ACADEMIC STUDIES
MATH
Support Materials and Exercises
for
FRACTIONS
Book 1
An Introduction to Fractions for the Adult Learner
SPRING 1999
2
FRACTIONS
Fractions are used in our everyday life.
We talk about fractions with regards to food (I cut my daughter’s
sandwich in 21 ) , we measure fractions when we are cooking
(First, you add 23 cup of flour), even our time has been divided
into fractions (I will be back in a 41 of an hour).
Since we talk in fractions so much, it is necessary that we
understand what fractions are.
A fraction is an equal part of a whole. When you cut an apple
pie into 5 equal sections, each section stands for an equal part of
the whole pie. Each section is a fraction of the whole pie. If you
cut the pie, but did not eat any of it you would have 55 of the pie or
1 whole pie. If you ate one section of the pie, you would have
eaten 15 of the pie.
3
Exercise 1: Anything that is divided into equal parts can be
referred to in terms of fractions. Look at the following chart.
Which of the figures can be divided into equal parts?
4
Parts of a Fraction
There are two parts of a fraction. The numerator and the
denominator. Let’s use the following example to help us
understand these two parts:
Joe has eaten
3
5
of the apple pie.
numerator -
the numerator is the top half of the fraction
which lets us know how many equal parts of
the whole we are dealing with or how many
pieces of the pie Joe has eaten. In our
example, the 3 is the numerator. It tells us
that Joe has eaten 3 equal parts of the pie.
denominator -
the denominator is the bottom half of the
fraction which tells us how many equal parts
there are in the whole. In our example, 5 is
the denominator. It tells us that the pie was
cut in five equal parts.
± this pie has been cut into five equal parts
Joe has eaten three of the five pieces
The fraction of the pie that was eaten
was 53 .
5
Writing Fractions:
We see:
We say: three fifths are coloured
We write: 3
-number of parts coloured
5
-number of parts in total
We see:
We say: one half is coloured
We write: 1
-number of parts coloured
2
-number of parts in total
We see:
We say: two thirds are coloured
We write: 2
-number of parts coloured
3
-number of parts in total
We see:
We say: five twelfths are coloured
We write: 5 -number of parts coloured
12 -number of parts in total
6
Exercise 2:
Write the correct fraction beside each number
word given.
Example:
five eights
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
5
8
one half
two thirds
four fifths
three quarters
six ninths
seven tenths
two fourths
three eights
fifty-one hundredths
ten twentieths
Exercise 3:
Example:
1)
2
5
2)
6
8
3)
3
4
4)
1
7
5)
4
4
6)
8
10
7)
5
9
Write the correct number word beside each
fraction given.
7
8
seven eights
7
Exercise 4:
Draw a line segment between each fraction and
number word that name the same number.
five eights
1
3
one half
5
9
two thirds
3
4
four fifths
4
5
six sixths
6
6
one quarter
1
2
one third
1
4
two fourths
5
8
five ninths
2
4
three quarters
2
3
8
Look at the following figures.
a)
b)
c)
d)
e)
Exercise 5:
Write a fraction for the coloured part of each figure above.
Exercise 6:
What fraction of each of the figures above was not coloured.
9
Exercise 7:
1
2
=
3
4
=
5
8
=
7
10
5
6
=
=
14
17
=
Try the following. The first one has been done for
you. Draw a diagram showing the fraction given.
10
Types of Fractions
There are three types of fractions: proper, improper, and mixed.
proper fraction -
in a proper fraction, the denominator is
always larger than the numerator, such
as 23 , 78 , or 136 because the fraction
represents less than the whole thing.
Ex: He ate 21 of the chocolate cake.
improper fraction -
in an improper fraction, the numerator is
equal to, or larger than, the
denominator; for example, 85 , 93 , or
7
7 because the fraction represents the
whole or more than the whole thing.
