this material.
The Basics
Order of Operations………………………………
Fractions………………………………………….
Adding Fractions……………………………..
Subtracting Fractions…………………………
Multiplying and Dividing Fractions………….
Decimals………………………………………….
Percents…………………………………………..
Ratio and Proportion……………………………..
Signed Numbers………………………………….
All Together Now, Parts I through III……………
Date Started
Date Completed
Date Started
Date Completed
Date Started
Date Completed
1
3
3
7
10
12
18
22
25
29
Applying Math to the Real World
Applications - Fractions………………………….
Applications - Decimals………………………….
Applications - Percents…………………………..
Applications - Ratio and Proportion……………..
Applications - Signed Numbers………………….
Putting It Together: Part I………………………..
Putting It Together: Part II……………………….
Putting It Together: Part III………………………
Putting It Together: Part IV………………………
Is It Reasonable? ………………………………...
35
39
43
48
54
59
64
69
74
79
Real World Math - Careers and Math
Part I: Math in the Real World and Daily Life…..
Part II: Careers and Math………………………..
84
85
Negotiated Completion Date
After working for a few days on this unit, sit down with your instructor and negotiate a
completion date for this unit.
Start Date:
_________________
Completion Date:
_________________
Instructor Signature: __________________________
Student Signature:
NSSAL
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__________________________
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The Big Picture
The following flow chart shows the five required units and the four optional units (choose two of
the four) in Level IV Graduate Math. These have been presented in a suggested order.
Math in the Real World Unit (Required)
• Fractions, decimals, percents, ratios, proportions, and
signed numbers in real world applications
• Career Exploration and Math
Solving Equations Unit (Required)
• Solve and check equations of the form Ax + B = Cx + D ,
A = Bx 2 + C , and A = Bx 3 + C .
Consumer Finance Unit (Required)
• Simple Interest and Compound Interest
• TVM Solver (Loans and Investments)
• Credit and Credit Scores
Graphs and Functions Unit (Required)
• Understanding Graphs
• Linear Functions and Line of Best Fit
Measurement Unit (Required)
• Imperial and Metric Measures
• Precision and Accuracy
• Perimeter, Area and Volume
Choose two of the four.
Linear
Functions and
Linear Systems
Unit
Trigonometry
Unit
Statistics Unit
ALP Approved
Projects
(Complete 2 of
the 5 projects.)
Note:
You are not permitted to complete four ALP Approved Projects and thus avoid selecting from
the Linear Functions and Linear Systems Unit, Trigonometry Unit, or Statistics Unit.
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Course Timelines
Graduate Level IV Math is a two credit course within the Adult Learning Program. As a two
credit course, learners are expected to complete 200 hours of course material. Since most ALP
math classes meet for 6 hours each week, the course should be completed within 35 weeks. The
curriculum developers have worked diligently to ensure that the course can be completed within
this time span. Below you will find a chart containing the unit names and suggested completion
times. The hours listed are classroom hours.
Unit Name
Minimum
Completion Time
in Hours
24
20
18
28
24
20
20
Total: 154 hours
Math in the Real World Unit
Solving Equations Unit
Consumer Finance Unit
Graphs and Functions Unit
Measurement Unit
Selected Unit #1
Selected Unit #2
Maximum
Completion Time
in Hours
36
28
24
34
30
24
24
Total: 200 hours
As one can see, this course covers numerous topics and for this reason may seem daunting. You
can complete this course in a timely manner if you manage your time wisely, remain focused,
and seek assistance from your instructor when needed.
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Order of Operations
There is a specific order one must follow when calculating the value of an expression.
1.
2.
3.
4.
Perform all operations within grouping symbols like brackets “[ ]”, and parentheses “( )”.
Evaluate expressions with exponents.
Perform multiplications and divisions in order from left to right.
Perform additions and subtractions in order from left to right.
People often remember this order using the acronym, BEDMAS.
B - brackets
E - exponents
DM - division and multiplication
AS - addition and subtraction
Example 1:
Evaluate 3 + 5(8 − 2) .
Example 2:
Evaluate 52 + (7 − 4 ) − 2 × 3 .
3 + 5(8 − 2)
3 + 5(6)
parentheses
3 + 30
multiplication
33
addition
52 + (7 − 4 ) − 2 × 3
52 + 3 − 2 × 3
25 + 3 − 2 × 3
25 + 3 − 6
22
Example 3:
2
Evaluate (17 − 8) − 23 ÷ 4 + 5 .
Example 4:
Evaluate
(17 − 8)2 − 23 ÷ 4 + 5
7 − 2×3+ 9
.
23 − 3
Here the fraction bar is like a grouping
symbol. We simplify above and below the
fraction bar separately.
7 − 2×3+ 9
23 − 3
7−6+9
8−3
10
5
2
92 − 23 ÷ 4 + 5 parentheses
81 − 8 ÷ 4 + 5
81 − 2 + 5
84
parentheses
exponents
multiplication
addition then subtraction
exponents
division
subtraction then addition
Questions
1. Evaluate each expression.
(a) 14 − 3(5 − 2 )
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(b) (3 + 4 )5 + 8
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(c) 100 − 32 × 2
(d) 10 ÷ (7 − 2 ) + 4
(e) 7 + (2 + 3) × 3 − 1
2
(f) 7 × 3 − 20 ÷ 5 − 1
(g) 2 + 33 + 1 ÷ 4
)
(h) 3(8 − 2 + 1) − 1
(i) 17 − 3(6 − 4 )
(j) 23 + (5 + 6 ) − 5 × 3
(k) (82 + 8) ÷ 32 + 4 × 6
(l) 30 + (6 − 4 + 1) − 2
(m) 2(16 − 13) − (4 + 1)
(n) 4(5 − 2 ) + 3(2 + 6 )
19 + 17
(o) 2
5 − 42
12 + (2 × 3)
(p)
1+ 7
(
2
2
2
(q)
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2
3 + 7(9 − 6 )
2×6 − 4
14 + (10 − 8)
18 − 24
3
(r)
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Adding Fractions
To add like fractions (fractions that have the same denominator), add the numerators and write
the sum over the common denominator. Express your answer in simplest form.
Example 1:
3 2
+
7 7
5
=
7
The two fractions have a common denominator of 7.
Example 2:
3
5
+
16 16
8
=
16
1
=
2
The two fractions have a common denominator of 16.
Example 3:
7 6
+
8 8
13
=
8
5
=1
8
The two fractions have a common denominator of 8.
You have 3 sevenths plus 2 sevenths. The answer will be 5
sevenths.
You have 3 sixteenths plus 5 sixteenths. The answer will be 8
sixteenths.
The fraction is changed to its simplest form by dividing both the
numerator and denominator by 8.
You have 7 eighths plus 6 eighths. The answer will be 13
eighths.
13
is called an improper fraction. Improper fractions occur
8
when the numerator is larger than the denominator. The
5
improper fraction can be changed to the mixed number 1 .
8
To add unlike fractions (fractions that have different denominators), identify the least common
multiple (LCM) of the denominators, write each fraction as an equivalent fraction whose
denominator is the LCM, add the like fractions, and express the sum in simplest form.
Example 4:
1 1
+
3 2
1× 2 1× 3
=
+
3× 2 2× 3
2 3
= +
6 6
5
=
6
The denominators (3 and 2) are different. List the multiples of 3
and 2 separately.
Multiples of Three: 3, 6, 9, 12, 15, …
Multiples of Two: 2, 4, 6, 8, 10, 12, 14, …
Notice that the least common multiple of 3 and 2 is 6. This
means that the common denominator will be 6.
Now you write each fraction as an equivalent fraction whose
denominator is 6.
You have 2 sixths plus 3 sixths. The answer is 5 sixths.
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Example 5:
1 3
+
8 4
1 3× 2
= +
8 4× 2
1 6
= +
8 8
7
=
8
The denominators (8 and 4) are different. List the multiples of 8
and 4 separately.
Multiples of Eight: 8, 16, 24, 32, 40, 48, …
Multiples of Four: 4, 8, 12, 16, 20, 24, …
Notice that the least common multiple of 8 and 4 is 8. This
means that the common denominator will be 8.
Now you write each fraction as an equivalent fraction whose
denominator is 8.
You have 1 eighth plus 6 eighths. The answer is 7 eighths.
Example 6:
4 2
+
5 3
4×3 2×5
=
+
5× 3 3× 5
12 10
= +
15 15
22
=
15
7
=1
15
The denominators (5 and 3) are different. List the multiples of 5
and 3 separately.
Multiples of Five: 5, 10, 15, 20, 25, …
Multiples of Three: 3, 6, 9, 12, 15, 18, …
Notice that the least common multiple of 5 and 3 is 15. This
means that the common denominator will be 15.
Now you write each fraction as an equivalent fraction whose
denominator is 15.
The improper fraction
7
22
is changed to the mixed number 1 .
15
15
To add mixed numbers, create a common denominator. Add the whole numbers, then add the
fractions. If the fractional component is an improper fraction, some additional work will have to
be done.
Example 7:
1
4
7 +5
6
9
1× 3
4× 2
=7
+5
6×3
9× 2
3
8
=7 +5
18
18
11
= 12 +
18
11
= 12
18
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Find the LCM of 6 and 9.
Multiples of Six: 6, 12, 18, 24, 30, …
Multiples of Nine: 9, 18, 27, 36, 45, …
Now you write each fraction as an equivalent fraction whose
denominator is 18.
Add the whole numbers 7 and 5. Add the fractions
4
3
8
and
.
18
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Example 8:
2
11
1 +7
3
12
2× 4
11
=1
+7
3× 4
12
8
11
=1 + 7
12
12
19
=8+
12
7
= 8 +1
12
7
=9
12
Find the LCM of 3 and 12.
Multiples of Three: 3, 6, 9, 12, 15, …
Multiples of Twelve: 12, 24, 36, 48, 60, …
Now you write each fraction as an equivalent fraction whose
denominator is 12.
Add the whole numbers 1 and 7. Add the fractions
Change the improper fraction
8
11
and
.
12
12
19
7
to the mixed number 1 .
12
12
Add the whole numbers 8 and 1.
Questions:
1. (a)
4 6
+
11 11
(b)
2 1
+
5 5
(c)
7
3
+
12 12
(d)
5 1
+
9 9
(e)
5 6
+
7 7
(f)
8
5
+
10 10
(g)
5 3
+
6 6
(h)
7 5
+
8 8
2. (a)
1 2
+
4 5
(b)
3 1
+
5 6
(c)
2 3
+
3 4
(d)
1 3
+
6 4
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(e)
3 1
+
8 6
(f)
4 5
+
9 6
(g)
2 1
+
3 12
(h)
4 8
+
5 15
2
1
3. (a) 3 + 6
5
4
1
5
(b) 4 + 7
3
8
1
7
(c) 1 + 2
6
12
4
1
(d) 6 + 1
7
2
5
3
(e) 2 + 4
6
8
1
3
(f) 4 + 2
5
10
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Subtracting Fractions
Like addition, you need a common denominator to subtract fractions. To subtract like fractions
(fractions that have the same denominator), subtract the numerators and write the difference over
the common denominator. Express your answer in simplest form.
Example 1:
5 2
−
7 7
3
=
7
The two fractions have a common denominator of 7.
Example 2:
9 5
−
16 16
4
=
16
1
=
4
The two fractions have a common denominator of 16.
You have 5 sevenths subtract 2 sevenths. The answer will be 3
sevenths.
You have 9 sixteenths subtract 5 sixteenths. The answer will be
4 sixteenths.
The fraction is changed to its simplest form by dividing both the
numerator and denominator by 4.
To add unlike fractions (fractions that have different denominators), identify the least common
multiple (LCM) of the denominators, write each fraction as an equivalent fraction whose
denominator is the LCM, add the like fractions, and express the sum in simplest form.
Example 3:
2 1
−
3 2
2 × 2 1× 3
=
−
3× 2 2 × 3
4 3
= −
6 6
1
=
6
Find the LCM of 3 and 2.
Example 4:
5 1
−
8 4
5 1× 2
= −
8 4× 2
5 2
= −
8 8
3
=
8
Find the LCM of 8 and 4.
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Multiples of Three: 3, 6, 9, 12, 15, …
Multiples of Two: 2, 4, 6, 8, 10, 12, 14, …
Now you write each fraction as an equivalent fraction whose
denominator is 6.
You have 4 sixths subtract 3 sixths. The answer is 1 sixth.
Multiples of Eight: 8, 16, 24, 32, 40, 48, …
Multiples of Four: 4, 8, 12, 16, 20, 24, …
Now you write each fraction as an equivalent fraction whose
denominator is 8.
You have 5 eighths subtract 2 eighths. The answer is 3 eighths.
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To subtract mixed numbers, create a common denominator. Subtract the fractions and then
subtract the whole numbers. If the larger of the two mixed numbers does not have the larger
proper fraction component, then borrow 1 from the whole number. Then add the 1 to the proper
fraction to create an improper fraction. This will have to be done before the subtraction can
occur.
