Fractions
Arab mathematicians
began to use the horizontal
fraction bar around the
year 1200.
They were the first to write
fractions as we do today.
Uses of Fractions
The numbers �12
�, �23
�, �54
�, �71
�, and �12050
� are all fractions. A fraction is
written with two whole numbers that are separated by a fraction
bar. The top number is called the numerator. The bottom
number is called the denominator. The numerator of a fraction
can be any whole number. The denominator can be any whole
number except 0.
When naming fractions, name the numerator first, then name
the denominator.
Fractions were invented thousands of years ago to name
numbers between whole numbers. People needed these
in-between numbers for making careful measurements.
Here are some examples of measurements that use fractions:
�23
� cup, �34
� hour, �190� km, and 13 �
12
� lb.
Fractions are also used to name parts of wholes. The whole
might be one single thing, like a pizza. Or, the whole might be
a collection of things, like all the students in a classroom. The
whole is sometimes called the ONE.
42 forty-two
Fractions
�34
� yard
�13
� mile
�58
� of a pizza
Mrs. Blake’s Classroom�12
� of the students are girls.
Name the whole, or ONE, for each statement.
“Derek ate �58
� of the pizza.”
The whole is the entire pizza.
The fraction, �58
�, names the part of the pizza that Derek ate.
“In Mrs. Blake’s classroom, �12
� of the students are girls.”
The whole is the collection of all students in Mrs. Blake’s classroom.
The fraction, �12
�, names the part of that collection that are girls.
In Everyday Mathematics, fractions are used in other ways that
may be new to you. Fractions are used in the following ways:
♦ to show rates (such as cost per ounce)
♦ to compare (such as comparing the weights of two animals)
♦ to name percents (�12
� is 50%)
♦ to show divisions (15 � 3 can be written �135�)
♦ to show the scale of a map or a picture
♦ to show probabilities
�12
� teaspoon
Number Sense 1.5; Number Sense 1.7
Here are some other examples of uses of fractions:
♦ Study the recipe shown at the right. Many of the amounts
listed in the recipe include fractions.
♦ This spinner has �13
� of the circle colored red, �14
� colored blue,
and �152� colored green.
If we spin the spinner many times, it will land on red about�13
� of the time. It will land on blue about �14
� of the time.
And it will land on green about �152� of the time.
The probability that the spinner will land on a color that
is not green is �172�.
♦ If a map includes a scale, you can use the scale to
estimate real-world distances. The scale on the map
shown here is given as 1:10,000. This means that every
distance on the map is �10,
1000� of the real-world distance.
A 1 centimeter distance on the map stands for a real-
world distance of 10,000 centimeters (100 meters).
forty-three 43
Fractions
Size Chart for Women’s Shoes
Heel-to-toe length (in.) Size
8�11
56� to 9�
116� 6
9�126� to 9�
136� 6 �
12
�
9�146� to 9�
166� 7
9�176� to 9 �
196� 7 �
12
�
9�11
06� to 9�
11
16� 8
9�11
26� to 9�
11
46� 8�
12
�
9�1156� to 10�
116� 9
♦ Fractions are often used to describe
clothing sizes. For example, women’s
shoes come in sizes 3, 3 �12
�, 4, 4 �12
�, and
so on, up to 14.
Part of a size chart for women’s
shoes is shown at the right. It gives
the recommended shoe size for
women whose feet are between 9
and 10 inches long.
♦ A movie critic gave the film Finding Cosmo a rating of
3�12
� stars (on a scale of 0 to 4 stars).
Number Sense 1.5; Number Sense 1.7
Look at the collection of counters.
What fraction of the counters is red?
There are 12 counters in all.
Five of the counters are red.
Five out of 12 counters are red.
This fraction shows what part of the collection is red.
Sally ate half a pizza. Is that a lot?
The answer depends on how big the pizza was. If the
pizza was small, then �12
� is not a lot. If the pizza was
large, then �12
� is a lot.
Name the fraction of counters that are each shape in the collection above.
1. Circles 2. Triangles 3. Squares
Check your answers on page 341.
Understanding the many ways people use fractions will help
you solve problems more easily.
Fractions for Parts of a Whole
Fractions are used to name a part of a whole thing that is
divided into equal parts. For example, the circle at the right has
been divided into 8 equal parts. Each part is �18
� of the circle.
Three of the parts are blue, so �38
� (three-eighths) of the circle
is blue.
In Everyday Mathematics, the whole thing that is divided into
equal parts is called the ONE. To understand a fraction used to
name part of a whole, you need to know what the ONE is.
Fractions for Parts of a Collection
A fraction may be used to name part of a collection of things.