(Improper fractions can always be
reduced.)
mixed number -
a number that includes both a whole
number and a fraction. 1 45 , 8 23 , and
939 21 are all mixed numbers. They
represent a whole number plus a
fraction. Ex: I will be back in 2 21 hours.
11
Exercise 8:
1)
2
3
2)
For each of the following fractions, write whether
they are proper, improper, or mixed. The first one
has been done for you.
= proper
11) 15 58 =
1
3
4
12)
15
8
3)
8
4
=
13)
100
101
4)
3
10
=
14) 7 53 =
62 115 =
15) 3 29 =
5)
=
=
=
6)
8
15
=
16)
5
21
=
7)
99
15
=
17)
25
10
=
8)
4
3
=
18) 18 45 =
9)
4
21
=
19)
4
3
=
10)
7
8 18
20)
8
31
=
=
12
Changing an Improper Fraction into a Mixed Number
In math, there are times when we need to change an improper
fraction into a mixed fraction such as when we reduce them to
lower terms.
To do this you simply divide the numerator by the denominator.
2 R2
8
= 2 23
Three goes into eight, 2 times;
3 = 3Ž 8
therefore, your whole number is 2.
The remainder (also a 2) is always
placed over the original
denominator which in this case is
3.
Let’s try another one:
3 R4
19
5Ž 19
=3 45
Five goes into nineteen, 3 times;
5 =
therefore, your whole number is 3.
The remainder is placed over the
original denominator which is 5.
Here is one more:
7
14
=7
In this case, the denominator goes
2 = 2Ž14
into the numerator an even amount
of times; so, the answer is just a
whole number. Notice that you do
not put a zero fraction ( 20 ) as this is
incorrect.
13
Exercise 9:
Now, you try. Change the following improper
fractions into mixed fractions.
1)
14
3
=
11)
43
2
=
2)
24
15
=
12)
81
5
=
3)
101
25
=
13)
12
7
=
4)
31
5
=
14)
67
31
=
5)
47
6
=
15)
213
15
6)
38
3
=
16)
15
6
=
7)
52
10
=
17)
39
12
=
8)
75
8
=
18)
24
5
=
9)
10
3
=
19)
50
3
=
10)
28
11
=
20)
85
14
=
=
14
Changing a Mixed Fraction into an Improper Fraction
There are other times, such as when you are multiplying and
dividing fractions (which you will learn about later), when you will
need to change a mixed fraction into an improper fraction. In
order to do this, you multiply the whole number by the
denominator, then add the numerator, and then you place your
answer over the original denominator.
Ex: 1 43 = 1 x 4 + 3 = 7 over 4 =
7
4
Let’s try one. 4 23 First, you multiply the whole number by the
denominator: 4 x 3 (12); then you add on the numerator 12 + 2
(14); and then you place your answer over the original
denominator which is 3: 143
Let’s do another. 3 283 = 3 x 28 + 3 =87 over 28 =
87
28
If you have only a whole number and you want to make this into a
fraction, you simply place the whole number over the
denominator one (1). Ex: 28 = 281
15
Exercise 10:
Change the following mixed numbers into
improper fractions:
1)
5 23 =
11) 25 23 =
2)
10 49 =
12) 6 95 =
3)
8 27 =
13) 4 254 =
4)
15 21 =
14) 2 143 =
5)
2 245 =
15) 5 127 =
6)
3 152 =
16) 32
7)
21 23 =
17) 18 43 =
8)
9 47 =
18) 2 507 =
9)
50 43 =
19) 6 29 =
10) 4 78 =
1
2
20) 4 1415 =
=
16
Equivalent Fractions
Fractions can appear different because they have different terms
(numerator and denominator), yet, they are the same number.
We call these fractions equivalent fractions. Equivalent fractions
are fractions that are equal in value; such as 23 , 128 , and 12
18 . All of
these are talking about the same amount or the same part of the
whole.
2
3
=
8
12
=
12
18
=
As you can see, the fractions all represent the same part of the
whole. The whole does not change, it is the same size. The
difference is that it has been divided into smaller or larger
portions.