Example 5:
5
2
7 −5
6
9
5×3
2× 2
=7
−5
9× 2
6×3
15
4
= 7 −5
18
18
11
=2
18
Example 6:
1
2
5 −2
3
4
1× 3
2× 4
=5
−2
4×3
3× 4
3
8
=5 −2
12
12
8
3

= 4 +1+  − 2
12
12 

12 3 
8

= 4 + +  − 2
12 12 
12

15
8
=4 −2
12
12
7
=2
12
Example 7:
3
7
9 −2
4
20
7
3× 5
=9 −2
20
4×5
7
15
=9 −2
20
20
15
7 

= 8 + 1 +  − 2
20
20 

20 7 
15

= 8 +
+ −2
20 20 
20

27
15
=8 −2
20
20
12
=6
20
3
=6
5
Questions:
9
2
−
10 10
(b)
11 3
−
12 12
(c)
7 1
−
8 8
(d)
9
3
−
16 16
2. (a)
3 2
−
4 3
(b)
4 1
−
5 2
(c)
5 1
−
6 4
(d)
5 1
−
8 6
1. (a)
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(e)
5 1
−
6 2
(f)
9 3
−
10 5
4
1
3. (a) 6 − 2
5
3
3
1
(b) 8 − 3
4
6
8 1
(c) 7 − 1
9
6
1
5
(d) 4 − 2
3
7
2
4
(e) 9 − 3
3
5
1
7
(f) 5 − 1
2
8
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Multiplying and Dividing Fractions
Multiplying Fractions
To multiply two fractions, multiply the numerators and multiply the denominators.
a c a×c
Therefore:
× =
Note: b and d are not equal to zero.
b d b×d
Example 1:
4 3
×
5 7
4×3
=
5× 7
12
=
35
Example 2:
3 2
×
10 5
3× 2
=
10 × 5
6
=
50
3
=
25
Example 3:
3
2 ×4
5
13 4
= ×
5 1
52
=
5
2
= 10
5
Example 4:
1
1
1 ×2
2
6
3 13
= ×
2 6
39
=
12
13
=
4
1
=3
4
Dividing Fractions
To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.
Therefore:
reciprocals
a c a d
÷ = ×
b d b c
a×d
=
b×c
Example 5:
3 4
÷
7 5
3 5
= ×
7 4
15
=
28
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Example 6:
3 7
÷
4 20
3 20
= ×
4 7
60
=
28
15
=
7
1
=2
7
Note:
b, c, and d are not equal to 0.
Example 7:
3
1 ÷5
4
7 5
= ÷
4 1
7 1
= ×
4 5
7
=
20
10
Example 8:
3
2
2 ÷1
4 5
11 7
= ÷
4 5
11 5
= ×
4 7
55
=
28
27
=1
28
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Questions
1. (a)
4 1
×
9 5
(b)
(d)
5 6
×
9 7
2
(e) 1 × 4
3
1
(f) 2 × 3
4
1 3
(h) 2 × 1
2 4
1 2
(i) 2 × 1
3 7
2 1
(g) 1 × 1
5 3
5 5
×
6 7
(c)
2 3
×
7 4
2. (a)
1 2
÷
4 5
(b)
3 5
÷
10 7
(c)
1 5
÷
6 12
(d)
3 5
÷
4 8
(e)
4 3
÷
5 10
(f)
3
÷6
7
4
(g) 1 ÷ 2
9
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1
1
(h) 3 ÷ 2
3
2
1
1
(i) 1 ÷ 2
4
7
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Decimals
Like fractions, decimals are used to represent part of a whole.
Area Model
Fraction
Decimal
3
4
0.75
When a number is written as a decimal, it is comprised on three parts: whole number part,
decimal point, and decimal part.
Whole
Number
Part
Decimal
Part
571.806
Decimal
Point
Place names and place values for the whole number and decimal parts of fractions are shown in
the chart below.
1000
100
10
1
Ten
Thousandths
Thousandths
Hundredths
.
Tenths
Decimal Part
Units
Tens
Hundreds
Thousands
Whole Number Part
1
1
1
1
10
100
1000
10000
Example 1:
Express the following numbers in expanded form.
(a) 165.32
(b) 67.891
Answers:
3
2
+
10 100
8
9
1
(b) 67.891 = 60 + 7 + +
+
10 100 1000
(a) 165.32 = 100 + 60 + 5 +
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Example 2:
Write each number in words.
(a) 35.6
(b) 165.32
(c) 67.871
(d) 307.05
(e) 2019.083
Answers:
(a) 35.6 is “thirty-five and six tenths.”
(b) 165.32 is “one hundred sixty-five and thirty-two hundredths.”
(c) 67.871 is “sixty-seven and eight hundred seventy-one thousandths.”
(d) 307.05 is “three hundred seven and five hundredths.”
(e) 2019.083 is “two thousand nineteen and eighty-three thousandths.”
Adding and Subtracting Decimals
Adding decimals is similar to adding whole numbers. We line up the decimal points so that we
can add corresponding place value digits (e.g. tenths with tenths, hundredths with hundredths,
and so on). As with whole numbers, we start from the right and carry when it is necessary.
Example 3:
Add: 42.08 + 208.95.
Example 4:
Add: 36.07 _ 9.065.
Answer:
1
Answer:
1
4 2
+ 2 0 8
1
.
.
1
0 8
9 5
1
3 6
9
+
2 5 1 . 0 3
.
.
0 7
0 6 5
4 5 . 1 3 5
Subtracting decimals is similar to subtracting whole numbers. We line up the decimal points so
that we can subtract corresponding place value digits (e.g. tenths from tenths, hundredths from
hundredths, and so on). As with whole numbers, we start from the right and borrow when it is
necessary.
Example 5:
Subtract: 57.62 - 6.18
Example 6:
Subtract: 98.04 - 32.801
Answer:
Answer:
5
5 7
−
6
.
.
12
7
6 2
1 8
9 8
− 3 2
5 1 . 4 4
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10
.
.
3
10
0 4 0
8 0 1
6 5 . 2 3 9
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C. D. Pilmer
Multiplying Decimals
Step 1: Initially ignore the decimal points and multiply as if both of the factors are whole
numbers.
Step 2: Now the decimal point must be positioned in the product. The number of decimal places
in the product is the sum of the number of places in the factors (count places from the
right).
Example 7:
Multiply: 6.32 × 2.4
Answer:
6. 3 2
×
2. 4
Example 8:
Multiply: 0.832 × 9.31
(2 decimal places)
(1 decimal place)
Answer:
0. 8 3 2
×
9. 3 1
(3 decimal places)
8 3 2
2 4 9 6 0
4 8 8 0 0
2 5 2 8
1 2 6 4 0
1 5. 1 6 8
7
7. 7 4 5 9 2
(3 decimal places)
( 2 decimal places)
(5 decimal places)
Dividing Decimals
Step 1: Move the decimal point to the right in the divisor until the devisor is
quotient
a whole number.
divisor dividend
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 as if you were dividing with whole numbers. Place the decimal point in
the quotient directly above the new decimal point in the dividend.
Example 9:
Divide: 1.792 ÷ 0.32
Example 10:
Divide: 3.612 ÷ 4.3
Answer:
1.792 ÷ 0.32 becomes 179.2 ÷ 32 because
we moved the decimal point in both the
dividend and divisor two places to the
right.
5.6
32 179.2
Answer:
3.612 ÷ 4.3 becomes 36.12 ÷ 43 because
we moved the decimal point in both the
dividend and divisor one place to the right.
0.84
43 36.12
344
172
172
0
160
192
192
0
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C. D. Pilmer
Questions:
1. Express the following numbers in expanded form.
(a) 2482.6
(b) 7.93
(c) 265.147
(d) 40.0562
2. Write each number in words.
(a) 14.9
(b) 3002.15
(c) 459.736
(d) 480.07
(e) 67.025
(f) 23.0578
3. Add.
(a) 42.13 + 30.65
(c) 6.93 + 34.68
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(b) 107.63 + 41.029
(d) 78.073 +105.96
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C. D. Pilmer
(e) 9.8562 + 6.2153
(f) 0.793 + 8.6254
(g) 32.06 + 7.42 + 11.23
(h) 0.645 + 1.39 + 2.0431
4. Subtract.
(a) 46.37 - 14.12
(b) 27.891 - 4.24
(c) 328.46 - 41.28
(d) 489.231 - 25.65
(e) 3.2935 - 0.326
(f) 8.03 - 5.56
(g) 15.064 - 9.38
(h) 2.050 - 0.462
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C. D. Pilmer
5. Multiply.
(a) 6.4 × 2.8
(c) 40.5 × 5.23
6. Divide
(a) 8.84 ÷ 2.6
(c) 0.279 ÷ 0.45
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©2009
(b) 3.52 × 4.6
(d) 0.453 × 6.21
(b) 1.674 ÷ 0.31
(d) 10.793 ÷ 4.3
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C. D. Pilmer
Percents
Percent means per one hundred. The % sign is used to show the number of parts out of one
hundred parts. For example, 37% means 37 parts out of 100 parts. This particular percent can
37
and the decimal 0.37.
also be expressed as the fraction
100
Area Model
Fraction
Decimal
Percent
37
100
0.37
37%
Changing Percents to Decimals
• Drop the % symbol and divide by 100 (i.e. move the decimal point two places to the left).
• Examples:
45% = 0.45
87% = 0.87
16% = 0.16
9% = 0.09
2% = 0.02
1.4% = 0.014
0.5% = 0.005
120% = 1.20
113% = 1.13
Changing Percents to Fractions
• Drop the % sign from the percent, place the number over 100, and simplify the fraction if
possible.
60
36
55
• Examples:
60% =
36% =
55% =
100
100
100
36 ÷ 4
55 ÷ 5
60 ÷ 20
=
=
=
100 ÷ 4
100 ÷ 5
100 ÷ 20
9
11
3
=
=
=
25
20
5
Changing Decimals to Percents
• Multiply by 100 (i.e. move the decimal point two places to the right) and add the % sign.
• Examples:
0.65 = 65%
0.19 = 19%
0.82 = 82%
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©2009
0.04 = 4%
0.07 = 7%
0.029 = 2.9%
0.009=0.9%
1.06 = 106%
1.13 = 113%
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C. D. Pilmer
Changing Fractions to Percents
• Convert the fraction to a decimal using division (by hand or with a calculator), and then
convert the decimal to a percent (i.e. move the decimal point two places to the right and
add the % symbol).
7
13
• Examples:
Convert
to a percent.
Convert
to a percent.
20
25
0.35
0.52
20 7.00
25 13.00
60
125
100
50
100
50
0
0
7
13
= 0.35 = 35%
= 0.52 = 52%
20
25
Taking the Percentage of a Number
• If you need to find a specific percentage of a number, convert the percentage to a decimal
and multiply that decimal by the number (by hand or with a calculator).
Examples: Find 65% of 180.
0.65 × 180 = 117
Find 32% 0f 2100.
0.32 × 2100 = 672
Find 7% of 342.
0.07 × 342 = 23.94
•
Find 113% of 46.
1.13 × 46 = 51.98
Some of these questions can be done quickly and without a calculator if you are dealing
with "friendly" percentages (e.g. 10%, 20%, 30%,…).
Examples: Find 30% of 150
We know that 10% of 150 is 15, therefore 30% of 150 must be 45 (3 × 15).
Find 20% of 320
We know that 10% of 320 is 32, therefore 20% of 320 must be 64 (2 × 32).
Find 70% of 90
We know that 10% of 90 is 9, therefore 70% of 90 must be 63 (7 × 9).
Questions:
1. Convert the following percents to decimals.
(a) 38% =
(b) 21% =
(c) 37% =
(d) 4% =
(e) 6% =
(f) 24.5% =
(g) 3.4% =
(h) 0.8% =
(i) 105% =
(j) 14.2% =
(k) 210% =
(l) 3.6% =
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C. D. Pilmer
2. Convert the following percents to fractions.
(a) 45%
(b) 70%
(c) 28%
(d) 80%
(e) 6%
(f) 42%
(g) 72%
(h) 120%
3. Convert the following decimals to percents.
(a) 0.92 =
(b) 0.47 =
(c) 0.32 =
(d) 0.07 =
(e) 0.7 =
(f) 0.007 =
(g) 0.042 =
(h) 0.206 =
(i) 1.51 =
(j) 1.8 =
(k) 1.06 =
(l) 1.034 =
4. Convert the following fractions to percents. You may use a calculator to complete this
question. ()
(a)
2
5
(b)
14
25
(c)
17
20
(d)
5
8
(e)
3
8
(f)
15
16
(g)
26
25
(h)
9
8
5. Complete each of the following. You may use a calculator to complete this question. ()
(a) Take 35% of 6280.
NSSAL
©2009
(b) Take 65% of 580.
20
(c) Take 9% of 1600.
Draft
C. D. Pilmer
(d) Take 15% of 74.
(e) Take 3.5% of 56.
6. Complete each of the following. Do not use a calculator.
(a) Take 10% of 430.
(b) Take 10% of 5800.
(f) Take 113% of 57.
(c) Take 20% of 60.
(d) Take 20% of 140.
(e) Take 30% of 70.
(f) Take 40% of 110.
(g) Take 70% of 2000.
(h) Take 60% of 400.
(i) Take 20% of 2500.
(j) Take 40% of 300.
(k) Take 60% of 7000.
(l) Take 30% of 1200.
NSSAL
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C. D. Pilmer
Ratio and Proportions
Ratios
A ratio is the quotient of two quantities. A ratio can be written in two different forms: fractional
notation and colon notation. For example, the ratio of 2 to 3 can be written as:
2
(fractional notation)
•
3
• 2:3 (colon notation)
Ratios are easier to understand when they are written in their lowest terms (also called simplest
form). For example, we can create a ratio of Ted’s new monthly savings to his monthly
earnings. Initially the ratio was expressed as 125:1500. When the ratio is changed to its lowest
terms, it is expressed as 1:12. That means that for every $12 that Ted earns, $1 is set aside as
savings.
Example 1:
Write the ratio of $35 to $15 in fractional notation and express in lowest terms.