44 forty-four
Fractions
Number Sense 1.5; Number Sense 1.7
When you pick a ball out of this jar without looking,
the chance of getting a red ball is �58
�. The chance of getting a
blue ball is �38
�.
probability of picking a red ball �
probability of picking a blue ball �
number of red balls
total number of balls�
number of blue balls
total number of balls�
�58
�
�38
�
To understand a fraction that is used to name part of a
collection, you need to know how big the whole collection is.
forty-five 45
Fractions
Only half of Sam’s cousins can come to his party.
Is that many people?
It depends on how many cousins Sam has. If Sam has only
4 cousins, then 2 cousins are coming; that’s not many
people. But if Sam has 24 cousins, then 12 cousins are
coming. That’s many people.
Fractions in Measuring
Fractions are used to make more careful measurements.
Think about the inch scale on a ruler. Suppose the spaces
between the whole-inch marks are left unmarked. With a
ruler like this, you can measure only to the nearest inch.
Now suppose the 1-inch spaces are divided into quarters by �12
�-inch and �14
�-inch marks. With this ruler, you can measure
to the nearest �12
� inch or to the nearest �14
� inch.
To understand a fraction used in a measurement, you need
to know what the unit is. To say, “Susan lives �12
� from here”
makes no sense. Susan might live half a block away or
half a mile. The unit in measurement is like the ONE when
fractions are used to name a part of a whole.
Fractions in Probability
A fraction may tell the chance that an event will happen.
This chance, or probability, is always a number from 0 to 1.
An impossible event has a probability of 0; it has no chance
of happening. An event with a probability of 1 is sure to happen.
An event with a probability of �12
� has an equal chance of
happening or not happening.
Ruler has inch marks only. You can measure to the nearest inch.
Ruler has �12
�- and �14
�-inch marks.
You can measure to the nearest
�12
� inch or to the nearest �14
� inch.
Number Sense 1.5; Number Sense 1.7
The word fraction is
derived from the Latin
word frangere, which
means “to break.”
Fractions are sometimes
called “broken numbers.”
Fractions and Division
Division problems can be written using a slash / instead of the
division symbol �. For example, 21 � 3 can be written 21 / 3.
Division problems can also be written as fractions. One of the
many uses of fractions is to show divisions. The example below
shows that 21 � 3 can be written as the fraction �231�.
46 forty-six
Fractions
21 � 3 � 7 Show that �231� � 7 also.
This is the whole, or ONE. This is the whole after dividing
it into 3 equal parts. Each part
is �13
� of the whole.
The picture below shows that 21 thirds make 7 wholes. So, �231� � 7.
You can rename any fraction by dividing on your calculator.
To rename �231�, think of it as a division problem and divide:
Press 21 3 . The answer in the display will show 7,
which is another name for �231�.
21 � 3 � 21 / 3
and
21 � 3 � �231�
Fractions less than 1 can also be thought of as divisions.
Show that 3 � 4 � �34
�.
Think of 3 � 4 as an equal-sharing problem.
Suppose 4 friends want to share 3 oranges.
They could cut or divide each orange into 4 equal parts.
Each person gets �34
� of an orange. So, 3 � 4 � �34
�.
Number Sense 1.5; Number Sense 1.7
A length of 6 centimeters on the original will be 3 centimeters on the copy.
� �
12
�
copy size
original size
Fractions in Rates and Ratios
Fractions are often used to name rates
and ratios.
A rate compares two numbers with
different units. For example, 30 miles per
hour is a rate that compares distance with
time. It can be written as �310
hmoiulers
�.
A ratio is like a rate, but it compares two
quantities that have the same unit.
Other Uses of Fractions
Fractions are used to compare distances on maps to distances
in the real world, and to describe size changes.
forty-seven 47
Fractions
Rate Example
speed (jogging)
price
conversion of units
Ratio Example
won / lost record
rainy days comparedto total days
Find the real-world distance from Clay St. to S. Lake St.
Measure this distance on the map. It is 6 cm.
Each distance on the map is �20,1000� of the real-world
distance. So, the real-world distance equals
20,000 times the map distance. The real-world
distance � 20,000 * 6 cm � 120,000 cm.
100 cm � 1 m. So, 1,200 cm � 12 m and
120,000 cm � 1,200 m.
The distance from Clay St. to S. Lake St.
is 1,200 m.
distance
time�
7 blocks
4 minutes
cost
quantity�
99¢
3 erasers
distance in yards
distance in feet�
1 yard
3 feet
games (won)
games (lost)�
6
8
(rainy) days
(total) days�
11
30
Number Sense 1.5; Number Sense 1.7
Note
Even though they are
called improper, there is
nothing wrong about
improper fractions.
Do not avoid them.
Mixed Numbers
Numbers like 1 �12
�, 2 �35
�, and 4 �38
� are called mixed numbers.
A mixed number has a whole-number part and a fraction part.
In the mixed number 2 �35
�, the whole-number part is 2 and the
fraction part is �35
�. A mixed number is equal to the sum of the
whole-number part and the fraction part: 2 �35
� � 2 � �35
�.
Mixed numbers are used in many of the same ways that
fractions are used.