The symbol = stands for “equal to;” in this case, you can say
12
2
8
12
8
18 , and 12 are equal to each other ( 3 = 12 = 18 ). They are
equivalent fractions.
2
3
,
We can create equivalent fractions quite easily. You simply
multiply the numerator and the denominator by the same number.
17
The most important thing to keep in mind from here on in is
that whatever you do to the denominator, you must also do
to the numerator.
To make an equivalent fraction for 83 , we choose a number to
multiply the terms (numerator and denominator) by and then do
the multiplication.
Let’s choose 5:
3
3× 5
15
So,
8 = 8× 5 = 40
15
40
Let’s try another one:
Exercise 11:
is an equivalent fraction for 83 .
Find an equivalent fraction for 45 .
First, choose a number. I will choose 6.
4
4×6
24
5 = 5× 6 = 30
Find an equivalent fraction for the following:
1)
6
9
=
10)
1
3
=
2)
3
4
=
11)
8
9
=
3)
10
11
=
12)
3
14
=
4)
3
25
=
13)
1
30
=
5)
2
75
=
14)
4
7
6)
4
5
=
15)
11
12
=
7)
7
8
=
16)
9
50
=
=
18
8)
1
10
=
17)
3
100
9)
2
15
=
18)
5
6
=
=
Sometimes, you are given the new denominator and you need to
find the missing numerator. This equation will look like this
2
3
= 21? .
The first thing that you must do is to figure out what number you
have multiplied the original denominator by to get the new
denominator. You do this by dividing the second denominator by
the original one:
21 ÷ 3 = 7
Now we know that our multiplier will be 7.
The second step is to multiply the original numerator by 7
(remember, whatever you do to the denominator, you must also
do to the numerator).
2
3
= 23×× 77 = 21?
? = 14
Therefore,
2
3
=
14
21
.
19
Let’s try another one.
3
11
=
?
132
.
First, we divide the new denominator by the original one:
132 ÷ 11 = 12
Then, we multiply the original numerator by the response to the
first step:
3
11
= 113××1212 =
?
132
? = 36
Therefore,
3
11
36
= 132
The same steps are followed if you are missing the new
denominator but you know the numerator. For example:
6
7
=
36
?
.
First divide the new numerator by the original one (36÷6=6); then,
multiply the original denominator by the response to step one
(7x6=42).
6
7
=
6× 6
7×6
=
36
?
Therefore,
6
7
=
36
42
20
Exercise 12:
Make equivalent fractions by finding the missing
numerator or denominator
=
11)
4
7
=
32
?
12)
1
2
=
?
36
?
45
13)
5
18
=
25
?
?
49
14)
9
10
=
180
?
?
30
15)
2
19
=
6
?
22
?
16)
12
17
=
?
85
17)
2
5
21
?
18)
9
20
=
70
?
19)
21
100
=
70
?
20)
1
3
1)
5
8
2)
10
17
3)
4
5
=
4)
3
7
=
5)
14
15
=
6)
11
12
=
7)
8
9
8)
7
15
=
9)
14
21
10)
7
10
=
=
?
40
?
51
72
?
=
82
?
=
=
?
240
=
19
?
?
600
21
You can also find out if two fractions are equivalent by following
similar steps. Let us look at the two fractions 21 and 1530 to see if
these two fractions are equivalent.
Step 1:
divide the second denominator (30) by the original
denominator (2) 30 ÷2 =(15)*
Step 2:
multiply the original numerator (1) by the product of
step 1 (15)*. If the numerator of the second fraction
equals this number, then the two fractions are
equivalent.
1
2
=
1× 15
2 × 15
= 1530 these two fractions are equivalent.
How about the following two fractions?
Step 1: 9÷3 = (3)*
Step 2: 2x(3)*=6
2
3
=
8
9
Does the numerator of the second fraction equal 6? No, it is an
8. Therefore, 23 and 89 are not equivalent. We can use the “not
equal to” symbol (…) to show that they are not equivalent.