Answer:
$35 35 35 ÷ 5 7
=
=
=
$15 15 15 ÷ 5 3
Example 2:
Write the ratio of 3.1 to 4.2 as a fraction in simplest form.
Answer:
3.1 3.1 × 10 31
=
=
4.2 4.2 × 10 42
Example 3:
Write the ratio of 0.14 to 0.2 as a fraction in simplest form.
Answer:
0.14 0.14 × 100 14 14 ÷ 2 7
=
=
=
=
0.2
0.2 × 100 20 20 ÷ 2 10
Proportions
A proportion states that two ratios are equal.
• For example, 5:7 = 15:21 is a proportion. We can read it as “5 is to 7 as 15 is to 21.”
13 26
• For example,
is a proportion. We can read it as “13 is to 9 as 26 is to 18.”
=
9 18
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C. D. Pilmer
When one number of the proportion is unknown, we can use cross products to find the unknown
number.
Procedure:
1. Find the cross products.
2. Set the cross products equal to each other.
3. Divide the number not multiplied by x by the number multiplied by x.
Example 4:
Find the value of the unknown number x.
x 7
=
12 3
Answer:
x
12
=
7
3
The cross products will be 3 × x and 12× 7 .
3 x = 84
Set the cross product equal to each other.
84
3
x = 28
Divide the number not multiplied by x by the number multiplied by x.
x=
Example 5:
Find the value of the unknown number x.
9 1.5
=
x
7
Answer:
9
x
=
1.5
7
The cross products will be 9 × 7 and x × 1.5 .
63 = 1.5 x
Set the cross product equal to each other.
63
1.5
x = 42
Divide the number not multiplied by x by the number multiplied by x.
x=
Questions:
1. Write each ratio as a ratio of whole numbers using fractional notation. Write the fraction in
simplest form.
(a) 12 to 16
(b) 45 to 25
(c) 18 to 30
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C. D. Pilmer
2.
(d) 27 to 21
(e) 130 to 160
(f) 16 to 18
(g) 32 to 24
(h) 21 to 35
(i) 72 to 45
(j) 5.1 to 3.7
(k) 4.6 to 6.5
(l) 6.12 to 2.53
(m) 6.7 to 9.47
(n) 0.11 to 5.2
(o) 6.2 to 4
(p) 0.5 to 3.5
(q) 0.27 to 0.15
(r) 0.3 to 0.42
For each proportion, find the unknown number, x. ()
x 2
8 24
(a)
(b) =
=
21 3
5 x
(d)
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©2009
28 24
=
x 18
(e)
15 2.5
=
6
x
24
(c)
10 x
=
18 45
(f)
x
24
=
7.5 30
Draft
C. D. Pilmer
Signed Numbers
Much of the mathematics we use in our daily lives requires that we understand positive
numbers (e.g. 0.4, 2, 127, 3/5). Negative numbers (e.g. -7, -0.5, -4/5), however, are extremely
important as one continues their study of mathematics.
negative numbers
-4
-3
-2
-1
positive numbers
0
1
2
3
4
All signed numbers have magnitude (i.e. size) and direction (i.e. sign: positive or negative).
• For example the number -4 has a negative direction (to the left of 0 on the number line)
and a magnitude of 4.
• For example the number -3.2 has a negative direction (to the left of 0 on the number line)
and a magnitude of 3.2.
• For example the number +5 or 5 has a positive direction (to the right of 0 on the number
line) and a magnitude of 5.
• For example the number +7.85 or 7.85 has a positive direction (to the right of 0 on the
number line) and a magnitude of 7.85.
Understanding the terms, magnitude and direction, is important when trying to explain the rules
for addition, subtraction, multiplication and division of signed numbers.
Adding Signed Numbers
• To add two numbers with the same sign, add their magnitudes, and keep the common
sign.
ex. (− 5) + (− 3) = −8
ex. (+ 4 ) + (+ 6 ) = +10
•
ex. (− 2) + (− 7 ) = −9
ex. (− 7.2) + (− 1.3) = −8.5
4
 1  3
ex.  +  +  +  = +
5
 5  5
5
 2  3
ex.  −  +  −  = −
7
 7  7
To add two numbers with different signs, subtract the smaller magnitude from the larger
magnitude, and use the sign of the number with the larger magnitude.
ex. (− 4) + (+ 7 ) = +3
ex. (+ 5) + (− 3) = +2
NSSAL
©2009
ex. (+ 3) + (− 9) = −6
ex. (− 8) + (+ 1) = −7
ex. (+ 4.8) + (− 1.2 ) = +3.6
ex. (+ 3.1) + (− 7.8) = −4.7
3
 2  5
ex.  −  +  +  = +
8
 8  8
5
 2  7
ex.  +  +  −  = −
9
 9  9
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C. D. Pilmer
Subtracting Signed Numbers
• To subtract one signed number from another, change the question from a subtraction
question to an addition question, and change the sign of the number that was originally
being subtracted. Once these changes have been made, follow the rules for adding signed
numbers.
ex. (+ 2) − (− 5)
ex. (− 3) − (− 4 )
(+ 2) + (+ 5) = 7
(− 3) + (+ 4) = +1
ex. (− 4 ) − (+ 6 )
(− 4) + (− 6) = −10
ex. (− 1.4 ) − (− 0.8)
(− 1.4) + (+ 0.8) = −0.6
 1  2
ex.  +  −  − 
 5  5
3
 1  2
+  + +  = +
5
 5  5
 2  6
ex.  −  −  − 
 7  7
4
 2  6
−  + +  = +
7
 7  7
Multiplying Signed Numbers
• To multiply two numbers with the same sign, multiply the magnitudes, and the resulting
sign will be positive.
ex. (− 5) × (− 4) = +20
ex. (+ 6 ) × (+ 3) = +18
•
ex. (− 2) × (− 12) = +24
ex. (− 0.4) × (− 0.3) = +0.12
2
 1  2
ex.  +  ×  +  = +
15
 3  5 
6
 3  2
ex.  −  ×  −  = +
35
 7  5
To multiply two numbers with the different signs, multiply the magnitudes, and the
resulting sign will be negative.
ex. (− 6) × (+ 2) = −12
ex. (+ 8) × (− 4 ) = −32
ex. (+ 0.2) × (− 12) = −2.4
ex. (− 0.4 ) × (+ 0.7 ) = −0.28
4
 1  4
ex.  +  ×  −  = −
45
 9  5
10
 5  2
ex.  −  ×  +  = −
21
 7  3
Dividing Signed Numbers
• To divide two numbers with the same sign, divide the magnitude of the first by the
magnitude of the second, and the resulting sign will be positive.
ex. (+ 6) ÷ (+ 2) = +3
ex. (− 8) ÷ (− 4) = +2
ex. (− 35) ÷ (− 7 ) = +5
•
ex. (− 3.6 ) ÷ (− 0.6 ) = +6
To divide two numbers with the different signs, divide the magnitude of the first by the
magnitude of the second, and the resulting sign will be negative.
ex. (+ 16) ÷ (− 4) = −4
ex. (− 45) ÷ (+ 9) = −5
ex. (+ 63) ÷ (− 7 ) = −9
NSSAL
©2009
ex. (− 12 ) ÷ (+ 0.6 ) = −20
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C. D. Pilmer
Exponents and Signed Numbers
When one writes a 4 it means a × a × a × a . The a is the base, the 4 is exponent, and the
entire expression a 4 is called the power.
ex. 23 = 2 × 2 × 2
ex. 32 = 3 × 3
=8
=9
The challenging aspect of signed numbers and exponents occurs when one is attempting to
understand if the base of the power is positive or negative. For example, many people make
4
the mistake of thinking that (− 2 ) and − 24 mean the same thing; this is not the case.
•
•
has a base of -2 and means (− 2 ) × (− 2 ) × (− 2 ) × (− 2 ) . Therefore (− 2 ) equals 16.
4
− 2 has a base of 2 and means − 1 × (2 × 2 × 2 × 2 ) . Therefore − 24 equal -16.
(− 2)4
4
ex. (− 5) = (− 5) × (− 5)
= 25
ex. (− 4 ) = (− 4 ) × (− 4 ) × (− 4 )
= −64
ex. (− 7 ) = (− 7 ) × (− 7 )
= 49
ex. (− 3) = (− 3) × (− 3) × (− 3) × (− 3)
= 81
ex. − 62 = −1 × (6 × 6 )
= −36
ex. − 34 = −1 × (3 × 3 × 3 × 3)
= −81
3
2
4
2
Order of Operations and Signed Numbers
Order of Operations (BEDMAS) applies to all numbers.
Example:
2
6 + [(− 8) + (+ 5)] − (− 4)
Example:
2
7 + [− 2 − (+ 4)] + 5 × (− 2)
= 6 + (− 3) − (− 4)
= 6 + (− 3) − (+ 16)
= 3 + (− 16)
= 7 + [− 2 + (− 4)] + 5 × (− 2)
2
= −13
2
= 7 + (− 6) + 5 × (− 2)
= 7 + 36 + 5 × (− 2)
= 7 + 36 + (− 10)
= 33
2
Example:
(− 2)3 − 18 ÷ (− 9) × (+ 2)
= (− 8) − 18 ÷ (− 9 ) × (+ 2 )
= (− 8) − (− 2 ) × (+ 2 )
= (− 8) − (− 4 )
= (− 8) + (+ 4 )
= −4
Questions
1. (a) (− 3) + (− 2 )
(b) (− 7 ) + (− 6 )
(c) (+ 3) + (+ 8)
(d) (− 5) + (+ 9 )
(e) (− 8) + (+ 2)
(f) (+ 12) + (− 5)
(g) (+ 2) + (− 15)
(h) (− 7 ) + (− 11)
(i)
(− 3) + (+ 10)
(k) (− 9 ) − (+ 2 )
(l)
(− 10) − (− 2)
(j)
(− 14) + (− 7 )
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C. D. Pilmer
(m) (+ 15) − (+ 9 )
(n) (+ 21) − (− 8)
(o) (− 9 ) − (+ 7 )
(p) (− 11) − (− 8)
(q) (− 3) − (− 9 )
(r) (− 7 ) × (− 5)
(s) (− 6) × (− 7 )
(t)
(− 8) × (+ 4)
(u) (+ 7 ) × (− 8)
(v) (− 9) × (− 6)
(w) (− 27 ) ÷ (− 9 )
(y) (− 28) ÷ (+ 7 )
(z) (+ 32) ÷ (− 4)
2. (a) (− 14) − (− 6)
(b) (− 48) ÷ (+ 8)
(c) (− 9) × (+ 7 )
(d) (+ 3) + (− 19)
(e) (+ 17 ) − (− 8)
(f) (− 7 ) × (− 10)
(g) (− 30) ÷ (− 10)
(h) (− 12) + (− 14)
(i)
(− 10) − (+ 8)
(k) (+ 4.5) + (− 9.1)
(l)
(− 2.5) ÷ (+ 0.5)
(m) (+ 1.4) + (− 1.1)
(n) (− 0.6 ) × (− 4 )
(o) (− 5.1) − (+ 4.3)
 5  1
(p)  +  ×  − 
 9  2
 1  4
(q)  −  +  − 
 6  6
 4  2
(r)  +  −  − 
 7  7
(b) (− 3)
(c) (− 5)
(e) − 14
(f) − 26
(j)
(− 9.7 ) − (− 3.5)
3. (a) (− 6 )
2
(d) − 82
(x) (− 36) ÷ (− 6)
3
3
4. (a) (− 10 ) + [(− 2 ) − (− 5)]× 3 (b) (− 3) − [(− 8) + (+ 3)]
2
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(c) (− 2 ) × [(− 1) + (− 2 )] + (+ 5)
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C. D. Pilmer
All Together Now, Part I
Complete all of the calculations without using a calculator. Show your work.
(a)
(+ 13) − (− 5)
(b)
78.14 - 42.8
(c)
Find x.
x 5
=
12 3
(d)
32 + (11 − 6 ) − 2 × 4
(e)
Take 40% of 120.
(f)
2 1
2 ÷1
3 2
(g)
5.4 × 7.2
(h)
(− 2)3
(i)
19.5 ÷ 5
(j)
(− 6)× (− 5)
(k)
2
1
5 +4
3
7
(l)
80.42 + 7.9
(o)
35 − (2 + 3)
6+4
(m)
3 1
5 −1
4 7
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(n)
(− 13) + (− 7 )
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All Together Now, Part II
Complete all of the calculations without using a calculator. Show your work.
(a)
Take 20% of 70.
(b)
6.3 × 2.5
(c)
2 × (7 − 4 ) + 7
(d)
1
4
2 +3
2
9
(e)
(− 5)2
(f)
(+ 8)× (− 4)
2
(g)
(− 9) + (+ 7 )
(h)
Find x.
2 6
=
7 x
(i)
(j)
168.45 - 92.8
(k)
5 2 + (10 − 3)
32 − 1
(l)
(− 3) − (+ 8)
(m)
41.4 ÷ 9
(n)
2
1
5 ÷2
3
2
(o)
18.45 + 9.813
NSSAL
©2009
30
3
1
6 −2
5
4
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C. D. Pilmer
All Together Now, Part III
Complete all of the calculations without using a calculator. Show your work.
(a)
(d)
1 1
4 ÷1
6 3
(+ 11) + (− 2)
(b)
(− 11) − (− 7 )
(c)
5
1
7 −2
8
3
(e)
1 + (7 + 4 ) × 2 − 4 2
(f)
16.2 ÷ 6
(g)
Find x.