Mixed numbers can be renamed as fractions. For example, if
a circle is the ONE, then 2 �35
� names 2 whole circles and �35
� of
another circle.
If you divide the 2 whole circles into fifths, then you can see
that 2 �35
� � �153�.
To rename a mixed number as a fraction, first rename 1 as a
fraction with the same denominator as the fraction part. Then
add all of the fractions.
For example, to rename 4 �38
� as a fraction, first rename 1 as �88
�.
Then 4 �38
� � �88
� � �88
� � �88
� � �88
� � �38
� � �385�.
Fractions like �153� and �
88
� are called improper fractions.
An improper fraction is a fraction that is greater than or equal
to 1. In an improper fraction, the numerator is greater than or
equal to the denominator.
A proper fraction is a fraction that is less than 1. In a proper
fraction, the numerator is less than the denominator.
48 forty-eight
Fractions
Check your answers on page 341.
Write a mixed number for each picture.
1. 2.
Write an improper fraction for each mixed number.
3. 1 �34
� 4. 2 �13
� 5. 3 �58
�
Number Sense 1.5; Number Sense 1.7
On Ms. Klein’s bus route, she picks up 24 students, 18 boys and 6 girls.
Equivalent Fractions
Two or more fractions that name the same number are called
equivalent fractions.
forty-nine 49
Fractions
The four circles below are the same size, but they are divided into different
numbers of parts. The green areas are the same in each circle. These circles show
different fractions that are equivalent to �12
�.
The fractions �12
�, �24
�, �36
� and �48
� are all equivalent. They are just different names for the
part of the circle that is green.
You can write: �12
� � �24
� �12
� � �48
� �24
� � �48
�
�12
� � �36
� �24
� � �36
� �36
� � �48
�
The fractions �264� , �
132� , and �
14
� are all equivalent.
You can write �264� � �
132� � �
14
�.
2 equal parts
1 part green
�12
� of the circle
is green.
24 equal groups
Each group is �214�
of the total.
6 groups of girls
�264� of the students
are girls.
12 equal groups
Each group is �112�
of the total.
3 groups of girls
�132� of the students
are girls.
4 equal groups
Each group is �14
�
of the total.
1 group of girls
�14
� of the students
are girls.
4 equal parts
2 parts green
�24
� of the circle
is green.
6 equal parts
3 parts green
�36
� of the circle
is green.
8 equal parts
4 parts green
�48
� of the circle
is green.
Number Sense 1.5; Number Sense 1.7
Find fractions that are equivalent to �12
84�.
� �192� � �
68
� � �34
�
1. a. What fraction of this rectangle is shaded?
b. Give two other fractions for the shaded part.
2. Name 3 fractions that are equivalent to �12
�.
3. Name 3 fractions that are equivalent to �24400
�.
Check your answers on page 341.
Rules for Finding Equivalent Fractions
Here are two shortcuts for finding equivalent fractions.
Using Multiplication
If the numerator and the denominator of a fraction are both
multiplied by the same number (not 0), the result is a fraction
that is equivalent to the original fraction.
Fractions
Change �25
� to an equivalent fraction.
Multiply the numerator and the denominator of �25
� by 3.
In symbols, you can write � �165� .
So, �25
� is equivalent to �165�.
�25
� is red.
�165� is red.
2 * 35 * 3
6 � 315 � 3
18 � 224 � 2
18 � 324 � 3
18 � 624 � 6
Using Division
If the numerator and the denominator of a fraction are both
divided by the same number (not 0), the result is a fraction that
is equivalent to the original fraction.
To understand why division works, use the example shown above.
But start with �165� this time and divide both numbers in the
fraction by 3: � �25
�
The division by 3 “undoes” the multiplication by 3 that we did
before. Dividing both numbers in �165� by 3 gives an equivalent
fraction, �25
�.
50 fifty Number Sense 1.5; Number Sense 1.7
1. True or false?
a. �12
� � �198� b. �
58
� � �23
02� c. �
25
� � �14
20� d. �
02
� � �1
020�
2. a. Use the table to find 3 other fractions that are equivalent to �15
�.
b. Add 2 more equivalent fractions that are not in the table.
Check your answers on page 341.
Table of Equivalent Fractions
This table lists equivalent fractions. All the fractions in a row
name the same number. For example, all the fractions in the
last row are names for the number �78
�.
Every fraction in the first column is in simplest form. A fraction
is in simplest form if there is no equivalent fraction with a
smaller numerator and smaller denominator.
fifty-one 51
Fractions
Note
Every fraction is either
in simplest form or is
equivalent to a fraction
in simplest form.
Lowest terms
means the same
as simplest form.
SimplestName Equivalent Fraction Names
Under normal conditions,�15
� of the length of a
telephone pole should
be in the ground.