2
3
≠
8
9
Let’s try a couple more.
Are the following fractions equivalent (equal =) or not equivalent
(not equal …)
4
5
24
, 35
Step 1: 35÷5=(7)*
Step 2: 4x(7)*=28
numerators are different
therefore, 45 ≠ 2435
1
2
8
, 16
Step 1: 16÷2=(8)*
Step 2: 1x(8)*=8
numerators are the same
Therefore, 12 = 168
22
Exercise 13:
Are the following fractions equivalent? Use = or …
to show their relationship.
1)
1
2
, 44
21
11)
2
4
16
2)
4
5
, 45
36
12)
14
25
, 125
3)
8
9
, 27
24
13)
2
5
, 70
28
4)
2
16
, 64
10
14)
2
3
, 24
5)
7
41
, 82
21
15)
4
10
, 80
6)
5
6
, 30
25
16)
9
19
, 95
7)
1
4
, 100
25
17)
5
6
35
8)
15
20
, 80
45
18)
3
10
, 130
9)
3
9
, 108
36
19)
18
50
, 150
10)
4
9
, 27
12
20)
14
15
, 60
, 24
70
14
36
45
, 42
39
54
58
23
Comparing Fractions with Common Denominators Using the
Symbols < and >.
We use many symbols in math to show relationships between
numbers. Here are two:
1. <
2. >
< means “smaller than”
> means “greater than”
Let’s look at some examples with fractions that use these
symbols.
1 3
<
4 4
Example 1:
The fraction
1
4
is smaller than the fraction
of the whole.
<
3
.
4
It represents less
24
Example 2:
The fraction
2
3
is greater than the fraction
1
.
3
The
first fraction represents more of the whole than the
second fraction.
Notice! You can read the expression from left to right, or
from right to left. It still reads the same.
2 4
¢
6 6
4 2
¦
6 6
Two sixths is smaller than four sixths , or...
Four sixths is greater than two sixths.
Usually though, the expression is read from left to
right.
Read the following expressions.
1)
3 1
>
5 5
Three fifths is greater than one fifth.
2)
2 4
<
6 6
Two sixths is smaller than four sixths.
3)
4 2
>
7 7
Four sevenths is greater than two sevenths.
5 8
4)
<
Five ninths is smaller than eight ninths.
9 9
How do I tell which fraction is the largest?
25
When comparing fractions that have the same denominators, the
largest fraction is the one that has the biggest numerator- what
we really want to know is, which fraction represents more of the
2
5
whole? The fraction
is larger than
1
,
5
because the numerator 2
is larger than the numerator 1.
Remember...the bottom number (denominator) tells us how many
equal parts there are in the whole. In this case, both have five
equal parts, so only the numerators are being compared.
Read the following expressions.
1)
3
5
is larger than
1
5
2)
5
8
is smaller than
3)
7
9
is larger than
4)
4
10
5)
6
8
6)
9
16
is smaller than
13
16
7)
5
10
is smaller than
8
10
4
9
is smaller than
is larger than
7
8
8
10
4
8
26
Exercise 14:
Write the word “true” or “false” for each set of fractions.
1)
3 5
>
7 7
6)
3 1
>
6 6
11)
12 10
>
20 20
2)
4 5
>
5 5
7)
20 15
>
30 30
12)
50 35
<
60 60
3)
2 4
<
8 8
8)
10 12
<
20 20
13)
2 3
>
4 4
4)
4 2
<
9 9
9)
40 10
<
50 50
14)
15 12
>
30 30
5)
5 3
>
10 10
10)
8 11
>
12 12
15)
9 10
<
10 10
Exercise 15:
Use the symbol “>” or “<“ to compare each of the following sets of
fractions.