5 x
=
4 12
(h)
Take 30% of 220.
(i)
28.56 - 6.902
(j)
(− 1)4
(k)
7.4 × 3.1
(l)
5
3
1 +4
6
8
(m)
9 + 3× 5
(8 − 2)2 − 30
(n)
(− 9)× (− 3)
(o)
5.61 + 63.815
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Reflect Upon Your Learning
Fill out this questionnaire after you have completed pages 1 to 31. Select your response to each
statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
I understand all of the concepts covered in the section, Order of
Operations.
(b) I do not need any further assistance from the instructor on the
material covered in the section, Order of Operations.
(c) I understand all of the concepts covered in the section,
Fractions.
(d) I do not need any further assistance from the instructor on the
material covered in the section, Fractions.
(e) I understand all of the concepts covered in the section,
Decimals.
(f) I do not need any further assistance from the instructor on the
material covered in the section, Decimals.
(g) I understand all of the concepts covered in the section,
Percents.
(h) I do not need any further assistance from the instructor on the
material covered in the section, Percents.
(i) I understand all of the concepts covered in the section, Ratio
and Proportions.
(j) I do not need any further assistance from the instructor on the
material covered in the section, Ratio and Proportions.
(k) I understand all of the concepts covered in the section, Signed
Numbers.
(l) I do not need any further assistance from the instructor on the
material covered in the section, Signed Numbers.
(m) I was able to complete all or most of the questions successfully
in the three sections titled All Together Now.
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1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
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C. D. Pilmer
Introduction to Application Questions
For the rest of this unit we are going to focus on application problems involving fractions,
decimals, percents, signed numbers, ratios and proportions. Most learners refer to these
application questions as word problems. Word problems can be easy to solve if we can
recognize what concept (e.g. fractions, ratios and proportions, …) and operation (e.g. addition,
substraction, multiplication,…) we are dealing with.
Don’t know where to start? Follow this plan of action.
1. Read the problem. Then, reread the problem.
2. Write down what information that has been given to you.
3. Write down what you need to find.
4. Identify the type of problem you have been given, is it a
(a) ratio and proportion problem
(b) whole number problem,
(c) percent problem,
(d) fraction problem,
(e) decimal problem, or
(f) signed number problem?
5. Are there key words that tell you what operation (addition, subtraction, …) should be used?
6. Be sure to reread the question, examine your answer, and determine whether your answer
seems reasonable given the situation described in the question.
In step 5, we mentioned key words. Here are some of the key words and phrases associated with
the operations of addition, subtraction, multiplication and division.
Words associated with ADDITION
• Combined
• Plus
• Sum
• More
• Total
• Altogether
• Increased by
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Words associated with SUBTRACTION
• Decreased
• Difference
• Minus
• Less
• Take away
• Taken from
• Removed
• Diminished by
• Subtracted from
• More than
• Amount left
• Fewer than
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Words associated with MULIPLICATION
• Product
• Times
• Factors of
• Total
• Of
• Multiplied by
Words associated with DIVISION
• Quotient
• Divisor
• Equal parts
• Into
• Divided up
• Over
• Per
• Ratio of
• How many of one thing “fits” into
another
• Out of
Example:
Determine the thickness of a board that was originally 1
3
5
inches thick then has
inch
4
16
removed by a planer.
5
inch
16
1
3
inch
4
Answer:
If we are "removing" or taking away some of the wood, this tells us that we are dealing with
the operation of subtraction. And since we are dealing with fractions, we have to know how
to subtraction fractions. (This is done by making common denominators.)
3 5
1 −
4 16
3× 4 5
−
=1
4 × 4 16
12 5
=1 −
16 16
7
inches
=1
16
(This is the new thickness of the board.)
Other Important Points:
• Average = means to add up all numbers and divide your sum by the number of items that
you have.
• Perimeter = means the total distance around the outside of a figure
• Ratio and Proportion Questions: Involve the comparison of two quantities (e.g. “for every
2 tablespoon of milk, you will need 3 tablespoons of sugar")
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Applications - Fractions
Example 1:
The cross-sectional view of a pipe is provided.
Based on the information in the diagram,
determine the outer diameter of the pipe.
3
4
2
3
4
"
5
8
Example 2:
3
inch counter top is comprised
4
1
of a particle board core with
inch laminate
16
glued on top. How thick is the particle board
core?
A particular
"
"
3
4
Answer:
3
5 3 3× 2
5 3× 2
+2 + =
+2 +
4
8 4 4× 2
8 4× 2
6
5 6
= +2 +
8
8 8
17
=2
8
1
= 2+2
8
1
= 4 inches
8
Example 3:
For each hour that an oil burner runs, it uses
3
1 litres of fuel. If the burner is only running
4
2
for
hour, then how much fuel is used?
3
Answer:
3 2 7 2
1 × = ×
4 3 4 3
14
=
12
7
=
6
1
= 1 litres
6
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"
core
Answer:
3 1 3× 4 1
− =
−
4 16 4 × 4 16
12 1
= −
16 16
11
=
inch
16
Example 4:
You have 10 pounds of flour in a bag. You are
removing it from the bag using a container that
2
can hold
pounds of flour. Assuming that
5
you are filling the container each time, how
many times will you use the container to
completely empty the bag?
Answer:
2 10 5
10 ÷ = ×
5 1 2
50
=
2
= 25 times
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Questions:
1. Determine the missing measurement.
7
5 inches
8
?
1
8 inches
2
3
1
3
inch thick plywood,
inch thick felt paper, and inch
4
16
8
thick shingles. What is the total thickness of the sheathing?
2. Roof sheathing is comprised of
3. Figure out the number of sheets of
4. If the diameter of a hole is 2
5
inch plywood in a stack 25 inches high?
8
7
inch, what is the radius of
8
the hole?
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radius
36
diameter
Draft
C. D. Pilmer
5. A motor is attached to a support block.
Using the diagram, determine how high
the motor’s shaft is off the ground?
Motor
6. How many pieces of pipe that are
6
5
8
2
3
4
″
″
Support Block
2
foot long can be laid end to end to create a pipe 8 feet
3
long?
7. Find the thickness of a 1
1
1
inch board after
inch is planed off?
4
16
8. A dog groomer uses a 36 gallon container to wash dogs. If she only fills it
3
full of water,
4
how many gallons of water does she use?
9. A baker used 2
1
2
cups of flour for rolls and 1 cups for cookies. How much flour was
2
3
used?
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10. Find the perimeter of this regular hexagon. With any
regular polygon, all the sides are of equal length.
11. How much longer is a 2
12. How many
2
3
inches
4
1
3
inch nail compared to a 1 inch nail?
2
4
2
cup sugar bowls can be filled using 10 cups of sugar?
3
13. Three quarters of a pizza is left over from last night’s party. You want half of the remaining
pizza. How much of the original pizza does your piece represent?
14. Figure out the length of the following shaft.
1
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3
4
"
2
3
8
"
1
1
2
"
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C. D. Pilmer
Applications - Decimals
Example 1:
A patient was given injections of 3.2 ml, 2.15
ml, and 1.9 ml of a particular medication.
How much medication did the patient receive
in total of that particular medication?
Example 2:
A school bus driver making $12.50 per hour
was given a raise to $14.05 per hour. How
much is the raise?
Answer:
Subtract the two numbers.
Answer:
Add the three numbers.
3
1
1 4
− 1 2
3 . 2
2 . 1 5
+ 1 . 9
10
.
.
0 5
5 0
1 . 5 5
The driver received a $1.55 raise.
7 . 2 5
The patient received 7.25 ml of medication.
Example 3:
Determine the area of
this rectangle.
5.1 m
2.7 m
Answer:
The area of a rectangle is found by
multiplying the length by the width.
5. 1
× 2. 7
1
Example 4:
A phone company is charging $0.06 per
minute for long distance calls within Canada.
If your bill for a long distance call within
Canada was $4.32, how many minutes was the
call?
Answer:
Divide 4.32 by 0.06.
Change 4.32 ÷ 0.06 to 432 ÷ 6 by moving
the decimal point two places to the right on
both the dividend and the divisor.
72
6 432
(1 decimal place)
(1 decimal place)
3 5 7
0 2 0
1 3. 7 7
42
12
12
0
(2 decimal places)
The area of the rectangle is 13.77 m2.
The call lasted 72 minutes.
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Questions:
1. A pair of jeans sells for $39.99 is marked down $7.49. What is the new price of the jeans?
2. A particular cut of meat costs $6.70 per kilogram. How much does 0.8 kilograms cost?
3. If Churchill Publishing shares can be bought for $5.80 per share, how many shares can be
purchased with $2030?
4. Determine the perimeter of this triangle.
0.69 m
1.15 m
0.92 m
5. A patient is given 0.075 mg tablets each day for 7 days. How much medication of this type
did the patient receive in total over that 7 day period?
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6. Four students decided to split the lunch bill equally among themselves. If the bill came to
$50.52, how much would each person pay?
7. Gasoline costs $1.12 per litre. How much will you pay for 18.5 litres?
8. When a particular piston in a car engine is cold, its diameter measures 8.579 centimetres.
After the engine has been running, the piston becomes hot and expands. At that time, its
diameter measures 8.593 centimetres. How much did the piston expand?
9. Akira and her partner traveled 526.3 km on day one, 488.5 km on day two, and 417.6 km on
day three. How far did they travel over those three days?
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10. A bottle initially containing 30 ml of medication has 2.7 ml removed. How much medication
remains in the bottle?
9.49 cm
11. Determine the missing dimension.
2.81 cm
? cm
4.67 cm
12. A metalworker is casting a particular part. He randomly selects five completed parts and
weighs them. Their masses are shown below.
134.5 g, 133.9 g, 134.2 g, 133.7 g, 133.7g
Determine the average mass of these five parts.
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Applications - Percents
Example 1:
Tanya wants to leave a 20% tip on a meal costing $36. How much should she leave?
Answer:
Take 20% of $36
0.20 × 36 = $7.20
Example 2:
Angela conducted a survey in her adult education class and discovered that 15 out of the 24
learners had at least one child between the ages of 6 years and 18 years.
(a) What percentage of learners in this class had at least one child between the ages of 6 years
and 18 years?
(b) What percentage of learners in this class did not have at least one child between the ages of 6
years and 18 years?
Answer:
15
(a)
= 0.625 = 62.5%
24
9
(b)
= 0.375 = 37.5%
24
24
23
22
21
20
19
18
17
16
Number of Individuals
Example 3:
Montez conducted a survey where he asked 40 individuals to
indicate the number of working televisions they had at their
place of residence. The results of this survey are displayed
on the bar graph.
(a) What percentage of the individuals has one television?
(b) What percentage of the individuals has two televisions?
(c) What percentage of the individuals has three or more
televisions?
(d) If we only considered individuals who had 2 or more
televisions, what percentage of those individuals has 4
televisions?
15
14
13
12
11
10
9
8
7
6
5
4
Answers:
22
(a)
= 0.55 = 55%
40
10
(b)
= 0.25 = 25%
40
8
(c)
= 0.2 = 20%
40
3
(d)
= 0.1666... ≈ 16.7%
18
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3
2
1
0
1
2
3
4
Num ber of Televisions
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Example 4:
Of 3600 vehicle accidents, only 2% were caused by mechanical failure. How many accidents
were caused by mechanical failure?
Answer:
Take 2% of 3600
0.02 × 3600 = 72 accidents
Example 4:
The circle graph shows the sales at fast-food
Hamburger Sales (USA)
hamburger chains as a percent of total fast-food sales.
It should be noted that $42 billion represents the total
Others, 19%
amount of money generated in one year by all of the
fast-food hamburger chains in the United States.
McDonald's,
43%
(a) How much money did McDonald’s generate from
Hardee's, 6%
hamburger sales in one year?
(b) How much money did Wendy’s generate from
Wendy's,
12%
hamburger sales in one year?
(c) If Wendy’s could increase its hamburger sales by
Burger King,
1% at the expense of MacDonald’s sales, how
20%
would that affect the money generated by the sale
of hamburger sales for both Wendy’s and
MacDonald’s?
(d) Presently what percentage of hamburger sales does Burger King and McDonald’s share
between them?
(e) If the total sales of hamburgers increased to $45 billion the following year, but the market
shares remained the same, how much money would Burger King generate from hamburger
sales?
Answers:
(a) Take 43% of $42 billion
0.43 × 42 = $18.06 billion
(b) Take 12% of $42 billion
0.12 × 42 = $5.04 billion
(c) Wendy’s: Now take 13% of $42 billion.
0.13 × 42 = $5.46 billion
(d) 20% + 43% = 63%
(e) Take 20% of $45 billion
0.20 × 45 = $9 billion
McDonald’s: Now take 42% of $42 billion.
0.42 × 42 = $17.64 billion
Example 6:
Janice is purchasing a coat that costs $69.99. What is the total cost including sales tax (13%)?
Answer:
Janice must pay 100% for the coat plus 13% for the tax. That gives us 113%.
Take 113% of $69.99
1.13 × 69.99 ≈ $79.09
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Example 7:
Nita usually spends $250 per month on entertainment (e.g. movies, concerts, clubs). With the
downturn in the economy, she has decided to reduce her spending on entertainment by 30%.
How much will she now spend on entertainment after she initiates this plan?
Answer:
If the spending is reduced by 30%, that means that 70% of the spending is retained.
Take 70% of $250.