0 (zero) �01
� �02
� �03
� �04
� �05
� �06
� �07
� �08
� �09
�
1 (one) �11
� �22
� �33
� �44
� �55
� �66
� �77
� �88
� �99
�
�12
� �24
� �36
� �48
� �150� �
162� �
174� �
186� �
198� �
12
00�
�13
� �26
� �39
� �142� �
155� �
168� �
271� �
284� �
297� �
13
00�
�23
� �46
� �69
� �182� �
11
05� �
11
28� �
1241� �
12
64� �
12
87� �
23
00�
�14
� �28
� �132� �
146� �
250� �
264� �
278� �
382� �
396� �
14
00�
�34
� �68
� �192� �
1126� �
12
50� �
12
84� �
2218� �
23
42� �
23
76� �
34
00�
�15
� �120� �
135� �
240� �
255� �
360� �
375� �
480� �
495� �
15
00�
�25
� �140� �
165� �
280� �
12
05� �
13
20� �
1345� �
14
60� �
14
85� �
25
00�
�35
� �160� �
195� �
1220� �
12
55� �
13
80� �
2315� �
24
40� �
24
75� �
35
00�
�45
� �180� �
11
25� �
1260� �
22
05� �
23
40� �
2385� �
34
20� �
34
65� �
45
00�
�16
� �122� �
138� �
244� �
350� �
366� �
472� �
488� �
594� �
16
00�
�56
� �11
02� �
11
58� �
2204� �
23
50� �
33
06� �
3452� �
44
08� �
45
54� �
56
00�
�18
� �126� �
234� �
342� �
450� �
468� �
576� �
684� �
792� �
18
00�
�38
� �166� �
294� �
1322� �
14
50� �
14
88� �
2516� �
26
44� �
27
72� �
38
00�
�58
� �11
06� �
12
54� �
2302� �
24
50� �
34
08� �
3556� �
46
04� �
47
52� �
58
00�
�78
� �11
46� �
22
14� �
2382� �
34
50� �
44
28� �
4596� �
56
64� �
67
32� �
78
00�
Number Sense 1.5; Number Sense 1.7
1. Name a fraction or mixed number for each mark labeled A, B, and C on the
ruler above.
2. What is the length of this nail?
a. in quarter inches b. in eighths of an inch
c. in sixteenths of an inch
Check your answers on page 341.
Equivalent Fractions on a Ruler
Rulers marked in inches usually have tick marks of different lengths. The
longest tick marks on the ruler below show the whole inches. The marks used
to show half inches, quarter inches, and eighths of an inch become shorter
and shorter. The shortest marks show the sixteenths of an inch.
Every tick mark on this ruler can be named by a number of sixteenths. Some
tick marks can also be named by eighths, fourths, halves, and ones. The
picture below shows the pattern of fraction names for a part of the ruler.
This pattern continues past 1 inch, with mixed numbers naming the tick marks.
52 fifty-two
Fractions
Number Sense 1.5; Number Sense 1.7
� is less than
� is greater than
� is equal to
�35
� � �38
�
�45
� � �35
� because 4 � 3. �29
� � �79
� because 2 � 7.
�12
� � �13
� because halves are bigger than thirds.
�38
� � �34
� because eighths are smaller than fourths.
Comparing Fractions
When you compare fractions, you have to pay attention to both
the numerator and the denominator.
Like Denominators
Fractions are easy to compare when they have the same
denominator. For example, to decide which is larger, �78
� or �58
�,
think of them as 7 eighths and 5 eighths. Just as 7 bananas
is more than 5 bananas, and 7 dollars is more than 5 dollars,
7 eighths is more than 5 eighths.
To compare fractions that have the same denominators, just
look at the numerators. The fraction with the larger numerator
is larger.
Like Numerators
If the numerators of two fractions are the same, then the
fraction with the smaller denominator is larger. Remember, a
smaller denominator means the ONE has fewer parts and each
part is bigger. For example, �35
� � �38
� because fifths are bigger
than eighths, so 3 fifths is more than 3 eighths.
fifty-three 53
Fractions
�58
� � �78
� or �78
� � �58
�
Note
Fractions with like
denominators have the
same denominator.
�14
� and �34
� have like
denominators.
Fractions with like
numerators have the
same numerator.
�23
� and �25
� have
like numerators.
Use of the symbol � for
“equal to” dates back to
1571. Use of the symbols
� and � for “greater
than” and “less than”
dates back to 1631.
Number Sense 1.7.2
Compare. Write �, �, or � in each box.
1. �35
� �37
� 2. �23
� �49
� 3. �38
� �58
� 4. �26
� �25
�
Check your answers on page 341.
Unlike Numerators and Unlike Denominators
Several strategies can help you compare fractions when
both the numerators and the denominators are different.
Comparing to �12
� Compare �37
� and �58
�.
Notice that �58
� is more than �12
�
and �37
� is less than �12
�.
So, �37
� � �58
�.
Comparing to Comparing fractions to 0 or 1 can also be helpful.