1)
2
6
____
4
6
7)
2
____ 3
3
3
2)
5
9
____
2
9
8)
12
20
_____
10
20
3)
4
5
____
2
5
9)
6
10
_____
9
10
4)
7
12
____
10
12
10)
13
30
_____
25
30
5)
6
8
____
1
8
11)
4
8
6)
3
6
____
6
6
12)
10
20
____
2
8
_____
16
20
27
Exercise 16:
Write a fraction that is smaller than the one given.
1)
3
6
2)
5
10
3)
7
9
4)
4
8
5)
5
6
6)
3
4
7)
9
12
8)
1
3
9)
4
6
10)
10
12
11)
13
20
12)
3
9
Exercise 17:
Write a fraction that is larger than the one given.
1)
2
3
2)
4
6
3)
1
8
4)
5
6
5)
8
10
6)
7
11
7)
3
7
8)
9
10
9)
4
9
10)
1
2
11)
50
60
12)
100
250
Exercise 18:
Draw a picture that represents more of the whole than the fraction
given.
1)
3
5
2)
7
8
3)
2
6
4)
2
4
5)
7
10
28
AN EASY WAY TO COMPARE FRACTIONS THAT HAVE
DIFFERENT DENOMINATORS.
Which fraction is larger -
1
2
or
1
3
Did you notice? The denominators
are different! Not to worry!!!
When two fractions have the same numerator but different
denominators, the largest fraction is the one with the
smallest denominator.
The fraction
The fraction
1
2
1
3
means that a whole amount has equal 2 parts.
means that a whole amount has 3 equal parts. So
of course, the parts will be smaller for the fraction
Compare:
The larger fraction is
1
4
and
1
,
4
1
.
3
1
5
because “4" is the smallest denominator,
which means that the whole is divided into fewer parts.
The only thing you really have to remember in order to use
this shortcut is that the numerators must be the same.
2
4
7
10
is greater than
is greater than
2
5
7
12
3
6
is greater than
3
7
4
8
is greater than
4
10
29
Exercise 19:
Put the following fractions in order from least to greatest.
Remember to compare the denominators since the numerators
are the same!
2
10
2
3
2
6
2
9
2
4
Exercise 20:
Put the following fractions in order from least to greatest.
6/10
6/7
6/6
6/9
Exercise 21:
Compare these fractions using “>” or “<“.
1)
2/4 — 2/3
4)
6/10 — 6/12
7)
5/8 — 3/8
2)
5/7 — 5/8
5)
6/7 — 5/7
8)
3/4 —3/5
3)
2/4 — 2/5
6)
2/5 — 4/5
9)
4/6 — 3/6
30
Exercise 22:
For each fraction given , write one that is smaller and one that is
larger.
Example:
2/5 < 2/4 < 2/3
1)
— < 4/8 < —
4)
— < 3/9 ---
7)
— < 4/12 < ---
2)
--- < 5/7 < —
5)
— < 5/8 —
8)
— < 6/9 < ---
3)
— < 3/6 < —
6)
—< 8/10 < — 9)
— < 7/11 < ---
Exercise 23
Draw a line to the fraction that is larger than the one given.
1)
3/5
3/4
3/6
4)
6/15
6/16
6/13
2)
6/10
6/12
6/8
5)
6/7
6/8
6/5
3)
4/9
4/7
4/8
6)
5/8
5/7
5/9
31
REDUCING FRACTIONS TO THEIR LOWEST TERMS
We already know that fractions can look entirely different and still
represent the same amount. We called these kinds of fractions
equivalent fractions. Here are two examples: 1 and 2 . One
2
4
half equals two quarters.
While we’re on the subject of reducing...
The word “reduce” means to make smaller.” We reduce our
mortgage by making regular payments, we reduce our weight by
eating less and exercising, and...we can reduce fractions too!
When we reduce a fraction, we create a new one using smaller
numbers, but... the value of the fraction remains the same ( we
are left with the same amount of the whole as what we started
with).
Why bother reducing fractions anyway????????