0.70 × 250 = $175
Questions:
You may use calculator to complete these questions. ()
1. Of the 128 people attending a movie, 96 bought popcorn. What percentage bought popcorn?
2. Jack wants to leave a 15% tip for a meal that cost $26. How much should he leave?
3. Nasrin brings home $2000 per month and budgets $350 for food. What percentage of her
earnings does she budget for food?
4. The new local contract states that employees will receive a 4.5% salary increase. If an
employee was making $24 000 a year, how much will he/she make under the new contract?
5. Of 120 adults interviewed at the mall, only 6 stated that they had used public transit to get to
the mall?
(a) What percentage of the people interviewed at the mall used public transit to get to the
mall?
(b) What percentage of the people interviewed at the mall did not use public transit to get to
the mall?
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7. F. Porter Contractors created a bar graph to illustrate
the age ranges of their employees.
(a) How many employees does the company have?
(b) What percentage of the staff is between 20 and 29
years of age?
(c) What percentage of the staff is 40 years of age or
older?
(d) What percentage of the staff is between 30 years
and 49 years?
(e) If two new employees were hired who were both
34 years of age, what percentage of employees
would be between 30 and 39 years of age.
Emergency Room Admissions
Auto
Accidents,
27%
Home and
Work
Injuries, 34%
Heart
Attacks, 17%
Other, 9%
Respiratory
Problems,
13%
13
12
11
10
9
Number of Employees
6. The circle graph shows emergency room
admissions at a particular hospital over a one
month period. During that time, a total of 420
were admitted.
(a) How many individuals were admitted for
respiratory problems?
(b) How many individuals were admitted for
heart attacks?
(c) How many people were admitted for
injuries associated with work injuries, home
injuries or automobile accidents?
8
7
6
5
4
3
2
1
0
20-29
30-39
40-49
50-59
Ages
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8. Jacob is presently paying $260 per month on electricity. The electricity is used for
appliances, lights, heating, and the hot water tank. He is told that he can reduce this cost by
40% if he installs a woodstove. If he installs the woodstove, what should he expect his
monthly electric bill to be?
9. (a) Jun is buying a video game that costs $49.99. What is the total cost including sales tax
(13%)?
(b) If Jun’s $49.99 video game was marked down 10%, how much would he have to pay in
total (including 13% sales tax)?
10. If a metal rod, initially measuring 37.45 cm in length, expands by 0.1% when it is heated,
how long will the rod be when it is heated?
11. You have $1600 in Digaflex stock.
(a) If the stock drops in value by 5%, how much is your stock worth?
(b) If the stock increases in value by 5%, how much is your stock worth?
12. A carpenter determines that he needs 840 board feet of flooring to complete a job. If an
additional 10% must be factored in for waste, how many board feet of flooring should the
carpenter order?
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Applications - Ratio and Proportion
Ratios
Example 1:
Given the following rectangle:
(a) Determine the ratio of its length to its width in fractional notation.
(b) Determine the ratio of its width to its perimeter in colon notation.
Answers:
length 10 cm 10 10 ÷ 2 5
(a)
=
=
=
=
width
6 cm
6
6÷2 3
(b)
6 cm
10 cm
width
6 cm
6
6÷2
3
=
=
=
=
perimeter 32 cm 32 32 ÷ 2 16
3:16
Example 2:
The Lake Fletchers canoe club entered a regatta. They took 48 female paddlers and 56 male
paddlers.
(a) Find the ratio of the male paddlers to female paddlers from this particular canoe club that
participated in the regatta.
(b) Find the ratio of the female paddlers to the total number of paddlers from this particular
canoe club that participated in the regatta.
Answers:
male paddlers
56 56 ÷ 8 7
(a)
=
=
=
female paddlers 48 48 ÷ 8 6
(b)
female paddlers
48
48 ÷ 8
6
=
=
=
total number of paddlers 104 104 ÷ 8 13
Example 3:
What is the alternator to engine ratio in colon notation if the alternator turns at 1150 rpm
(revolutions per minute) when the engine is idling at 500 rpm?
Answer:
alternator 1150 rpm 1150 1150 ÷ 50 23
=
=
=
=
engine
500 rpm
500
500 ÷ 50 10
23:10
Example 4:
You need 15 ft3 of cement to make 80 ft3 of concrete. Determine the ratio of cement to concrete
in simplest form.
Answer:
cement 15 ft 3 15 15 ÷ 5 3
=
=
=
=
concrete 80 ft 3 80 80 ÷ 5 16
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Proportions
Example 1:
A farmer uses 70 kg of chemical on a 20 acre field. How many kilograms of the same chemical
will the farmer need to use on a 65 acre field?
Answer:
kilograms
acres
70 x
=
20 65
20 × x = 70 × 65
20 x = 4550
4550
20
x = 227.5
Set the cross products equal to each other.
x=
The farmer needs 227.5 kg of the chemical.
Example 2:
If you need 826 bricks to construct a wall that is 14 feet long, how many bricks will you need to
construct a wall that is 48 feet long? Assume that the walls are the same height?
Answer:
bricks
feet
826 x
=
14 48
14 × x = 826 × 48
14 x = 39648
39648
14
x = 2832
Set the cross products equal to each other.
x=
You need 2832 bricks.
Example 3:
For a particular medication, the label reads “1.6 g in 20 cm3.” How many cubic centimeters are
required to obtain 7.2 g of the active ingredient?
Answer:
grams
cm 3
1.6 7.2
=
20
x
1.6 × x = 20 × 7.2
1.6 x = 144
144
1.6
x = 90
Set the cross products equal to each other.
x=
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You need 90 cm3.
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Example 4:
To estimate the number of fish in a lake, a fisheries officer catches 160 fish from the lake, tags
them, and throws them back. He returns the next day and catches 70 fish. He notes that of these
70 fish, 28 have the tags that he attached on the previous day. Estimate how many fish are in the
lake.
Answer:
tagged fish
number of fish
28 160
=
70
x
28 × x = 70 × 160
28 x = 11200
11200
28
x = 400
Set the cross products equal to each other.
x=
We estimate that there are 400 fish in the lake.
Example 5:
Ethan is using a map of the city to determine the distance between his house and his girlfriend’s
house. The scale on the map indicates that every 4 centimetres on the map is equivalent to an
actual distance of 1 kilometre. If the distance on the map between Ethan’s house and that of his
girlfriend is 18.4 cm, what is the actual distance between the houses?
Answer:
map distance
actual distance
4 18.4
=
1
x
4 × x = 1 × 18.4
4 x = 18.4
18.4
4
x = 4.6
Set the cross products equal to each other.
x=
The houses are 4.6 km apart.
Questions:
You are permitted to use a calculator on these questions. () For questions for the first three
questions, express all ratios using whole numbers and in simplest form.
1. Use the following triangle to answer parts (a), (b), and (c).
(a) Find the ratio of the longest side to the shortest side.
(b) Find the ratio of the longest side to the perimeter.
(c) Find the ratio of the shortest side to the perimeter.
20 m
12 m
16 m
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2. A poll at a local college found that 3220 students out of 4800 students were single.
(a) Find the ratio of single students to the total number of students.
(b) Find the ratio of married students to single students.
(c) Find the ratio of married students to the total number of students
3. For every 100 tons of the earth’s crust there is 28 tons of silicon. What is the ratio of the
weight of silicon to the weight of the earth’s crust?
4. A particular fruit punch asks that you mix 3 parts grapefruit juice to 4 parts cranberry juice.
(a) How much grapefruit juice should be mixed with 1200 ml of cranberry juice?
(b) How much cranberry juice should be mixed with 1500 ml of grapefruit juice?
5. The ratio of a quarterback’s completed passes to attempted passes is 3 to 8.
(a) If he attempted 32 passes, how many would he have completed?
(b) If he completed 15 passes, how many would he have attempted?
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6. Jane can waterproof 450 ft2 of decking with 4 litres of sealant. How many litres will she
need to seal a 1575 ft2 deck?
7. A quality control inspector examined 100 circuit boards and found 3 to be defective. At this
rate, how many defective circuit boards would one expect to find in a lot of 2400?
8. The recommended daily allowance of protein for adults is 0.8 grams for every 2.2 pounds of
body weight.
(a) If someone weights 160 pounds, how much protein should he/she eat each day?
(b) If someone is consuming 44 grams of protein each day, how much should that individual
weight if they are attempting to follow the recommended daily allowance?
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9. A sump pump discharges 320 litres of water in 5 minutes. How much time would it take for
the same sump pump to discharge 800 litres of water?
10. The paper needed for a printing job weighs 5.5 kg per 500 sheets. How many kilograms of
paper will be needed for a job requiring 13 500 sheets?
11. A builder sells a 1500 ft2 house for $187 500. If a 2200 ft2 house of the same quality is built
by the same builder, how much would you expect it to cost?
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Applications - Signed Numbers
There are several applications where signed numbers are used.
•
Temperature is the most familiar application. If the temperature is reported as -10oC, you
know that the temperature is 10oC below OoC.
•
Sea level is another application. For example, the Badwater Basin in Death Valley has a
reported elevation of -282 feet (or -86 metres). That means Badwater Basin is 282 feet
(86 metres) below sea level. By contrast, Regina has a reported elevation of 1894 feet (or
577 metres). That means that Regina is 1894 feet (577 metres) above sea level.
•
Businesses report their financial health using signed numbers. If a company reports a
gain of $300 000, it would be recorded as +$300 000. If a company reports losses of
$200 000, it is recorded as -$200 000.
•
Financial statements from banks or credit institutions use signed numbers. If you deposit
$50, it is recorded as +$50. If you withdraw $40, it is recorded as -$40.
•
In golf, scores are often stated in terms of par. Suppose two golfers, Bill and Annette, are
playing the same par 4 hole. If Bill completes the hole in 6 shots, his score on that hole is
+2 because he made two shots above par. If Annette completes the hole in 3 shots, her
score on the hole is -1 because she made one shot below par.
When dealing with application, there are some key words that help identify the sign of a number.
Negative
below
loss
withdrawal
decrease
past
below
Positive
above
gain
deposit
increase
future
above
Example 1:
A football team gained 7 yards on the first down, lost 3 yards on the second down, and lost 6
yards on the third down. What was the overall change in position after the third down?
Answer:
(+ 7 ) + (− 3) + (− 6)
= (+ 4 ) + (− 6 )
= −2
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The position is -2 yards. This means that over the three
downs they lost 2 yards.
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Example 2:
The lowest elevation in Africa is Lake Assal at -512 feet. The highest elevation is Mount
Kilimanjaro at 19 340 feet. What is the difference in elevation between these two locations?
Answer:
19340 − (− 512 )
= 19340 + (+ 512 )
= 19852
They differ in elevation by 19 852 feet.
Example 3:
Submarine A is at a depth of -62 metres relative to sea level. Submarine B is three times deeper
than submarine A. What is the depth of submarine B?
Answer:
3 × (− 62 ) = −186
Submarine B is at a depth of -186 metres.
Example 4:
Kelly has been paying off her car loan. Last year she made monthly payments totaling $4944.
How much were her monthly payments? Express your answer as a signed number.
Answer:
(− 4944) ÷ (+ 12) = −412
The monthly payments can be expressed as -$412.
Example 5:
The high temperature values over one week in January are found in the chart below.
Day of the Week
Temperature (oC)
S
-5
M
-1
T
+3
W
+2
T
-6
F
-3
S
-4
Determine the average high temperature over that one week period.
Answer:
(− 5) + (− 1) + (+ 3) + (+ 2) + (− 6) + (− 3) + (− 4)
7
− 14
=
7
The average high temperature over the week was -2oC.
= −2
Questions:
Answer the following questions and express your answers as signed numbers.
1. On the game show Jeopardy, contestants are penalized the value of the question when it is
answered incorrectly. Suppose a contestant has a score of $800 and proceeds to answer a
$1000 question incorrectly. What will the contestant’s new score be?
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2. On Saturday, the temperature in Halifax was 3oC and the temperature in Sydney was -5oC.
What was the difference in temperature between these two locations?
3. Ajay lost an average of 1.2 kilograms per week for 8 weeks. What was his net change in
weight?
4. On the first nine holes, Mike Wier received 1 eagle (i.e. two strokes below par), 3 birdies (i.e.
one stroke below par), 4 pars, and 1 bogie (i.e. one stroke above par). How far is Mike over
or under par after the first nine holes?
5. During the first quarter of 2009, a company reported a net income of -$30 million. If this
continued, what would the company’s net income be after 4 quarters?
6. Leo initially has $300 in his account. He then deposits a cheque worth $220. Over the next
few days he writes a cheque for $260, and makes two ATM withdrawals, each of $140.
What is the balance in Leo’s account?
7. The temperature of a particular object was initially -6oC. The object’s temperature increased
by 2oC every minute. What was the temperature of the object after five minutes?
8. A scuba diver is initially at a depth of 7 m below the surface. He then dives down 4 m more.
Find the diver’s present depth.
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9. The temperature in the morning is -7oC. By noon, the temperature has risen by 3oC.
Between noon and the evening, the temperature drops 6oC. What is the temperature in the
evening?
10. A company’s net income for each quarter of a fiscal year is reported below.
Quarter
1
2
3
4
Income
(in millions)
$1.2
-$0.9
-$1.7
$0.2
(a) What is the company’s net income for the year?
(b) What is the company’s average quarterly income?
11. Jorell has been supplied with the high temperature reading for his town on January 10 for the
last six years.
Year
2003
2004
2005
2006
2007
2008
Temperature
-10oC
-6oC
1oC
-4oC
3oC
-8oC
He figures that if he finds the average temperature over the last six years, he can use this
value to predict the temperature on January 10, 2009. Find the average temperature.
