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CHAPTER 3: FRACTIONS
CHAPTER 3 CONTENTS
3.1 Introduction to Fractions
3.2 Multiplication of Fractions and of Mixed Numbers
3.3 Division of Fractions and of Mixed Numbers
3.4 Addition of Fractions and of Mixed Numbers
3.5 Subtraction of Fractions and of Mixed Numbers
3.6 Order of Operations
3.7 U.S. Measurement Conversions
3.8 Application Problems
Image from Microsoft Office Clip Art
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
138
3.1 Introduction to Fractions
Before we begin, we will review some ideas that will be used in our studies of fractions.
Prime Number: A prime number is a number greater than 1 that is only divisible by 1 and
itself.
The number 3 is a prime number; it is only divisible by the number 3 and 1. The number 4 is not
a prime number because it is divisible by 1, 2 and 4.
Prime Factorization: The prime factorization of a number is the product of all of the prime
numbers that equals the number.
Example 1: Write the prime factorization of the numbers 32 and 40.
One way to determine the prime factorization of a number is to create a factor tree. We will
create factor trees for both 32 and 40 below. To create a factor tree, split each number into two
factors that multiply to equal the number above. Continue to split the factors into factors until
each factor is prime. A prime number is only divisible by 1 and itself. The prime factors in
these factor trees have a box around them. The product of all of the prime fractors at the bottom
of the factor tree is the prime factorization.
32 4 8
2 2 2 4
2 2
40 4 10
2 2 2 5
The prime factorization for 32 is: The prime factorization for 40 is:
32 2 2 2 2 2 40 2 2 2 5
Practice 1: Write the prime factorization of the numbers 36 and 24.
Answer: 36 2 2 3 3 24 2 2 2 3
Watch it: http://youtu.be/aXxzz99urEU
Greatest Common Factor: The Greatest Common Factor (GCF) of two numbers is the greatest
number that divides both numbers evenly. One way to determine the GCF is to use the prime
factorization. The product of all of the prime factors that are common in both numbers will give
the GCF.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
139
Example 2: Determine the Greatest Common Factor of 32 and 40. Denoted as GCF(32, 40).
The product of the prime factors the two numbers have in common equals the GCF.
GCF(32,40) = 2 2 2 8
Therefore, 8 is the largest number that divides 32 and 40.
Practice 2: Determine the Greatest Common Factor of 36 and 24. Denoted as GCF(36, 24).
Answer: 12
Watch it: http://youtu.be/102S-l4-9pQ
Least Common Multiple: The Least Common Multiple (LCM) of two numbers is the smallest
number that both numbers divide evenly. One way to determine the LCM of two numbers is to
multiply the two numbers together and divide that product by the GCF.
Example 3: Determine the Least Common Multiple of the numbers 32 and 40. Denoted
LCM(32, 40).
LCM(32,40) 32 40 GCF(32,40)
32 40 8
1280 8
160
Therefore, 160 is the smallest number that both 32 and 40 divide evenly.
FORMULA FOR DETERMINING THE LEAST COMMON MULTIPLE
LCM( , ) GCF( , )a b a b a b
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
140
Practice 3: Determine the Least Common Multiple of the numbers 36 and 24. Denoted
LCM(36, 24).
Answer: 72
Watch it: http://youtu.be/JBKzWeZvsBM
We will be using GCF, LCM, and prime factorization as we work with fractions. You may need
to refer back to this explanation as you work through this chapter.
Fraction: Fractions are used in mathematics for many purposes. Most commonly fractions
express the number of equal parts out of a whole (or out of a part), a division problem, a
remainder, or a ratio (a comparison of two numbers).
If a pizza is cut into eight equal pieces and you take three pieces you have the fraction3
8, which
is read “three eighths” or “3 divided by 8” or “3 out of 8.”
Equivalent Fractions: Changing the numerator and denominator of the fraction without
changing its value forms an equivalent fraction. There are two ways to accomplish this: by
expanding the fraction or by reducing the fraction.
PARTS OF A FRACTION
Fractions have two numbers that are separated by a fraction bar.
Numerator This number indicates how many equal parts we have.