0 or 1 For example, �78
� � �34
� because �78
� is closer to 1.
( �78
� is �18
� away from 1 but �34
� is �14
�
away from 1. Since eighths are
smaller than fourths, �78
� is closer to 1.)
Using Equivalent One way to compare fractions that
Fractions always works is to find equivalent fractions that
have the same denominator. For example, to
compare �58
� and �35
�, look at the table of equivalent
fractions on page 51. The table shows that both
fifths and eighths can be written as 40ths:
�58
� � �24
50� and �
35
� � �24
40�. Since �
24
50� � �
24
40�, you
know that �58
� � �35
�.
Using Decimal Using decimal equivalents is another
Equivalents way to compare fractions that always
works. For example, to compare �25
� and �38
�,
use a calculator to change both fractions
to decimals:
�25
�: Key in: 2 5 Answer: 0.4
�38
�: Key in: 3 8 Answer: 0.375
Since 0.4 � 0.375, you know that �25
� � �38
�.
54 fifty-four
Fractions
Note
Remember that fractions
can be used to show
division problems.
� a � bab
Number Sense 1.9
�38
� � �18
� � �48
� � � �12
�4 � 48 � 4
�170� � �
130� � �
140� � � �
25
�4 � 2
10 � 2
Adding and Subtracting Fractions
Like Denominators
Adding or subtracting fractions that have the same
denominator is easy: Just add or subtract the numerators,
and keep the same denominator.
You can use division to put the answer in simplest form.
Unlike Denominators
When you are adding and subtracting fractions that have unlike
denominators, you must be especially careful. One way is to
model the problem with pattern blocks. Remember that
different denominators mean the ONE is divided into different
numbers (and different sizes) of parts.
fifty-five 55
Fractions
�13
� � �16
� � ?
If the hexagon is ONE, then the rhombus is �13
� and the triangle is �16
�.
When you put one rhombus and one triangle together, you
will find that they make a trapezoid. If the hexagon is ONE,
then the trapezoid is �12
�.
So, �13
� � �16
� � �12
�.
�56
� � �23
� � ?
If the hexagon is ONE, then �56
� is 5 triangles and �23
� is 2 rhombuses.
To take away �23
� (2 rhombuses) from �56
� (5 triangles), you would
need to take away 4 triangles.
Then there would be 1 triangle or �16
� left.
So, �56
� � �23
� � �16
�.
�13
� � �16
� � �12
�
�56
� � �23
� � �16
�
�23
��56
�
Number Sense 1.7
Clock Fractions
A clock face can be used to model fractions with 2, 3, 4, 5, 6, 10,
12, 15, 20, 30, or 60 in the denominator.
Fractions
�15
� hour �
12 minutes
�23
� hour �
40 minutes
�610� hour �
1 minute
�152� hour �
25 minutes
Note
Thousands of years ago,
the ancient Babylonians
divided the day into
24 hours, the hour into
60 minutes, and the
minute into 60 seconds.
This system for keeping
time is a good model
for working with
many fractions.
A clock face can help in solving simple fraction addition and
subtraction problems.
�13
� � �16
� � ?
�13
� hour � 20 minutes
�16
� hour � 10 minutes
�13
� � �16
� � �12
�
�34
� � �13
� � ?
�34
� hour � 45 minutes
�13
� hour � 20 minutes
�34
� � �13
� � �152�
Using a Calculator
Some calculators can add and subtract fractions.
�38
� � �14
� � ?
Key in: 3 8 1 4 ; or 3 8 1 4 Answer: �58
�
Solve. Use pattern blocks or clock faces to help you.
1. �56
� � �12
� 2. �23
� � �14
� 3. �172� � �
14
� 4. �12
� � �23
�
Check your answers on page 341.
56 fifty-six Number Sense 1.7
A furlong is a unit of
distance, equal to �18
� mile.
It is often used to
measure distances in
horse and dog races.
To add a distance in
furlongs (eighths of a
mile) and a distance
given in tenths of a
mile, you could rename
the fractions using
8 * 10 � 80 as a
like denominator.
�14
� � �23
� � ?
A quick way to find a like denominator for these fractions
is to multiply the denominators: 4 * 3 � 12.
Rename �14
� and �23
� as 12ths:
�14
� � � �132�
�23
� � � �182�
So, �14
� � �23
� � �132� � �
182� � �
11
12�.
�34
� � �25
� � ?
A like denominator for these fractions is 4 * 5 � 20.
Rename �34
� and �25
� as 20ths:
�34
� � � �12
50�
�25
� � � �280�
So, �34
� � �25
� � �12
50� � �
280� � �
270�.
Sometimes tools like pattern blocks or clock faces are not
helpful for solving a fraction addition or subtraction problem.
Here is a method that always works.
Using a Like Denominator
To add or subtract fractions that have different denominators,
first rename them as fractions with a like denominator. A quick
like denominator to use is the product of the denominators.