Its easier to understand fraction amounts when they are written
using smaller numbers. Besides, would you rather cut a pizza in
2 pieces and eat 1 piece or have to cut it in eight pieces and eat
4? The second choice is a lot more work!
32
Reducing Fractions
Which is easier to understand?
“I will call you in
6
8
of an hour.” or, “I will call you in
3
4
of an hour”.
Both of these fractions represent the same amount of time, but
the second example makes more sense to us. The fraction 3 is
4
in its reduced form.
To create a smaller, but equivalent fraction, just think of a
number that both the numerator and denominator can be
divided by evenly.
Example:
6
18
Both numbers can be divided by 6 evenly to
create the fraction 1 .
3
6 divided by 6 = 1
18 divided by 6 = 3
Watch out!
If you do not divide
the numerator and denominator by
the SAME number, you will not end up with an equivalent fraction.
Instead, you will be left with a different amount than when you
started!
ex:
10
20
10 divided by 5 = 2
The new fraction is
20 divided by 4 = 5
In the fraction 10 ,10 is half of 20.
20
In the fraction
2
,
5
2 is less than half of 5.
2
.
5
33
Example 2
4
10
To create an equivalent but smaller fraction, divide
both numbers by 2.
4÷2=2
10 ÷ 2 = 5
The new fraction is
2
.
5
4 2
'
10 5
Example 3
7
21
Both numbers can be divided by 7 evenly.
7 ÷7 = 1
21 ÷ 7 = 3
The new fraction is
1
.
3
7 1
'
21 3
Example 4
6
8
Both numbers can be divided by 2 evenly.
6÷2=3
8÷2=4
The new fraction is 3 .
4
6 3
'
8 4
33
Greatest Common Factor
Sometimes when you divide to reduce a fraction, the new numerator
and denominator can still be reduced.
Example:
18
48
new fraction
Both numbers can be divided by 2 to create the
9
.
24
This fraction can be reduced again. Both numbers
can be divided by 3. We then end up the fraction
3
.
8
You can save yourself a lot of time and effort by finding the greatest
common factor or GCF to start with. This is the largest common
number that both numerator and denominator can be divided by
evenly.
Example:
18
48
The factors of 18 are:
{1,2,3,6,9,18}
The factors of 48 are:
{1,2,3,4,6,12,16,24,48}
Six is the greatest common factor (GCF). Now just divide.
18 divided by 6 = 3
48 divided by 6 = 8
The fraction
18
48
has now been reduced to its simplest form
3
.
8
34
Exercise 24:
Reduce the fractions below to their simplest form.
4)
4
10
11)
60
100
5)
6
12
12)
24
40
6)
12
24
13)
12
30
7)
10
30
14)
150
500
8)
16
24
15)
3
18
9)
12
18
16)
10
25
10)
6
28
17)
4
24
11)
9
27
18)
12
30
12)
16
24
19)
12
36
13)
20
25
20)
6
14
Exercise 25:
1)
2)
3)
4)
5)
3 and 9
2 and 6
20 and 50
5 and 10
35 and 18
Think of a number that can be divided evenly by...
6)
7)
8)
9)
10)
12 and 18
30 and 40
49 and 21
35 and 50
100 and 1000
35
Exercise 26:
For each group of fractions, circle the ones that can be reduced.
a.
5
8
4
10
3
6
6)
4
9
3
7
14
18
b.
6
14
4
7
2
5
7)
3
8
5
20
4
15
c.
5
9
14
20
25
35
8)
3
9
9
14
7
12
d.
12
15
50
70
12
25
9)
2
16
4
10
6
28
e.
8
30
17
20
10
15
10)
25
55
2
3
8
28
Exercise 27:
True or False.
a.
If I spend
3
6
b.
If I buy
6
10
of the books I want, I buy
c.
If I find
12
15
of the cards I lost, I find
d.
If I earn
8
12
1
2
of my day at work, I spend
4
5
2
5
of my day at work.
of the books.
of the cards.
of the money you do, I earn
4
7
of the money.