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Reflect Upon Your Learning
Fill out this questionnaire after you have completed pages 33 to 57. Select your response to each
statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
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I understand all of the concepts covered in the section,
Applications - Fractions.
I do not need any further assistance from the instructor on the
material covered in the section, Applications - Fractions.
I understand all of the concepts covered in the section,
Applications - Decimals.
I do not need any further assistance from the instructor on the
material covered in the section, Applications - Decimals.
I understand all of the concepts covered in the section,
Applications - Percents.
I do not need any further assistance from the instructor on the
material covered in the section, Applications - Percents.
I understand all of the concepts covered in the section,
Applications - Ratio and Proportions.
I do not need any further assistance from the instructor on the
material covered in the section, Applications - Ratio and
Proportions.
I understand all of the concepts covered in the section,
Applications - Signed Numbers.
I do not need any further assistance from the instructor on the
material covered in the section, Applications - Signed Numbers.
58
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
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Putting It Together, Part I
In the previous sections, you have looked at a variety of application questions but they have
always been grouped together based on a particular mathematic concept (fractions, decimals,
percents, ratio, proportion, and signed numbers). In this activity and those that follow, you will
be expected to solve these same types of questions but you will have to figure out which
particular mathematical concept will be used to answer the question. For the first activity sheet,
hints have been provided in brackets so that you can go back in this resource and look up
previously completed questions.
Questions:
Calculators can only be used with questions involving percentages or proportions.
1. Of 72 people who applied to a particular college program, the college accepted 55. What
percentage of applicants did the college accept into this program? (Percents) ()
2. Determine the perimeter of this rectangle. (Decimals)
45.2 cm
23.4 cm
3. On the game show Jeopardy, contestants are penalized the value of the question when it is
answered incorrectly. Suppose a contestant has a score of $600 and proceeds to answer a
$1000 question incorrectly. What will the contestant’s new score be? (Signed Numbers)
4. A motor brush is 1
7
3
inches long. How long is it after
inch wears away? (Fractions)
8
16
5. The surface temperature on Mars varies greatly between night and day, and from the poles to
the equator. At the equator, the temperature can reach at high of 27oC at midday. At the
poles, the temperature can reach a low of -128oC at night. Determine the difference between
the highest and lowest temperature found on Mars. (Signed Numbers)
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6. A car depreciates (drops in value) as soon as it leaves the dealer’s lot. In the first year it
typically depreciates by 20%. If the car initially cost $18 000, how much is it worth after one
year? (Percents) ()
7. Barbara earns $9.52 per hour. How much will she make before deductions if she works 23
hours? (Decimals)
8. The scale on a particular map indicates that 1 centimeter corresponds to 30 kilometres. How
far apart are two towns if they are 16.3 centimetres apart on the map? (Ratio and Proportion)
()
3
cup of vinegar. If you wanted to
4
triple the recipe, how much vinegar would you need? (Fractions)
9. A recipe for a homemade cleaning solution requires
10. Kelly has been paying off a loan. Last year she made monthly payments totaling $1560.
How much were her monthly payments? Express your answer as a signed number. (Signed
Numbers)
11. How much commission does a salesman receive on sales totaling $6200 at a 9% rate of
commission? (Percents) ()
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12. A 180 pound individual burns 10.9 calories per minute while playing tennis. If this
individual plays for 90 minutes, how many calories will he/she burn? (Decimals)
13. A fuel pump delivers 35 ml of fuel in 420 strokes. How many strokes are needed to pump
100 ml of fuel? (Ratio and Proportion) ()
14. Partition walls separate the rooms of a house. If the partition wall is separating the garage
1
from main living space of the house, it must be clad with
inch drywall on the one side and
2
5
inch drywall on the other side (i.e. garage side). If the studs used to construct the partition
8
1
walls are 3 inches thick, what is the total thickness of the partition wall when it clad on
2
both sides with drywall? (Fractions)
15. If 100 grams of ice cream contains 14 grams of fat, how many grams of fat are found in a
350 gram serving of the same ice cream? (Ratio and Proportion) ()
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16. A contractor estimates that it will cost him $127 000 to build a house for a client. If he needs
to make 8% profit on the build, what should he charge the client on the build? (Percents) ()
17. Four new snow tires cost $509.96. What was the price per tire? (Decimals)
18. The space shuttle completes one orbit every 1
1
hours. How many orbits does it complete in
2
24 hours? (Fractions)
19. Kate has been supplied with the high temperature reading for her small town on January 15
for the last five years.
Year
2004
2005
2006
2007
2008
Temperature
-6oC
1oC
-4oC
2oC
-8oC
She figures that if she finds the average temperature over the last five years, she can use this
value to predict the temperature on January 15, 2009. Find the average temperature. (Signed
Numbers)
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Test Results
11
10
9
8
Number of Students
20. A college instructor was looking at her students’ test
results.
(a) How many students wrote the test?
(b) What percentage of the class received a mark of
90% or greater?
(c) What percentage of the class received a mark
between 70% and 79%?
(d) What percentage of the class received a failing
grade (less than 60%)?
(e) What percentage of the class received a mark of
80% or greater?
(Percents) ()
7
6
5
4
3
2
1
0
40-49
50-59
60-69
70-79
80-89
90-99
Mark on Test (%)
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Putting It Together, Part II
Questions:
Calculators can only be used with questions involving percentages or proportions.
1. James buys a T-shirt that costs $15.80 (taxes already included). If he pays with a $20 bill,
how much change will he receive?
2. One third of a pizza is left over from last night’s party. You want half of the remaining
pizza. How much of the original pizza does your piece represent?
3. A basketball player made 37 out of 46 free throws. What percentage is this? ()
4. The community college accepts 3 out of every 7 applicants for their carpentry program. ()
(a) In 2006 the school received 371 applicants for carpentry. How many did they accept?
(b) In 2007 the school accepted 147 applicants for carpentry. How many applicants applied?
5. The temperature of a particular object was initially -10oC. The object’s temperature
increased by 2oC every minute. What was the temperature of the object after three minutes?
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6. Figure out the length of side A.
13
16
7
8
"
5
8
"
3
1
8
"
"
Side A
7. A phone company charges $0.07 cents per minutes for long distance calls to the United
States. If you talk to a friend in the US for 46 minutes, how much will you have to pay for
the call?
8. A student answered 90% of the questions on a test correctly. ()
(a) If the test was comprised of 60 questions, how many questions were answered correctly?
(b) If the test was comprised of 60 questions, how many questions were answered
incorrectly?
9. It takes 38 litres of maple sap to produce 2 litres of maple syrup. ()
(a) How much sap is required to produce 11 litres of syrup?
(b) How much syrup is produced using 142.5 litres of sap?
10. The water level in Fletchers Lake changed significantly over four months. In May it went up
by 15 cm. In June, it went up by 5 cm. In July it went down by 7 cm. In August it went
down 16 cm. How much had the water level changed by in that 4 month period? Express
your answer as a signed number.
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11. The inside diameter of a particular piece of copper tubing is
3
inch. The outside diameter is
8
9
inch. What is the wall thickness of the tubing?
16
12. A copper wire that is 750 feet long has a resistance of 1.89 ohms. What is the resistance in a
wire of the same type measuring 2000 feet in length? ()
13. Determine the unknown length, x, in the
figure.
2.4 cm
x
2.4 cm
12.53 cm
14. In a town of 6500 people, approximately 1200 are under the age of 25 years. ()
(a) What percentage of the town is under 25 years of age?
(b) What percentage of the town is 25 years of age or older?
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15. For each hour that an oil burner runs, it uses
for 2
3
gallon of fuel. If the burner is only running
4
2
hours, then how much fuel is used?
3
16. Kimi initially has $200 in her account. She then deposits a cheque worth $120. Over the
next few days she writes a cheque for $60, and makes an ATM withdrawal of $340. What is
the balance in Kimi’s account? Express your answer as a signed number.
17. Determine the length of the shaft?
5
1
4
″
9
13
16
″
18. A phone company is charging $0.13 per minute for overseas long distance calls. If your bill
for an overseas long distance call was $4.68, how many minutes was the call?
19. The yearly tuition at one college is $2700. The following year the tuition is supposed to
increase by 4%. What will the new tuition be? ()
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20. Roof pitch is a way to describe the steepness of a roof. If a roof has an 8:12 pitch, it means
that the roof rises 8 feet over a horizontal run of 12 feet. If a roof had an 8:12 pitch, what
would be the height for a horizontal run of 21 feet? ()
3
inches are cut from the
8
corners of a rectangular sheet of metal measuring 12 inches by 10
inches. After the four squares are removed, the sheet metal can be
folded to create a box that is open on the top. What are the
dimensions of the box?
21. Four squares having side lengths of 2
22. Kiana was making $11.40 per hour and then given a raise such that her new hourly wage is
$13.15. How much is the raise?
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Putting It Together, Part III
Questions:
Calculators can only be used with questions involving percentages or proportions.
1. If three sheets of
3
inch plywood are glued together, how thick will the new sheet be?
8
2. A metal casting weighted 35 kg out of the mold. It weighed 28 kg after finishing. What
percentage of the weight was lost in finishing? ()
3. Jun had a car loan which he paid off by making 36 equal monthly payments. If his monthly
payments were $204.36, how much did he ultimately pay back to the lending institution?
4. During the first quarter of 2008, a company reported a net income of -$1.5 million. If this
continued, what would the company’s net income be after 4 quarters?
5. A salesperson is paid a $75 commission for selling $800 worth of goods. What is the
commission on $1360 of sales? ()
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6. What is the difference in thickness between a
5
7
inch steel plate and a
inch steel plate?
16
8
7. Nickel silver is 55% copper (Cu), 15% nickel (Ni), and the rest is zinc (Zn). ()
(a) What percentage of nickel silver is Zn?
(b) If you had 60 kg of nickel silver, find the individual weights of the Cu, Ni, and Zn used in
the alloy.
8. A scuba diver is initially at a depth of 5 m below the surface. He then dives down 3 m more.
Find the diver’s present depth. Express your answer as a signed number.
9. A metal component for an engine must have 0.05 cm ground off. If the part initially
measured 3.71 cm, what will be the measurement after the metal is ground off.
10. If the diameter of a hole is 2
7
inch, what is the radius of the hole?
8
11. On average, an adult’s heart beats 8 times in 6 seconds. How many times should an adult’s
heart beat in 2 minutes? ()
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12. Jorell has $2800 in PanCanada stock. ()
(a) If the stock drops in value by 6%, how much is his stock worth?
(b) If the stock increases in value by 6%, how much is his stock worth?
13. A company’s net income for each quarter of a fiscal year is reported below.
Quarter
1
2
3
4
Income
(in millions)
$11
-$5
-$2
$8
(a) What is the company’s net income for the year?
(b) What is the company’s average quarterly income?
15
15
inch by 1
inch has to have a hole
16
16
1
drilled in the center. If there must be
of an inch of metal between the hole
4
and the side of metal, what is the diameter of the hole?
14. A small piece of metal measuring 1
15. A paramedic notes that the temperature of a patient is 101.7oF. After fifteen minutes, the
temperature drops to 99.5oF. How much did the temperature drop?
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16. A machinist can produce 6 parts in 40 minutes. If she can keep up this pace, how many parts
can she complete in 6 hours? ()
17. Tanya’s tuition was $2400. A loan for
3
of the tuition was obtained. How much was the
5
loan?
18. Pat DeCoste is a professional golfer from Shubenacadie, Nova Scotia. On the first nine
holes, Pat obtained 1 eagle (i.e. two strokes below par), 2 birdies (i.e. one stroke below par),
5 pars, and 1 double bogie (i.e. two strokes above par). How far is Pat over or under par after
the first nine holes? Express your answer as a signed number.
19. Determine the area of this rectangle.
7.3 m
5.2 m
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20. The circle graph shows the different expenses
associated with a business trip.
Assuming that the individual spent $1200 in total for
the business trip, answer each of the following
questions. ()
(a) How much money was spent on lodging?
(b) How much money was spent on transportation?
(c) How much money was spent on food and
entertainment?
Business Trip Expenses
Other
10%
Entertainment
15%
Lodging
30%
Food
20%
Transportation
25%
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Putting It Together, Part IV
Questions:
Calculators can only be used with questions involving percentages or proportions.
1. Determine the perimeter of this figure.
5.5 m
5.1 m
1.8 m
4.4 m
2. A 1.9 kg bag of grass seed covers 350 square feet of ground. How many kilograms of grass
seed are needed to cover 4340 square feet of ground? ()
3. The temperature at noon is 3oC. By supper time, the temperature has dropped by 4oC.
Between supper time and midnight, the temperature drops 5oC. What is the temperature at
midnight?
4. Janice earned $72 for working 7.5 hours. How much is her hourly wage?
5. A recipe calls for
1
1
cup of sugar. How much sugar should be used if only of the recipe is
2
3
being made?
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6. A football team gained 8 yards on the first down, lost 2 yards on the second down, and lost 7
yards on the third down. What was the overall change in position after the third down?
Express your answer as a signed number.
7. The property taxes for a particular city are described as $1.25 tax for every $100 of house
value. ()
(a) If the house’s value is $228 000, how much would the property taxes be?
(b) If you paid $2562.50 in property tax, what is the value of the house?
8. Determine the missing measure.
1
11
4
2
7
16
″
″
4
5
8
″
9. Tanya has to buy 27 feet of rope. If the rope costs $0.67 per foot, how much will it cost
before taxes?
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10. The human body contains 208 bones. Of those, 56 bones are found in the hands and feet.