Denominator This number indicates how many equal parts make up a whole.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
141
You can multiply the numerator and denominator by the same number and produce an equivalent
fraction, and you can divide the numerator and denominator by the same number and produce an
equivalent fraction.
A fraction whose numerator and denominator are the same number has the value 1 unless the
number is 0. For instance, the fraction, 8
8, means there are 8 pieces that make a whole and we
have 8 pieces. Therefore, we have 1 whole.
Note: The fraction 0
0 is undefined.
Reducing Fractions to Lowest Terms
In some cases, fractions may be reducible (also called simplifying fractions); however, the
numerator and the denominator of the fraction must have a common factor, which will evenly
divide both numbers without a remainder. For example, if an egg carton that holds 12 eggs has
EQUIVALENT FRACTIONS
a a c
b b c
and
a a c
b b c
for b, 0c
A FRACTION THAT EQUALS 1
1a
a when a is not 0.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
142
exactly 6 eggs in the container, you might say you have 6
12 of a carton of eggs. However, you
might just as easily remark that you have 1
2 of a carton of eggs. Both observations are correct
since 6
12 is equivalent to
1
2. In fact,
1
2 is
6
12 reduced to its lowest terms.
Let’s look at this more closely and determine how to do this mathematically so that we can
reduce other fractions that we are not as familiar with.
Example 4: Simplify 6
12
First, determine the prime factorization for the numbers 6 and 12. We will show how to do this
again by factor trees as follows.
6
2 3
12
4 3
2 2
The prime factorization for 6 is: The prime factorization for 12 is:
6 2 3 12 2 2 3
6 2 3
12 2 2 3
Rewrite the fraction using the prime factorization.
2
2
3
2
3 Divide out a 2 in the numerator and denominator.
3
2 3
1
1
Note: 2
12
3
2 3 Divide out a 3 in the numerator and denominator.
1
12
Note: 3
13
1
2
Note: If all factors in the numerator or denominator are divided out, the number that will
remains is a 1.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
143
Practice 4: Simplify 8
28. Answer:
2
7
Watch it: http://youtu.be/QiIP5PwDaqM
Example 5: Simplify 6
12
Another way to simplify fractions is to divide the numerator and the denominator by the Greatest
Common Factor (GCF). First determine the GCF for 6 and 12 is 6. To simplify the fraction
divide both the numerator and denominator by 6.
6 6 6 1
12 12 6 2
Since you cannot simplify1
2any further, that is there are no common factors for 1 and 2 other
than 1, the fraction is reduced to its lowest terms.
Sometimes at first we do not choose the largest common factor to divide into both the numerator
and denominator. In that case, just keep reducing until you cannot reduce any further. This
problem could also be simplified in the following way.
6 6 2
12 12 2
First, divide both the numerator and denominator by 2.
3
6 Simplify.
3 3
6 3
Then, divide both the numerator and denominator by 3.
1
2
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
144
Practice 5: Simplify 8
28. Answer:
2
7
Watch it: http://youtu.be/CM6a7Lc1q6A
Example 6: Simplify 28
42
We will first do this problem by using the prime factorization of the numerator and denominator.
28 2 2 7
42 2 3 7
Determine the prime factorization of the numerator and
denominator.
2
72
2 73 Divide out common factors in the numerator and
denominator.
2
3
We can also simplify this fraction by diving the numerator and denominator by common factors.
28 28 2 14
42 42 2 21
Since both 28 and 42 are even, they can be divided by 2.
without a remainder. So, divide 28 by 2 and 42 by 2
14 14 7 2
21 21 7 3
Both 14 and 21 can be divided by 7 without a remainder; so
the fraction is not fully simplified. Divide both numbers by
7.
28
42=
2
3 The fraction is now fully simplified.
Note that you could have gotten the answer in one step by dividing the original numerator and
denominator by 14 (the greatest common factor for 28 and 42).
28 28 14 2
42 42 14 3
Divide the numerator and denominator by the GCF.
Unless directions indicate otherwise, you should simplify a fraction to its lowest terms.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
145
Practice 6: Simplify 24
36 Answer:
2
3
Watch it: http://youtu.be/Y5GixJsD-es
Expanding Fractions
At times, especially when adding or subtracting fractions, we need to rewrite fractions in an
equivalent form. To do this we will multiply the numerator and the denominator by the same
number (usually a positive integer).