36
Exercise 28:
Which of the following fractions have been reduced to their simplest
form?
1)
5
9
6)
3
7
11)
35
45
2)
4
6
7)
4
8
12)
4
9
3)
2
4
8)
9
11
13)
100
150
4)
7
9
9)
5
10
14)
13
20
5)
8
10
10)
7
12
15)
12
14
Exercise 29:
Tell which GCF you would use to reduce each fraction.
1)
4
12
5)
10
25
9)
12
24
13)
5
15
2)
12
30
6)
3
12
10)
6
18
14)
20
35
3)
8
40
7)
14
20
11)
20
40
15)
50
75
4)
24
36
8)
18
36
12)
12
16
16)
10
20
37
Answer Key
Exercise 1
page 2:
Exercise 4
page 6:
five eights
5
8
one quarter
1
4
one half
1
2
one third
1
3
two thirds
2
3
two fourths
2
4
four fifths
4
5
five ninths
5
9
six sixths
6
6
three quarters
3
4
Exercise 5 and 6
see instructor
page 7:
a)
1 3
,
4 4
b)
3 4
,
7 7
c)
5 7
,
12 12
d)
4 5
,
9 9
5)
7 4
,
11 11
Exercise 7
page 8:
see instructor
38
Exercise 8
1) proper
5) mixed
9) proper
13) proper
17) improper
page 10:
2) mixed
6) proper
10) mixed
14) mixed
18) mixed
3)
7)
11)
15)
19)
improper
improper
mixed
mixed
improper
Exercise 9
1) 4 23 2)
page 12:
1 159 3) 4 251
4)
6 15
7)
2
8)
5
14) 2 31
5 10
13) 1 7
3
9)
5
15) 14 15 16) 2 6
98
2
1
33
5)
proper
improper
improper
proper
proper
6)
12 23
6
11) 21 2 12) 16 5
1
3
17) 3 12
10) 2 11
3
7 56
4)
8)
12)
16)
20)
3
1
4
18) 4 5
1
19) 16 3 20) 6 14
Exercise 10
17
1)
2)
3
page 14:
94
3)
9
58
7
4)
31
2
5)
53
24
6)
47
15
7)
65
3
8)
67
7
9)
203
4
10)
39
8
11)
77
3
12)
59
9
13)
104
25
14)
31
14
15)
67
12
16)
65
2
17)
75
4
18)
107
50
19)
56
9
20)
74
15
Exercise 11
page 18: see instructor
Exercise 12
page 20:
1)
5
8
2)
10
17
3)
4
5
=
=
=
25
40
30
51
36
45
11)
4
7
=
32
56
12)
1
2
=
18
36
13)
5
18
=
25
90
39
=
14)
9
10
=
180
200
28
30
15)
2
19
=
6
57
22
24
16)
12
17
=
17)
2
5
21
45
18)
9
20
=
70
105
19)
21
100
=
70
100
20)
1
3
=
11)
2
4
… 24
4)
3
7
5)
14
15
=
6)
11
12
=
7)
8
9
8)
7
15
=
9)
14
21
10)
7
10
=
21
49
72
81
Exercise 13
=
1
2
… 44
21
6)
5
6
= 30
2)
4
5
= 45
36
7)
1
4
= 100
25
12)
14
25
3)
8
9
= 27
24
8)
15
20
… 80
45
13)
2
5
4)
2
16
… 10
64
9)
3
9
36
= 108
14)
5)
7
41
… 82
10)
4
9
= 27
15)
Exercise 14
82
205
=
108
240
=
126
600
19
57
page 20:
1)
21
60
85
25
12
16
16)
9
19
= 95
= 125
17)
5
6
35
= 70
28
18)
3
10
= 130
2
3
… 14
24
19)
18
50
54
= 150
4
10
… 80
20)
14
15
… 60
70
36
page 24:
1)
false
6)
true
11) true
2)
false
7)
true
12) false
3)
true
8)
true
13) false
4)
false
9)
false
14) true
5)
true
10) false
15) true
45
= 42
39
58
40
Exercise 15
page 24:
1)
<
7)
<
2)
>
8)
>
3)
>
9)
<
4)
<
10) <
5)
>
11) >
6)
<
12) <
Exercises 16,17, and 18 page 25: see instructor
Exercise 19
Page 27:
2 2 2 2 2
, , , ,
10 9 6 4 3
Exercise 20
Page 27:
6 6 6 6
, , ,
10 9 7 6
Exercise 21
Page 27:
1.