What percentage of the bones is found in the hands and feet? ()
11. On 14 January 2009, 1 Canadian dollar was worth 0.6130 European euros. ()
(a) On this particular date, how much was $850 Canadian worth in euros?
(b) On this particular date, how much was 700 euros worth in Canadian dollars?
5
inch plywood. If he stacks 10 sheets of plywood on top of each other,
8
how high is the stack?
12. Brian is stacking
13. Out of 250 circuit boards, 12 were identified as defective. What percentage of circuit boards
was defective? ()
14. The specifications for a drainage pipe are that the pipe must drop 1 inch for every 4 feet of
pipe. How many inches should the pipe drop if it is 34 feet long? ()
15. A grocery store sells their own brand of pop. During a sale you can purchase 5 bottles (2
litres) of pop for $4.00. Normally an individual bottle of pop costs $1.19. How much will
you save per bottle if you purchase 5 bottles at the sale price?
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16. A video gaming store tracked the sales of video games to different age groups over a
weekend. The data was plotted on a bar graph. ()
Number of Individuals
Weekend Video Sales
11
10
9
8
7
6
5
4
3
2
1
0
5 to 9
10 to 14
15 to 19
20 to 24
25 to 29
30 to 34
35 to 39
40 to 44
45 and
over
Ages
(a) How many individuals purchased video games?
(b) What percentage of video games was sold to individuals who are less than 10 years of
age?
(c) What percentage of video games was sold to individuals between the ages of 20 and 29
years of age?
(d) What percentage of video games was sold to individuals who are 30 years of age or
older?
17. Kendrick was making $12.75 per hour and then given a raise such that his hourly wage is
$14.20. How much is the raise?
18. Determine the perimeter of this figure.
3
5
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8
1
2
″
″
2
1
4
″
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19. The following table illustrates how a particular stock fluctuated over a five day period.
Day
Monday
Tuesday
Wednesday
Thursday
Friday
Change in Stock
Price in Cents
-52
+33
-18
-23
+15
(a) How much did the stock change in price over the five day period?
(b) What was the average daily change in the price of the stock over that period of time?
20. The bill before taxes for a meal is $48.50. ()
(a) What is the cost of the meal after taxes (13%)?
(b) How much would you leave as a tip (15% of the complete bill)?
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Is It Reasonable?
In the last few sections you have been completing a variety of word problems using a variety of
mathematical concepts. When you complete a problem, do you take the time to consider
whether your answer is reasonable given the problem or situation? Checking the reasonableness
of an answer is an important skill in mathematics.
Example 1:
Maureen is organizing a field trip for five classes of grade 5 and 6 students. Buses will need to
be rented to transport the students, teachers and chaperones. They will be taking 129 students, 5
teachers and 10 chaperones. If a bus can carry a maximum of 45 individuals, how many buses
are required? Maureen completes the calculations.
129 + 5 + 10 144
=
= 3.2 (round 3.2 down to 3)
45
45
Maureen concludes that 3 buses are needed. Is this answer reasonable? Why or why not?
Answer:
Although we would typically round any number below 3.5 down to 3, it doesn’t make sense
in the context of this question. Since each bus can only carry 45 students, we would end up
having to leave a few individuals behind. In this case we must round 3.2 up to 4. Maureen
should have stated that 4 buses are needed.
Example 2:
1
5
inch in thickness is being glued to sheet of plywood that is
16
8
inch in thickness. How thick is the new sheet? Two individuals, Ken and Paul, attempted to
answer the question separately.
A laminate surface measuring
Ken’s Answer:
9
5 1 5 × 2 1 10 1
− =
− = − =
8 16 8 × 2 16 16 16 16
Ken concludes that the new sheet will be
9
inch in thickness.
16
Paul’s Answer:
6÷6 1
5 1
6
+
=
=
=
8 16 24 24 ÷ 6 4
Paul concludes that the new sheet will be
1
inch in thickness.
4
Is either of the answers reasonable? Why or why not?
Answer:
Neither of the answers is reasonable. When you glue the two pieces of material together, the
thickness should increase because the thicknesses are added together (not subtracted).
Although Ken probably understands how to add and subtract fractions, he didn’t realize that
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the fractions should be added in this case. Paul, by contrast, knew that the thicknesses
needed to be added but he didn’t know how to do this with fractions. He ended up with a
1
thickness of
inch, which is far less than just the thickness of the plywood. This should
4
have told him that he made a mistake.
Example 3:
A triangle has side lengths measuring 2.15 m, 4.9 m, and 5.06 m. Three students were asked to
determine the perimeter of the triangle. Their answers are supplied below. Which is a
reasonable answer and why?
Jeff’s Answer: 12.14
Dave’s Answer: 5.2675
Shirley’s Answer: 25.6036
Answer:
To determine the perimeter of a triangle, we must add the three side lengths. If we round the
numbers to the nearest unit, we get 2 + 5 + 5 = 12 . Jeff’s answer is the only reasonable
answer.
Example 4:
Jun and his girlfriend, Nasrin, are dining out. The bill for the meal comes to $41.78. They
decide to leave approximately a 15% tip. Jun thinks that they should leave $8.50. Nasrin thinks
that they should leave $6.25. Which answer is more reasonable? Why is this so?
Answer:
We will round $41.78 down to $40. Ten percent of $40 is $4. Twenty percent of $40 is $8.
That means that 15% of $40 will be halfway between $4 and $8. That gives us $6. We can
conclude that Nasrin’s tip of $6.25 is more reasonable.
Based on examples 3 and 4, we can see that having strong estimation skills is very useful in
judging the reasonableness of an answer.
Example 5:
The ratio of female students in the college to male students in the college is 6 to 5. If there are
810 male students in the college, how many females are attending? Hassan works out that there
are 675 females attending. Is his answer reasonable? Why or why not?
Answer:
Hassan’s answer is unreasonable. The ratio states that there are 6 female students for every 5
male students. This means the college has more female students than male students yet
Hassan concluded that for the 810 male students there were 675 female students (the exact
opposite).
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Questions:
It should be noted that the most important aspect of your responses to these questions is your
ability to explain in full sentences why an answer is reasonable or unreasonable. Do not
complete the actual calculations to determine the reasonable answer.
1. A rectangle measures 2.1 cm by 6.3 cm. Three students were asked to determine the area of
the rectangle. Their answers are supplied below. Which is a reasonable answer and why?
Hinto’s Answer: 132.3
Kimi’s Answer: 16.8
Taylor’s Answer: 13.23
2. Three students are asked to figure out 52% of $38. Their answers are supplied below.
Which is a reasonable answer and why?
Jake’s Answer: $19.76
Nita’s Answer: $57.76
Pam’s Answer: $11.76
3. A carton of eggs contains a dozen eggs. Angela is hosting a large family reunion and has
agreed to supply quiche for the meal. She needs to purchase 74 eggs for the recipe. How
many cartons should she purchase? Three students completed the same calculations (shown
below) but supplied different answers.
74
= 6.17
12
Which is a reasonable answer and why?
Mary’s Answer: 7
Jim’s Answer: 6.17
Deangelo’s Answer: 6
4. Kirk is running a piece of lumber measuring
3
inch in thickness through a planer. The
4
1
inch. How thick is the lumber after it has passed through the planer?
16
Three students have attempted to answer the question. Which is a reasonable answer and
why?
7
11
1
Bart’s Answer:
Lisa’s Answer:
Homer’s Answer:
8
16
6
planer will remove
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5. If asked to evaluate 3.01 × (1.99 + 2.13) − 0.975 , which one of these answers appears
reasonable and why?
Answer A: 7.1449
Answer B: 9.46645
Answer C: 11.4262
6. The temperature in the morning is -3oC. It rises 5 degrees by noon and then drops by 7
degrees by supper time. What is the temperature at supper time? Which one of these
answers appears reasonable and why?
Answer A: 9oC
Answer B: -5oC
Answer C: -15oC
7. The price of a vehicle depreciates by about 20% one year after it is sold. If the car was
initially worth $22 000, how much is it worth in one years time? Four students have
attempted to answer the question.
Jake’s Answer: $20 000
Shelly’s Answer: $4400
Bashir’s Answer: $26 400
Tylena’s Answer: $17 600
(a) Which is a reasonable answer and why?
(b) Explain why the other three answers are unreasonable.
8. The ratio of the sales of red liquorish to black liquorish is 9 to 2. If the company sold 6300
units of red liquorish, how many units of black liquorish were sold? Three students have
attempted to answer the question.
Monica’s Answer: 1400
Julie’s Answer: 28 350
Montez’s Answer: 5800
(a) Which is a reasonable answer and why?
(b) Explain why the other two answers are unreasonable.
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Reflect Upon Your Learning
Fill out this questionnaire after you have completed pages 59 to 82. Select your response to each
statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
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I understand all of the concepts covered in the sections, Putting
It Together.
I do not need any further assistance from the instructor on the
material covered in the sections, Putting It Together.
I can recognize which operation or operations (addition,
subtraction, multiplication, or division) should be used to solve
the application problem.
I remember how to use the necessary mathematical concept
(fractions, decimals, percents, signed numbers, ratio and
proportion) to complete the application problem.
I do not need any more practice questions.
I can judge the reasonableness of an answer.
83
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
1
2
2
3
3
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Real World Math - Careers and Math
Over the last few weeks you have been working on a variety of real world application problems
dealing with fractions, decimals, percents, signed numbers, ratio and proportion. There are many
more applications that relate to these and other mathematical concepts. You will encounter
many of these as you complete this course.
In this section of the unit, we want you to consider how mathematics affects your daily life and
your plans for a future career.
Part I: Math in the Real World and in Daily Life
Read and view the following web pages and videos.
1. The Universal Language (webpage)
• Google Search: Interactives Math in Daily Life Universal Language
• http://www.learner.org/interactives/dailymath/language.html
2. Real Life Math (video)
• YouTube Search: Real Life Math
• http://www.youtube.com/watch?v=HtqlIVN9bh8
3. Numb3rs Scene: Everything is numbers, Math is Everywhere (video)
• YouTube Search: Everything is Numbers
• http://www.youtube.com/watch?v=vFRTgr7MfWw
4. Math is What You Do! (video: rap)
• YouTube Search: Math Is What You Do
• http://www.youtube.com/watch?v=VraI21bgVtc
Question:
In two or three paragraphs, list at least three daily activities where three distinctly different
mathematical concepts are used. Also explain how these concepts are used to deal with these
daily activities. (Answer in the space provided on this page and the next page.)
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Part II: Careers and Math
Consider reading the following web pages and pdf documents when answering the questions
associated with this section of the activity.
1. More Career Connections (webpage)
• Google Search: PBS Teachers Mathline More Career Connections
• http://www.pbs.org/teachers/mathline/career/morecareer.shtm
2. Math on the Job - How You Use Math at Work (webpage)
• Google Search: Vocational Information Center Math on the Job
• http://www.khake.com/page56.html
3. Math is Everywhere (webpage)
• Google Search: “You Can Do Maths” “Maths is Everywhere”
• http://www.youcandomaths.com.au/maths-careers.php
4. Math in the Workplace: Overview (webpage)
• Google Search: Micron Math in the Workplace
• http://www.micron.com/k12/math/
• Investigate each of the following.
- Numbers and Operations
- Measurements
- Algebra
- Geometry
- Data Analysis and Probability
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5. Examining How Mathematics is Used in the Workplace (webpage)
• Google Search: “MAA Online” “Examining How Mathematics is Used in the
Workplace”
• http://www.maa.org/t_and_l/sampler/rs_6.html
6. Core Subjects and Your Career (pdf document)
• Google Search: “Bureau of Labor Statistics” “Core Subjects and Your Career”
• http://stats.bls.gov/opub/ooq/1999/summer/art03.pdf
7. Jobs That Use Geometry (webpage)
• Google Search: “Ask Dr. Math” “Jobs That Use Geometry”
• http://mathforum.org/library/drmath/view/54697.html
Questions:
1. List three jobs or careers that you are interested in pursuing.
2. Explain briefly why you choose each of these potential careers.
3. Choose one of these potential careers and investigate the math that you would need in that
career. The web pages and pdf documents listed above should assist with this investigation.
You will also be expected to conduct an interview with an individual employed in this career,
or an instructor who prepares students for such a career, and integrate this information into a
final report. You should include some sample questions that illustrate the type of math
problems one might expect to encounter. The report should be between 1 to 2 pages in
length.