1
2
2
1
2
2
2
4
3
1
2
3
3
6
4
1
2
4
4
8
Example 7: Rewrite 3
5 by multiplying both the numerator and the denominator by 4.
3 3 4 12
5 5 4 20
Practice 7: Rewrite 5
7 by multiplying both the numerator and the denominator by 3.
Watch it: http://youtu.be/_PDJ9HwYeh0 Answer: 15
21
EXPANDING FRACTIONS
a a c
b b c
for b, 0c
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
146
Example 8: Rewrite 3
5 by multiplying both the numerator and denominator by 2.
3 3 2 6
5 5 2 10
Practice 8: Rewrite 5
7 by multiplying both the numerator and denominator by 4.
Watch it: http://youtu.be/tlLTgE6xylw Answer: 20
28
As long as you multiply both the numerator and the denominator by the same number (except 0),
you will obtain a fraction that is equal in value to the original fraction.
There are three types of numbers that involve fractions: proper fractions, improper fractions and
mixed numbers.
Proper fraction: A proper fraction is any fraction whose numerator is smaller than its
denominator. Some examples of proper fractions include:
1 16 4 2, , ,
3 17 5 11
Improper fraction: A fraction whose numerator is equal to or greater than the denominator is
an improper fraction. Some examples of improper fractions include:
4 17 7 53 7, , , ,
4 17 6 19 1
Mixed number: When a number consists of both a whole number and a fraction number it is
called a mixed number or a mixed fraction. The fraction part of a mixed number is a proper
fraction. Some examples of mixed numbers include:
3 1 5 35 , 1 , 2 , 23
4 7 6 19
PARTS OF A DIVISION STATEMENT
Dividend Divisor Quotient
8 2 4
Dividend
Quotient
Divisor
84
2
Quotient
Divisor Dividend
42 8
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
147
Example 9: Rewrite the division problem 5 3 as a fraction.
5 is the dividend and 3 is the divisor, so as a fraction we have:
5
3
Practice 9: Rewrite the division problem 6 5 as a fraction.
Watch it: http://youtu.be/NcuJE2zLGeA Answer: 6
5
Example 10: Rewrite the fraction 5
3 as you would perform long division.
3 5
In previous discussions regarding division problems, when there was a remainder you simply
wrote the remainder as a whole number; however, we often use fractions to express that
remainder.
3
Whole number part
Denominator 5
3
Nume
2
rator
1
12
3 (See, it’s as easy as 1, 2, 3)
What does 5
3mean? Picture a whole candy bar that has 3 pieces. You have 5 of these pieces.
1 2
3
Practice 10: Rewrite the fraction 6
5 as you would perform long division.
Watch it: http://youtu.be/uH6f2zagJjI Answer: 5 6
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
148
Example 11: Rewrite 17
5 as a mixed number.
Divide the denominator into the numerator.
5
Whole number part
Denominator 17
15
Nume
2
rat r
3
o
32
5
Therefore, 17
5 53
2 .
Practice 11: Rewrite 21
8 as a mixed number. Answer:
52
8
Watch it: http://youtu.be/YIL3CA-FBtA
Example 12: Rewrite 27
6 as a mixed number.
Divide the denominator into the numerator.
6
Whole number part
Denominator 27
24
Nume
3
rat r
4
o
43
6
This answer is not finished. Look at the fraction on the mixed number. It can be simplified by
dividing 3 into the 3 in the numerator and the 6 in the denominator. This gives the final answer:
33 3 1
4 4 46 6 23
Therefore, 27 1
46 2 .
Always simplify fractions (when possible) before finishing a problem.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
149
The previous problem could have been simplified first and then the long division performed as
we will show below.
27 3 3 3
6 2 3
Write prime factorization of numerator and denominator.
33 3
2 3
Divide out a 3 in the numerator and denominator.
9
2 Simplify, and then complete long division using the
simplified fraction.
Divide the denominator into the numerator.
Whole number part
Denominator 9
8
Numerat
4
1
o
2
r
41
2
As we saw before, 27
6 24
1 .