2 2
<
4 3
4.
6 6
>
10 12
7.
5 5
<
8 5
2.
5 5
>
7 8
5.
6 6
>
7 8
8.
3 3
>
4 5
3.
2 2
>
4 5
6.
2 2
<
5 3
9.
4 4
>
6 9
41
Exercise 22
Page 28: see instructor
Exercise 23
Page 28:
1)
3
3
&&&&
5
4
4.
6
6
&&&&
15
13
2)
6
6
&&&&
10
8
5.
7
7
&&&&
12
10
3)
4
4
&&&&
9
7
6.
5
5
&&&&
8
7
Exercise 24
page 33:
1)
4 2
'
10 5
divide by 2
11)
16 2
'
24 3
divide by 8
2)
6 1
'
12 2
divide by 2
12)
12 2
'
18 3
divide by 6
3)
12 1
'
24 2
divide by 2
13)
6 3
'
28 14
divide by 2
divide by 3
14)
9 3
'
27 9
divide by 3
4)
10 1
'
30 3
5)
16 2
'
24 3
divide by 8
15)
3 1
'
18 6
divide by 3
6)
20 4
'
25 5
divide by 5
16)
10 2
'
25 5
divide by 5
7)
60 3
'
100 5
divide by 20
17)
4 1
'
24 6
divide by 4
8)
24 3
'
40 5
divide by 8
18)
12 2
'
30 5
divide by 6
9)
12 2
'
30 5
divide by 6
19)
12 1
'
36 3
divide by 12
10)
150 3
'
500 10
divide by 50
20)
6 2
'
14 5
divide by 2
42
Exercise 25
1)
2)
3)
4)
5)
page 33:
3
2
2,5,10
5
6
6)
7)
8)
9)
10)
Exercise 26
page 34:
2,6
2,5,10
7
5
10,20,50,100
1)
4 3
,
10 6
6)
14
18
2)
6
14
7)
5
20
3)
14 25
,
20 35
8)
3
9
4)
12 50
,
15 70
9)
2 4 6
, ,
16 10 28
5)
8 10
,
30 15
10)
25 8
,
55 28
Exercise 27
page 34:
1)
2)
true
Exercise 28
5 7 3 9 7 4 13
, , , , , ,
9 9 7 11 12 9 20
false
page 34:
3)
true
4)
false
43
Exercise 29
page 35
1)
2)
3)
4)
5)
6)
7)
8)
4
5
12
5
6
3
6
5
9)
10)
11)
12)
8
2
10
25
13)
14)
15)
16)
8
9
4
10
44
FEEDBACK PROCESS
For feedback, please forward your comments to:
New Brunswick Community College - Woodstock
100 Broadway Street
Woodstock, NB
E7M 5C5
Attention: Kay Curtis
Tel.: 506-325-4866 Fax.: 506-328-8426
*
In case of errors due to typing, spelling, punctuation or any proofreading errors,
please use the enclosed page to make the proposed correction using red ink and send
it to us.
*
For feedback regarding the following items, please use the form below:
-
Page
number
insufficient explanations;
insufficient examples;
ambiguity or wordiness of text;
relevancy of the provided examples;
others...
Nature of the problem
Proposed solution
(include your text if possible)
45
FEEDBACK PROCESS
Page
number
Comments:
Nature of the problem
Proposed solution
(include your text if possible)