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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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Answers:
Order of Operations (pages 1 to 2)
1. (a) 5
(g) 9
(m) 13
(b) 43
(h) 20
(n) 36
(c) 82
(i) 5
(o) 4
(d) 6
(j) 4
(p) 6
(e) 81
(k) 34
(q) 3
(f) 16
(l) 37
(r) 11
Adding Fractions (pages 3 to 6)
10
11
4
(e) 1
7
1. (a)
13
20
13
(e)
24
2. (a)
3. (a) 9
13
20
(b)
3
5
3
(f) 1
10
23
30
5
(f) 1
18
(b)
(b) 11
5
6
1
(g) 1
3
(c)
23
24
5
12
3
(g)
4
(c) 1
3
4
(c) 3
2
3
1
(h) 1
2
(d)
11
12
1
(h) 1
3
(d)
(d) 8
1
14
5
24
(e) 7
(f) 6
1
2
Subtracting Fractions (pages 7 to 9)
1. (a)
7
10
(b)
2
3
(c)
3
4
(d)
3
8
2. (a)
1
12
(b)
3
10
(c)
7
12
(d)
11
24
3. (a) 4
7
15
(b) 5
7
12
(c) 6
13
18
(d) 1
13
21
(e)
1
3
(f)
3
10
(e) 5
13
15
(f) 3
5
8
(e) 6
2
3
(f) 6
3
4
Multiplying and Dividing Fractions (pages 10 to 11)
4
45
13
(g) 1
15
1. (a)
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42
3
(h) 4
8
(b)
(c)
3
14
(d)
10
21
(i) 3
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5
8
13
(g)
18
2. (a)
21
50
1
(h) 1
3
2
5
7
(i)
12
(b)
(c)
(d) 1
1
5
(e) 2
2
3
(f)
1
14
Decimals (pages 12 to 17)
6
10
1
4
7
(c) 200 + 60 + 5 + +
+
10 100 1000
9
3
+
10 100
5
6
2
(d) 40 +
+
+
100 1000 10000
1. (a) 2000 + 400 + 80 + 2 +
2. (a)
(b)
(c)
(d)
(e)
(f)
(b) 7 +
fourteen and nine tenths
three thousand two and fifteen hundredths
four hundred fifty-nine and seven hundred thirty-six thousandths
four hundred eighty and seven hundredths
sixty-seven and twenty-five thousandths
twenty-three and five hundred seventy-eight ten thousandths.
3. (a) 72.78
(e) 16.0715
(b) 148.659
(f) 9.4184
(c) 41.61
(g) 50.71
(d) 184.033
(h) 4.0781
4. (a) 32.25
(e) 2.9675
(b) 23.651
(f) 2.47
(c) 287.18
(g) 5.684
(d) 463.581
(h) 1.588
5. (a) 17.92
(b) 16.192
(c) 211.815
(d) 2.81313
6. (a) 3.4
(b) 5.4
(c) 0.62
(d) 2.51
(c) 0.37
(g) 0.034
(k) 2.1
(d) 0.04
(h) 0.008
(l) 0.036
Percents (pages 18 to 21)
1. (a) 0.38
(e) 0.06
(i) 1.05
9
20
3
(e)
50
(b) 0.21
(f) 0.245
(j) 0.142
7
10
21
(f)
50
7
25
18
(g)
25
4
5
6
(h)
5
2. (a)
(b)
(c)
(d)
3. (a) 92%
(e) 70%
(i) 151%
(b) 47%
(f) 0.7%
(j) 180%
(c) 32%
(g) 4.2%
(k) 106%
(d) 7%
(h) 20.6%
(l) 103.4%
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4. (a) 40%
(e) 37.5%
(b) 56%
(f) 93.75%
(c) 85%
(g) 104%
5. (a) 2198
(d) 11.1
(b) 377
(e) 1.96
(c) 144
(f) 64.41
6. (a)
(d)
(g)
(j)
(b)
(e)
(h)
(k)
(c)
(f)
(i)
(l)
43
28
1400
120
580
21
240
4200
(d) 62.5%
(h) 112.5%
12
44
500
360
Ratios and Proportions (pages 22 to 24)
3
4
4
(g)
3
670
(m)
947
9
5
3
(h)
5
11
(n)
520
3
5
8
(i)
5
31
(o)
20
9
7
51
(j)
37
1
(p)
7
13
16
46
(k)
65
9
(q)
5
8
9
612
(l)
253
5
(r)
7
1. (a)
(b)
(c)
(d)
(e)
(f)
2. (a) 14
(b) 15
(c) 25
(d) 21
(e) 1
(f) 6
(d)
(j)
(p)
(v)
(e) -6
(k) -11
(q) 6
(w) 3
(f)
(l)
(r)
(x)
Signed Numbers (pages 25 to 28)
1. (a) -5
(g) -13
(m) 6
(s) 42
(y) -4
(b)
(h)
(n)
(t)
(z)
2. (a) -8
(g) 3
(m) 0.3
(b) -6
(h) -26
(n) 2.4
(c) -63
(i) -18
(o) -9.4
(d) -16
(j) -6.2
(p) -5/18
(e) 25
(k) -4.6
(q) -5/6
(f) 70
(l) -5
(r) 6/7
3. (a) 36
(b) -27
(c) -125
(d) -64
(e) -1
(f) -64
4. (a) -1
(b) -28
(c) 59
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-18
29
-32
-8
(c)
(i)
(o)
(u)
11
7
-16
-56
90
4
-21
-3
54
7
-8
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All Together Now, Part I (page 29)
(a) 18
(b) 35.34
(d) 6
(e) 48
(g) 38.88
(h) -8
17
(k) 9
21
(j) 30
(m) 4
17
28
(c) 20
16
7
(f)
or 1
9
9
(i) 3.9
(l) 88.32
(n) -20
(o) 1
All Together Now, Part II (pages 30)
(a) 14
17
(d) 5
18
(b) 15.75
(c) 25
(e) 25
(f) -32
(g) -2
(h) 21
(i) 3
(j) 75.65
(k) 4
(m) 4.6
(n) 2
13
20
(l) -11
1
16
(o) 28.263
All Together Now, Part III (pages 31)
1
25
or 3
8
8
(d) 9
(g) 15
(a)
7
24
2.7
21.658
5
6
24
69.425
(b) -4
(c) 5
(e) 7
(h) 66
(f)
(i)
(j) 1
(k) 22.94
(l)
(m) 4
(n) 27
(o)
Applications - Fractions (pages 35 to 38)
3
16
1. 2
5
8
2. 1
5. 9
3
8
6. 12
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7. 1
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4. 1
7
16
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9. 4
13.
1
6
3
8
10. 16
14. 5
1
2
11.
3
4
12. 15
5
8
Applications - Decimals (pages 39 to 42)
1. $32.50
2. $5.36
3. 350 shares
4. 2.76 m
5. 0.525 mg
6. $12.63
7. $20.72
8. 0.014 cm
9. 1432.4 km
10. 27.3 ml
11. 2.01 cm
12. 134 g
Applications - Percents (pages 43 to 47)
1. 75%
2. $3.90
3. 17.5%
4. $25 080
5. (a) 5%
(b) 95%
6. (a) 55 individuals
(b) 71 individuals
(c) 256 individuals
7. (a) 30 employees
(b) approximately 23.3%
(c) approximately 36.7%
(d) approximately 66.7% (e) 43.75%
8. $156
9. (a) $56.49
(b) $50.84
10. 37.49 cm (rounded off)
11. (a) $1520
(b) $1680
12. 924 board feet
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Applications - Ratio and Proportion (pages 48 to 53)
1. (a)
5
or 5:3
3
(b)
5
or 5:12
12
(c)
1
or 1:4
4
2. (a)
161
or 161:240
240
(b)
79
or 79:161
161
(c)
79
or 79:240
240
3.
7
or 7:25
25
4. (a) 900 ml
(b) 2000 ml
5. (a) 12 completed passes
(b) 40 attempted passes
6. 14 litres of paint
7. 72 defective circuit boards
8. (a) 58.2 grams
(b) 121 pounds
9. 12.5 minutes
10. 148.5 kilograms
11. $275 000
Applications - Signed Numbers (pages 54 to 57)
1. -$200
2. 8oC
3. -9.6 kg
4. -4
5. -$120 million
6. -$20
7. 4oC
8. -11 m
9. -10oC
10. (a) -$1.2 million
(b) -$0.3 million
11. -4oC
Putting It Together, Part 1 (pages 59 to 63)
1. 76%
4. 1
11
inches
16
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2. 137.2 cm
3. -$400
5. 155oC
6. $14 400
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1
cups
4
7. $218.96
8. 489 km
9. 2
10. -$130
11. $558
12. 981 calories
13. 1200 strokes
14. 4
16. $137 160
17. $127.49
18. 16 orbits
(b) 12.5%
(e) 37.5%
(c) 31.25%
5
inches
8
15. 49 grams
19. -3oC
20 (a) 32
(d) 12.5%
Putting It Together, Part II (pages 64 to 68)
1
of a pizza
6
3. 80%
1. $4.20
2.
4. (a) 159 applicants
(b) 343 applicants
5. -4oC
6. 1
8. (a) 54 questions
(b) 6 questions
9. (a) 209 litres
(b) 7.5 litres
10. -3 cm
11.
7
inches
16
7. $3.22
3
inch
32
12. 5.04 ohms
13. 7.73 cm
14. (a) 18%
(b) 82%
15. 2 gallons
16. -$80
17. 15
18. 36 minutes
19. $2808
20. 14 feet
1
inches
16
3
1
1
21. 7 inches by 5 inches by 2 inches
8
4
4
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22. $1.75
Putting It Together, Part III (pages 69 to 73)
1. 1
1
inches
8
2. 20%
3. $7356.96
4. -$6 million
5. $127.50
6.
7. (a) 30%
(b) Cu: 33 kg, Ni: 9 kg, Zn: 18 kg
8. -8 m
9. 3.66 cm
9
inch
16
10. 1
7
inches
16
11. 160 times
12. (a) $2632
(b) $2968
13. (a) $12 million
(b) $3 million
7
inches
16
15. 2.2oF
16. 54 parts
17. $1440
18. -2
19. 37.96 m2
20. (a) $360
(b) $300
(c) $420
14. 1
Putting It Together, Part IV (pages 74 to 78)
1. 16.8 m
2. 23.56 kg
4. $9.60
5.
7. (a) $2850
(b) $205 000
8. 4
3
inches
16
11. (a) 521.05 euro
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3. -6oC
1
cup
6
6. -1 yards
9. $18.09
10. 27%
(b) $1141.92
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12. 6
1
inches
4
13. 4.8%
14. 8.5 inches
(b) 4%
(c) 38%
15. $0.39
16. (a) 50
(d) 32%
5
inches
8
17. $1.45
18. 11
19. (a) -45¢
(b) -9¢
20. (a) $54.81
(b) $8.22
Is It Reasonable? (pages 79 to 82)
1. Taylor’s answer of 13.23 is reasonable because 6 × 2 = 12 .
2. Jake’s answer of $19.76 is reasonable because 50% of 40 is 20.
3. Mary’s answer of 7 is reasonable because you have to round up from 6.17.
11
3  12 
is reasonable because it slightly less than
 or  . One could also
16
4  16 
argue that Lisa is much smarter than Homer and Bart so there is a greater likelihood that her
answer is right.
4. Lisa’s answer of
5. Round to nearest whole number and use order of operations.
3 × (2 + 2 ) − 1
3× 4 −1
12 − 1
11
Answer C (11.4262) is the reasonable answer.
6. If temperature initially starts below zero, rises slightly and drops back a little more than it
rose. This means that the suppertime temperature should be close to the morning
temperature. Based on this, Answer B (-5oC) is the reasonable answer.
7. (a) If the vehicle depreciates (goes down in value) by 20%, that means that 80% of the
original price is retained. Based on this Tylena’s answer ($17 600) seems reasonable
because it looks to be 80% of $22 000.
(b) Although Jake’s answer is showing a drop in the value of the car, the drop isn’t even
10%. Shelly’s answer is extremely low. No one would buy a $22 000 vehicle if would
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only be worth $4400 in one years time. It appears that she figured out how much the
value will drop, but didn’t work out the actual value. Bashir’s answer indicates that the
value of the car appreciated (went up in value) which is not the case.
8. (a) According to the ratio there are 9 red liquorish sales for every 2 black liquorish sales. If
one sold 6300 units of red liquorish, then you would expect that black liquorish sales
would be much smaller; possibly between 1000 and 2000 units. That means that
Monica’s answer (1400) is the only one that seems reasonable.
(b) Julie’s answer is unreasonable because she is showing that black liquorish sales are far
greater than red liquorish sales. Montez’s answer is unreasonable because she is showing
that red liquorish sales are only slightly larger than black liquorish sales.
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Online Supports
Order of Operations
Learnalberta Exploring Order of Operations Use It
Math Goodies Order of Operations with Exponents
YouTube: Order of Operations
YouTube: Watch Video on Order of Operations PEMDAS
Fractions
Algebra Help Fraction Operations Calculator
Hot math Fraction Operations
Shodor Interactive Fraction Quiz
YouTube: Operations with Fractions
YouTube: Fractions & Proportions Subtracting Mixed Numbers
YouTube: Watch Video on Adding Mixed Numbers Pre Algebra Help
YouTube: Multiplying & Dividing Fractions and Mixed Numbers
Decimals
Math Goodies Solving Decimal Word Problems
Teachnology Decimal Math Worksheets
Sinclair Decimal Word Problems
Super Teacher Worksheets Decimals
YouTube: Operations with Decimals
YouTube: Adding Decimals
YouTube: Subtracting Decimals
YouTube: Multiplying Decimals
YouTube: Dividing Decimals
Percents
Math Goodies Percent Unit 4
Sweethaven Working with Percents
YouTube: Watch Video on Percent of a Number Math Help
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Ratio and Proportion
Purplemath Ratios Proportions
BBC Skillswise Ratio and Proportion
CSG Ratio Proportion Calculator
Teachers TV Math 4 Real Set A ratio and Proportion
Algebra Lab Word Problems Proportions
Introductory Consumer Math Proportion Word Problem Examples
Sinclair Proportion Word Problems Handouts
YouTube: Fractions & Proportions Understanding Math Ratios
Signed Numbers
Regents Mathematics A Signed Numbers
Sweethaven Operations with Signed Numbers
Southern Maine Community College Signed Numbers
Amby Integers Operations with Signed Numbers
YouTube: Lesson 101 Adding Signed Numbers
YouTube: Lesson 102a Subtracting Signed Numbers
YouTube: Lesson 102b Multiplying and Dividing Signed Numbers
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