Practice 12: Rewrite 52
8 as a mixed number. Answer:
16
2
Watch it: http://youtu.be/KYepikF1DvU
Integers as Fractions
Integers such as 0 or 7 or -12 or 38 sometimes need to be written as fractions. Since fractions
represent division and dividing by 1 does not change the value of the number, whole numbers
can be written as a fraction “over” 1. That is, divided by 1.
HOW TO WRITE AN INTEGER AS A FRACTION
For any integer, a , 1
aa .
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
150
Example 13: Write 0, 7, –12, and 38 as fractions.
0 =0
1 7 =
7
1
1212
1
38 =
38
1
Practice 13: Write 0, 3, –23, and 5 as fractions. Answer: 0 3 23 5
, , ,1 1 1 1
Watch it: http://youtu.be/g5-JTiVxRKs
Mixed Numbers
When you have a mixed number, you may need to rewrite it as an improper fraction in order to
work with it. In this section you will be introduced to an algorithm (procedure) for changing a
mixed number to an improper fraction. In section 3.5, you will learn why this procedure works.
To change a mixed number into an improper fraction, the numerator of the improper fraction will
be determined by multiplying the whole number times the denominator of the mixed number and
then adding this product to the numerator of the mixed number. The denominator of the
improper fraction will be the same as the mixed number.
Note:
b bA A
c c
Example 14: Rewrite 15
8 as an improper fraction.
Multiply the denominator with the whole number, and add that product to the numerator. Then
place that resulting sum in the numerator, and keep the same denominator.
5 (8 1) 5 8 5 13
18 8 8 8
HOW TO CHANGE A MIXED NUMBER TO AN IMPROPER FRACTION
( )b A c bA
c c
( )b b A c bA A
c c c
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
151
Practice 14: Rewrite 3
34
as an improper fraction. Answer: 15
4
Watch it: http://youtu.be/UrXadoqagVM
Example 15: Rewrite 34
9 as an improper fraction.
4 (9 3) 4 27 4 31
39 9 9 9
Practice 15: Rewrite 3
27
as an improper fraction. Answer: 17
7
Watch it: http://youtu.be/YJrl5nlu-K8
Example 16: Rewrite 3
25
as an improper fraction.
The number 3
25
means 3
25
. We will first determine the improper fraction in the
parentheses and then that number will be negative.
3 (5 2) 3 10 3 13 13
25 5 5 5 5
Note: Disregard the negative sign until the mixed number has been turned into an improper
fraction.
Practice 16: Rewrite 2
43
as an improper fraction. Answer: 14
3
Watch it: http://youtu.be/oYFuIXgLLNI
Mixed numbers are sometimes needed when finding the mean and median value of a set of data.
Recall that to compute an arithmetic mean (an average), add all of the values and divide that sum
by the number of values. If the answer is not a whole number we can write the result as a mixed
number or fraction.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
152
Example 17: Find the mean of 75, 45, 80, 94, 65. Add all of the values and divide by the number of values.
75 + 45 + 80 + 94 + 65
5=
359
5 This is the answer as an improper fraction.
359
35
09
5
7
5
1
4
5
359
571
4
Since 359 does not divide evenly by 5, we write
the mean as a mixed number by performing long
division.
Practice 17: Determine the mean of 85, 76, 55, 98. Answer: 1
782
Watch it: http://youtu.be/lxJkvDjfQ0o
Also recall that to find the median of a set of data values, arrange the values in numerical order
and locate the middle. If there are an odd number of data values, then the median is the number
in the middle of the list. If there is an even number of data values, then there are two data values
in the middle of the list: compute the mean (the average) of these two middle values to
determine the median. Again, if it is not a whole number we can write the result as a mixed
number or a fraction.
Example 18: Find the median of 20, –49, 31, –56, 67, –97.
We begin by listing the values in ascending order: –97, –56, –49, 20, 31, 67.
Note that there is an even number (six) of data values. The two middle data
values are -49 and 20. We find the mean (average) of these:
49 20 29
2 2
This is the answer as an improper fraction.
29
2
0
1
2
9
4
8
1
129
2 214
Since -29 does not divide evenly by 2, we can
write the result as a mixed number by performing
long division.
A negative divided by a positive is a negative. Therefore our answer is negative.