If two fractions are renamed so that they have the same
denominator, that denominator is called a common
denominator.
fifty-seven 57
Fractions
Add or subtract.
1. �24
� � �15
� 2. �58
� � �12
� 3. �172� � �
14
� 4. �12
� � �13
� � �14
�
Check your answers on page 341.
1 * 34 * 32 * 43 * 4
3 * 54 * 5
2 * 45 * 4
Number Sense 1.5
Use any method to solve these problems.
1. 6 * �23
� 2. 3 * �45
� 3. �34
� * 6 4. 4 * �34
� 5. �45
� * 5
Check your answers on page 341.
Multiplying Fractions and Whole Numbers
There are several ways to think about multiplying a whole
number and a fraction.
Using a Number Line
One way to multiply a whole number and a fraction is to think
about “hops” on a number line. The whole number tells how
many hops to make, and the fraction tells how long each hop
should be. For example, to solve 4 * �23
�, imagine taking 4 hops on
a number line, each �23
� unit long.
Using Addition
You can use addition to multiply a fraction and
a whole number. For example, to find 4 * �23
�, draw
4 models of �23
�. Then add up all of the fractions.
58 fifty-eight
Fractions
Using Fraction of an Area
You can think of multiplying with a fraction as finding the
fraction of an area. For example, to solve 4 * �23
� (which is the
same as �23
� * 4), find �23
� of an area that is 4 square units.
The rectangle on the left has an area of 4 square units.
The shaded area of the rectangle on the right has an area of �83
� square units (8 small rectangles, each with an area of �13
�.)
So, �23
� of the rectangle area � the shaded area � �83
�.
The word “of” in problems likethese means multiplication.
4 * �23
� � 2 �23
�
4 * �23
� = �23
� + �23
� + �23
� + �23
� � �83
�
4 squares �23
� * 4 � �83
�
�35
� of 20 means �35
� * 20.
�16
� of 18 means �16
� * 18.
�34
� of 24 means �34
� * 24.
4 * �23
� � �23
� * 4 � �83
�
Number Sense 1.5; Number Sense 1.9
Finding a Fraction of a Set
You can think of multiplication with fractions as finding a
fraction of a set. For example, think of the problem �25
� * 30 as
“What is �25
� of 30¢?” One way to solve this problem is first to
find �15
� of 30, and then use that answer to find �25
� of 30.
fifty-nine 59
Fractions
�25
� of 30 means �25
� * 30.
5 equal groups, with 6 in each group
�25
� * 30 � ? Think of the problem as “What is �25
� of 30?”
Step 1: Find �15
� of 30.
To do this, divide the 30 pennies into 5 equal groups.
Then count the number of pennies in one group.
30 � 5 � 6, so �15
� of 30 is 6.
Step 2: Next find �25
� of 30.
Since �15
� of 30 is 6, �25
� of 30 is 2 * 6 � 12.
�25
� * 30 � �25
� of 30 � 12
3 equal groups, with 5 in each group
�23
� * 15 � ? Think of the problem as “What is �23
� of 15?”
Step 1: Find �13
� of 15.
Divide 15 pennies into 3 equal groups.
15 � 3 � 5, so �13
� of 15 is 5.
Step 2: Next find �23
� of 15.
Since �13
� of 15 is 5, �23
� of 15 is 2 * 5 � 10.
�23
� * 15 � �23
� of 15 � 10
Find each answer.
1. �14
� * 28 � ? 2. �35
� of 20 � ? 3. 16 * �58
� � ?
4. Rita and Hunter earned $12 raking lawns. Since Rita did most of the work,
she got �23
� of the money. How much did each person get?
Check your answers on page 341.
Number Sense 1.5
Rename as fractions: 0, 12, 15.3, 3.75, and 25%.
0 � �
01
� 12 � �
112� 15.3 � �
11503
� 3.75 � �
31
70
50
� 25% � �
12050
�
Negative Numbers and Rational Numbers
People have used counting numbers (1, 2, 3, and so on) for
thousands of years. Long ago people found that the counting
numbers did not meet all of their needs. They needed numbers
for in-between measures such as 2�
12
� inches and 6�
56
� hours.
Fractions were invented to meet these needs. Fractions can
also be renamed as decimals and percents. Most of the numbers
you have seen are fractions or can be renamed as fractions.
However, even fractions did not meet every need. For example,
problems such as 5 � 7 and 2�
34
� � 5�
14
� have answers that are less
than 0 and cannot be named as fractions. (Fractions, by the way
they are defined, can never be less than 0.) This led to the
invention of negative numbers. Negative numbers are numbers
that are less than 0. The numbers ��
12
�, �2.75, and �100 are
negative numbers. The number �2 is read “negative 2.”