Practice 18: Determine the median of 3, -9, 2, -5, 6, -6. Answer: 1
12
Watch it: http://youtu.be/Rr3QHvccn8Y
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
153
Order of Fractions
When fractions have the same denominator they can be placed in order from least to greatest by
comparing their numerators. The following fractions all have the denominator 7. Since they
have a common denominator they are arranged from least to greatest according to their
numerators.
1
7,2
7,3
7,4
7,5
7,6
7,7
7
When fractions do not have a common denominator, we will need to first determine the common
denominator and then make all of the fractions into equivalent fractions with the common
denominator. The Least Common Denominator will be the Least Common Multiple of all the
denominators.
Example 19: Order the following fractions from least to greatest. 1 3 5 2 1
, , , ,3 6 6 3 6
To put fractions in order we must first write all of the fractions with a common denominator.
The Least Common Multiple of the denominators 3 and 6 is the smallest number they divide.
Multiples of 3: 3, 6, 9, 12…
Multiples of 6: 6, 12, 18…
The LCM(3, 6) = 6
Now, write an equivalent fractions for each fraction with the denominator of 6.
1 1 2
3 3
2
2 6
Multiply the numerator and denominator by 2.
3
6 This fraction already has a denominator of 6.
5
6 This fraction already has a denominator of 6.
2 2 4
3 3
2
2 6
Multiply the numerator and denominator by 2.
1
6 This fraction already has a denominator of 6.
2 3 5 4 1, , , ,
6 6 6 6 6
These are the equivalent fraction for 1 3 5 2 1
, , , ,3 6 6 3 6
with a
common denominator of 6.
1 2 3 4 5, , , ,
6 6 6 6 6
Put the fractions in order from least to greatest from the
smallest numerator to the largest numerator.
1 1 3 2 5, , , ,
6 3 6 3 6 Then write the fractions in their original form
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
154
Practice 19: Order the following fractions from least to greatest. 1 3 5 2 1
, , , ,4 4 8 4 8
Watch it: http://youtu.be/AsBhn5LUIU4 Answer: 1 1 2 5 3
, , , ,8 4 4 8 4
Example 20: Determine the median of the following numbers. 1 3 3 2 5
, , , ,2 8 4 4 8
To determine the median, we must first put the numbers in order. To put fractions in order we
must first write all of the fractions with a common denominator. The Least Common Multiple of
the denominators 2, 4 and 8 is the smallest number they all divide.
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16…
Multiples of 4: 4, 8, 12, 16, 20, 24…
Multiples of 8: 8, 16, 24…
The LCM(2, 4, 8) = 8
Now, write an equivalent fractions for each fraction with the denominator of 8.
1 1 4
2 2
4
4 8
Multiply the numerator and denominator by 4.
3
8 This fraction already has a denominator of 8.
3 3 6
4 4
2
2 8
Multiply the numerator and denominator by 2.
2 2 4
4 4
2
2 8
Multiply the numerator and denominator by 2.
5
8 This fraction already has a denominator of 8.
The equivalent fraction for
1 3 3 2 5, , , ,
2 8 4 4 8with a common denominator of 8 are:
4 3 6 4 5
, , , ,8 8 8 8 8
Put the fractions in order from least to greatest from the smallest numerator to the largest
numerator.
3 4 4 5 6
, , , ,8 8 8 8 8
3 4 4 5 6
, , , ,8 8 8 8 8
Determine the middle number.
Therefore, 4 1
8 2 is the median value.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
155
Practice 20: Determine the median of the following numbers. 1 3 4 2 1
, , , ,3 4 12 4 4
Watch it: http://youtu.be/VXknaa77Iys Answer: 1
3
Example 21: Determine the mode of the following numbers. 1 3 3 2 5
, , , ,2 8 4 4 8
To determine the mode we must first look at all of the fractions in order from least to greatest. In
Example 20, we have already put this data set in order from least to greatest.
3 4 4 5 6
, , , ,8 8 8 8 8
The mode is the data value that is repeated the most often. As you can see the data value 4
8is
repeated the most often. The fraction 4
8is equal to
1
2. Therefore,
1
2 is the mode.
Practice 21: Determine the mode of the following numbers. 1 3 4 2 1
, , , ,3 4 12 4 4
Answer: http://youtu.be/P6zxxx0khg8 Watch it: 1
3
Zero and Fractions
Earlier in your studies, you discovered that the number 0 has some special qualities. When you
are working with fractions, there are a couple of things that you need to keep in mind.
When the numerator is 0 (and the denominator is not 0), the value of that fraction is 0. Thus,
0
6= 0. To illustrate why this is true, let’s look closely at this equation.
Example 22: Simplify 0
6
We know that this is a division problem of 0 6 , which has a quotient of 0. We can express this
as 0
6= 0 .
On the other hand, if the denominator is 0, you have the case of division by 0. As you learned
earlier in this book, you cannot divide by 0: the answer is “undefined.” Fractions give us a
numeric way to demonstrate why you cannot divide by 0.
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
156
Practice 22: Simplify 0
12 Answer: 0
Watch it: http://youtu.be/8LfUW4AghoE
Example 23: Simplify 7
0
70
0 because multiplying the denominator, 0, by the whole number quotient, 7, does not
produce, 7: 0 7 7 .
Similarly, 7
00 because multiplying the denominator, 0, by the whole number quotient, 0, does
not produce, 7: 0 0 7 .
As you can see, division by 0 is not defined; we say that division by 0 is undefined.
Practice 23: Simplify 20
0 Answer: undefined
Watch it: http://youtu.be/KsS3Vkkv5yA
Watch All: http://youtu.be/56TSha9Row0
RULES FOR FRACTIONS WITH ZERO
If 0 is divided by any number (except 0),
the answer is 0.
00
n
If any number is divided by 0,
the answer is undefined.
0
nundefined
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
157
3.1 Introduction Exercises
1. The fraction 3
4 is equivalent to ___ ÷ ___
2. The fraction 6
5 is equivalent to ___ ÷ ___
3. Write 35 19 as a fraction. 4. Write 9 8 as a fraction.
5. Write 15 as a fraction. 6. Write 3 4 as a fraction.
7. The numerator of 7
10 is ___ and the denominator is ___.
8. The numerator of
100
13 is ___ and the denominator is ___.
9. Simplify 4
12 by dividing 4 and 12 by 4.
10. Simplify 36
42 by dividing 36 and 42 by 6.
11. Simplify 55
77 by dividing 55 and 77 by 11.
12. Simplify 6
8 13. Simplify
15
60
14. Simplify 20
64 15. Simplify
12
66
16. Simplify 16
56 17. Simplify
48
98
18. Convert 51
2 to an improper fraction. 19. Convert 3
4
7 to an improper fraction.
20. Convert 1
23
to an improper fraction.
21. Arrange from smallest to largest: 2 1 1 3 1
, , , ,3 12 3 4 2
22. Arrange from smallest to largest: 2 1 5 3
, , ,9 2 72 8
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
158
23. Write 14
3 as a mixed number in simplest form.
24. Write 78
8 as a mixed number in simplest form.
25. Find the mean of 15, 12, 3, 11, 7, 9.
26. Find the mean of –31, 20, –10, 17, –5.
27. Find the mean of 18, 21, 8, 11.
28. Find the median of 1 3 2
, ,2 4 3
29. Find the median of 1 10 4 3 7
, , , ,11 11 11 11 11
30. Find the mode of 3 5 2 1 5 10 1 4
, , , , , , ,8 7 6 2 15 14 10 12
CCBC Math 081 Introduction to Fractions Section 3.1
Third Edition 23 pages
159
3.1 Introduction Exercises Answers
1. 3 4 16. 2
7
2. 6 5 17. 24
49
3. 35
19 18.
11
2
4. 8
9 19.
25
7
5. 15
1 20.
7
3
6. 3
4 21.
1 1 1 2 3, , , ,
12 3 2 3 4
7. numerator: 7, denominator: 10 22. 5 2 3 1
, , ,72 9 8 2
8. numerator: 100; denominator: 13 23. 42
3
9. 1
3 24. 9
3
4
10. 6
7 25.
57
6= 9
1
2
11. 5
7 26.
9 41
5 5
12. 3
4 27.
29 114
2 2
13. 1
4 28.
2
3
14. 5
16 29.
4
11
15. 2
11 30.
1
3