Negative numbers serve several purposes:
♦ To express locations such as temperatures below zero on
a thermometer and depths below sea level
♦ To show changes such as yards lost in a football game
♦ To extend the number line to the left of zero
♦ To calculate answers to many subtraction problems
The opposite of every positive number is a negative number,
and the opposite of every negative number is a positive
number. The number 0 is neither positive nor negative;
0 is also its own opposite.
The diagram at the right shows this relationship.
The rational numbers are all the
numbers that can be written or renamed
as fractions or as negative fractions.
60 sixty
Fractions
Note
Every whole number
(0, 1, 2, and so on) can
be renamed as a fraction.
For example, 0 can be
written as �
01
�. And 8 can
be written as �
81
�.
Numbers like �2.75 and
�100 may not look like
negative fractions, but
they can be renamed as
negative fractions.
�2.75 � � �
141�, and
�100 � ��
101
0�
Note
Number Sense 1.8
This method will work
for most of the decimal
numbers you see. But it
will not work for every
decimal number. For
example, 0.4444…
cannot be written as
a fraction with a
denominator of 10, 100,
1,000, or any other
power of 10.
Note
�12
� � � �
15000
� � 0.50
�
45
� � � �
180� � 0.8
�
34
�: Key in: 3 4 Answer: 0.75 �
58
�: Key in: 5 8 Answer: 0.625
�
49
�: Key in: 4 9 Answer: 0.4444… �
131�: Key in: 3 11 Answer: 0.2727…
Fractions, Decimals, and Percents
Fractions, decimals, and percents are different ways to write
numbers. Sometimes it is easier to work with a fraction instead
of a decimal or a percent. Other times it is easier to work with
a decimal or a percent.
Renaming a Fraction as a Decimal
You can rename a fraction as a decimal if you can find an
equivalent fraction with a denominator of 10, 100, or 1,000.
This only works for certain fractions.
Another way to rename a fraction as a decimal is to divide
the numerator by the denominator. You can use a calculator
for this division.
Renaming a Decimal as a Fraction
To change a decimal to a fraction, write the decimal as a
fraction with a denominator of 10, 100, or 1,000. Then you can
rename the fraction in simplest form.
sixty-one 61
Fractions
Remember that
� a � b is true for
any fraction .
Note
The U.S. Constitution did
not take effect until 9 of
the 13 original states had
approved it.
�193� � 0.69 and
�
193� � 69%
(decimal and percent
rounded to 2 digits)
Write each decimal as a fraction.
For 0.5, the rightmost digit is 5, which is in the 10ths place.
So, 0.5 � �
150�, or �
12
�.
For 0.307, the rightmost digit is 7, which is in the 1,000ths
place. So, 0.307 � �
13,00070
�.
For 4.75, the rightmost digit is 5, which is in the
100ths place. So, 4.75 � �
41
70
50
� (a fraction) or 4 �
17050
� or 4 �
34
�
(mixed numbers).
1 * 502 * 50
4 * 25 * 2
a
ba
b
Number Sense 1.6
0
Renaming a Decimal as a Percent
To rename a decimal as a percent, try to write the decimal as a
fraction with a denominator of 100. Then use the meaning of
percent (number of hundredths) to rename the fraction as a percent.
Fractions
Rename each decimal as a percent.
0.5 � 0.50 � �15000
� � 50% 0.01 � �1
100� � 1% 1.2 � 1.20 � �
11
20
00
� � 120%
Rename each percent as a fraction in simplest form.
50% � �15000
� � �12
� 75% � �17050
� � �34
� 1% � �1
100� 200% � �
21
00
00
� �
Rename each percent as a decimal.
45% � �14050
� � 0.45 120% � �11
20
00
� � 1.20, or 1.2 1% � �1
100� � 0.01
Rename each fraction as a percent.
�12
� � 0.50 � �15000
� � 50% �35
� � 0.60 � �16000
� � 60% �38
� � 0.375 � �3170.05
� � 37.5%
Renaming a Percent as a Decimal
To rename a percent as a decimal, try to rename it as a fraction
with a denominator of 100. Then rename the fraction as a decimal.
Renaming a Percent as a Fraction
To rename a percent as a fraction, try to write it as a fraction
with a denominator of 100.
Renaming a Fraction as a Percent
To rename a fraction as a percent, try to rename it as a fraction
with a denominator of 100. Then rename the fraction as a percent.
Write each number as a fraction, a decimal, and a percent.
1. �12
� 2. 0.75 3. 10% 4. �45
�
Check your answers on page 341.
62 sixty-two Number Sense 1.6
��21
Musicians make patterns of sound to create music. Mathematics
can help us understand how both sound and music are created.
Sound, Music, and Mathematics
Sound
Every sound you hear begins with a vibration—a back and forth
motion. For musical instruments to produce sound, something must
be set in motion.
The rate at which a string, a drum head,
or a column of air vibrates is called the
frequency. Higher frequency vibrations
produce higher-pitched notes. Frequency
is measured in Hertz (Hz), or “vibrations
per second.” The human ear can hear
vibrations from about 15 Hz to 20,000 Hz.
The sound of a guitar starts whena person plucks or strums the strings.Each vibrating string moves backand forth at the same rate until itstops moving. When the stringsstop vibrating, the sound stops.
➤
The sound of a drumstarts when a personbeats the drum head.When the drum headstops vibrating, thesound stops.
The sound of a flute starts when a person blowsacross the mouthpiece. A column of air movesback and forth inside the flute. When the playerstops blowing, the columnof air stops vibratingand the sound ofthe flute stops.
sixty-three 63
➤
➤
Measurement and Geometry 3.2; Mathematical Reasoning 1.1
64 sixty-four
Instrument Length and Pitch
Many instruments rely on a vibrating column of air to make
sound. A longer column of air vibrates at a lower frequency and
makes a deeper- or lower-pitched note. Shorter vibrating air
columns make higher-pitched notes.
Here are some instruments you may haveheard, along with the frequency of thelowest note that can be played on theinstrument. What happens to the frequencyas the instruments get shorter?
➤
bassoon, 58 Hz clarinet, 139 Hz oboe, 233 Hz piccolo, 587 Hz
Measurement and Geometry 3.2; Mathematical Reasoning 1.1
sixty-five 65
A recorder can play a range ofpitches. By covering all of the fingerholes on a recorder, the musiciancreates the longest possible columnof air, and the lowest-pitched note.With all holes uncovered, a highnote is produced.
The piccolo has a very shortcolumn of air within it, so itproduces high-pitched notes.Piccolos produce notes in therange of about 600 to 4,000 Hz,which humans can hear easily.
This pan flute, from Peru, is playedby blowing across the edges ofhollow tubes of different lengths.Short tubes produce high-pitchednotes, and long tubes producelow-pitched notes. The playerslides the instrument from sideto side to change notes.
➤
Because the alto saxophone is muchlonger than the piccolo, its sound islower-pitched. Saxophones use areed, which is a carefully-shaped pieceof cane. The musician blows into themouthpiece, which causes the reed tovibrate. This starts the vibration ofthe column of air.
➤
➤
➤
Measurement and Geometry 3.2; Mathematical Reasoning 1.1
66 sixty-six
Percussion Instruments
Drums are percussion instruments. The size of the instrument
affects the pitch it can play. The size and tightness of the drum
head and the materials that the drum head is made from also
affect the pitch.
A drummer holds a West Africantalking drum, or donno, between theupper arm and the body. Squeezingthe strings with the upper armtightens the drumhead and raises the pitch of the drum. Releasing thestrings loosens the drumhead andlowers the pitch of the drum.
➤
➤
In a trap set, the largest drum—thebase or “kick” drum—produces thelowest-pitched notes. Each drum canbe tuned up or down by tighteningor loosening the heads.
The steel drum, from the Caribbeanisland of Trinidad, is made by cutting offthe top of a steel oil barrel. Each smallrounded section of the drum head isshaped to play a different pitched note.The pitch of the instrument can be veryhigh because the small metal sectionsvibrate rapidly.
➤
Measurement and Geometry 3.2; Mathematical Reasoning 1.1
sixty-seven 67
Stringed Instruments
The pitch of the notes that a stringed instrument can play is
related to the length, diameter, and tension of the strings.
This man is tuning his stringedinstrument. Tightening a stringraises the pitch. Loosening astring lowers the pitch.
Compared to the violin, the cellohas longer strings of greaterdiameters. It is designed to playlow-pitched notes.
When a musician winds a stringtighter around its tuning peg,the string is tightened and thepitch becomes higher.
The violin, the smallest member ofthe string family, has short stringswith small diameters. It is designedto play high-pitched notes. Whena player presses down on a string,the vibrating part is shortenedand the pitch becomes higher.
➤
➤
➤
➤
Measurement and Geometry 3.2; Mathematical Reasoning 1.1
The Piano
Looking closely at the way a piano works can help you see some
of the mathematical relationships in music.
68 sixty-eight
An octave begins and ends on a note with the samename. For example, the keys between “Middle C”and the C to the right of it represent 1 octave. Thereare 8 octaves on most pianos. The names of thewhite and black keys in an octave repeat eight times.
➤
A piano’s sound begins when a player presses a key.This causes a felt-covered wooden hammer to hit thestrings for that key. The strings then vibrate to producesound. Each key produces a note with a different pitch.
CMiddle
Cone octave
What patterns can you find in music? How
have you seen mathematics used in music?
➤
As you move to the right on thepiano keyboard, the frequenciesget higher. What patterns do yousee in the frequencies?
➤ This tuning fork vibrates 440 times per second. A piano tuner tightens or loosens the A4 stringuntil its pitch exactly matches the pitch of thevibrating tuning fork. Then all other strings aretightened or loosened based on that note.
➤
Measurement and Geometry 3.2; Mathematical Reasoning 1.1