fractions - edison park elementary

Fractions

Arab mathematiciansbegan to use the horizontalfraction bar around the year 1200.

They were the first to writefractions as we do today.

Uses of Fractions

The numbers �12�, �

23�, �

54�, �

71�, and �1

2050� are all fractions. A fraction is

written with two whole numbers that are separated by a fractionbar. The top number is called the numerator. The bottomnumber is called the denominator. The numerator of a fractioncan be any whole number. The denominator can be any wholenumber except 0.

When naming fractions, name the numerator first, then namethe denominator.

Fractions were invented thousands of years ago to namenumbers 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, �1

90� km, and 13 �

12� lb.

Fractions are also used to name parts of wholes. The wholemight be one single thing, like a pizza. Or, the whole might be a collection of things, like all the students in a classroom. Thewhole 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 thatmay 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

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 �1

72�.

♦ 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 everydistance 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 ShoesHeel-to-toe length (in.) Size

8�1156� 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�1106� to 9�

11

16� 8

9�1126� to 9�

11

46� 8�

12

�

9�1156� to 10�

116� 9

♦ Fractions are often used to describeclothing sizes. For example, women’sshoes 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 Nemo a rating of 3�

12� stars (on a scale of 0 to 4 stars).

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 thepizza 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 helpyou solve problems more easily.

Fractions for Parts of a WholeFractions are used to name a part of a whole thing that isdivided into equal parts. For example, the circle at the right hasbeen 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 intoequal parts is called the ONE. To understand a fraction used toname part of a whole, you need to know what the ONE is.

Fractions for Parts of a CollectionA fraction may be used to name part of a collection of things.

44 forty-four

Fractions

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 ballstotal number of balls

�

number of blue ballstotal number of balls �

�58

�

�38

�

To understand a fraction that is used to name part of acollection, 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 manypeople. But if Sam has 24 cousins, then 12 cousins arecoming. That’s many people.

Fractions in MeasuringFractions are used to make more careful measurements.

Think about the inch scale on a ruler. Suppose the spacesbetween the whole-inch marks are left unmarked. With aruler 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 whenfractions are used to name a part of a whole.

Fractions in ProbabilityA 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.

The word fraction isderived from the Latinword frangere, whichmeans “to break.”Fractions are sometimescalled “broken numbers.”

Fractions and DivisionDivision problems can be written using a slash / instead of thedivision symbol . For example, 21 3 can be written 21 / 3.

Division problems can also be written as fractions. One of themany uses of fractions is to show divisions. The example belowshows 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�.

A length of 6 centimeters on the original will be 3 centimeters on the copy.

� �12

�copy size

original size

Fractions in Rates and RatiosFractions are often used to name rates and ratios.

A rate compares two numbers withdifferent units. For example, 30 miles perhour is a rate that compares distance withtime. It can be written as �31

0hmoiulers

�.

A ratio is like a rate, but it compares twoquantities that have the same unit.

Other Uses of FractionsFractions 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

NoteEven though they arecalled improper, there isnothing wrong aboutimproper 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 thatfractions 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 seethat 2 �

35� � �

153�.

To rename a mixed number as a fraction, first rename 1 as afraction with the same denominator as the fraction part. Thenadd 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 equalto 1. In an improper fraction, the numerator is greater than orequal to the denominator.

A proper fraction is a fraction that is less than 1. In a properfraction, the numerator is less than the denominator.

48 forty-eight

Fractions


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�

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 calledequivalent fractions.

forty-nine 49

Fractions

The four circles below are the same size, but they are divided into differentnumbers of parts. The green areas are the same in each circle. These circles showdifferent 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� , �1

32� , and �

14� are all equivalent.

You can write �264� � �1

32� � �

14�.

2 equal parts1 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.


112�

of the total.

3 groups of girls�132� of the students

are girls.


14

�

of the total.

1 group of girls�14� of the studentsare girls.

4 equal parts2 parts green�24

� of the circleis green.





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�.


Rules for Finding Equivalent Fractions

Here are two shortcuts for finding equivalent fractions.

Using MultiplicationIf the numerator and the denominator of a fraction are bothmultiplied by the same number (not 0), the result is a fractionthat 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 �1

65�.

�25� is red.

�165� is red.

2 * 35 * 3

6 315 3

18 224 2

18 324 3

18 624 6

Using DivisionIf the numerator and the denominator of a fraction are bothdivided by the same number (not 0), the result is a fraction thatis equivalent to the original fraction.

To understand why division works, use the example shown above. But start with �1

65� this time and divide both numbers in the

fraction by 3: � �25�

The division by 3 “undoes” the multiplication by 3 that we didbefore. Dividing both numbers in �1

65� by 3 gives an equivalent

fraction, �25�.

50 fifty

1. True or false? a. �

12� � �1

98� 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 rowname the same number. For example, all the fractions in thelast 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 asmaller numerator and smaller denominator.

fifty-one 51

Fractions

NoteEvery fraction is eitherin simplest form or isequivalent to a fractionin simplest form.

Lowest terms means the same as simplest form.

SimplestName Equivalent Fraction Names

Under normal conditions,�15� of the length of atelephone 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� �

1200�

�13

� �26

� �39

� �142� �

155� �

168� �

271� �

284� �

297� �

1300�

�23

� �46

� �69

� �182� �

1105� �

1128� �

1241� �

1264� �

1287� �

2300�

�14

� �28

� �132� �

146� �

250� �

264� �

278� �

382� �

396� �

1400�

�34

� �68

� �192� �

1126� �

1250� �

1284� �

2218� �

2342� �

2376� �

3400�

�15

� �120� �

135� �

240� �

255� �

360� �

375� �

480� �

495� �

1500�

�25

� �140� �

165� �

280� �

1205� �

1320� �

1345� �

1460� �

1485� �

2500�

�35

� �160� �

195� �

1220� �

1255� �

1380� �

2315� �

2440� �

2475� �

3500�

�45

� �180� �

1125� �

1260� �

2205� �

2340� �

2385� �

3420� �

3465� �

4500�

�16

� �122� �

138� �

244� �

350� �

366� �

472� �

488� �

594� �

1600�

�56

� �1102� �

1158� �

2204� �

2350� �

3306� �

3452� �

4408� �

4554� �

5600�

�18

� �126� �

234� �

342� �

450� �

468� �

576� �

684� �

792� �

1800�

�38

� �166� �

294� �

1322� �

1450� �

1488� �

2516� �

2644� �

2772� �

3800�

�58

� �1106� �

1254� �

2302� �

2450� �

3408� �

3556� �

4604� �

4752� �

5800�

�78

� �1146� �

2214� �

2382� �

3450� �

4428� �

4596� �

5664� �

6732� �

7800�

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 inchc. in sixteenths of an inch


Equivalent Fractions on a Ruler

Rulers marked in inches usually have tick marks of different lengths. Thelongest tick marks on the ruler below show the whole inches. The marks usedto show half inches, quarter inches, and eighths of an inch become shorterand shorter. The shortest marks show the sixteenths of an inch.

Every tick mark on this ruler can be named by a number of sixteenths. Sometick marks can also be named by eighths, fourths, halves, and ones. Thepicture 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

� 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 boththe numerator and the denominator.

Like DenominatorsFractions are easy to compare when they have the samedenominator. 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, justlook at the numerators. The fraction with the larger numeratoris larger.

Like NumeratorsIf the numerators of two fractions are the same, then thefraction with the smaller denominator is larger. Remember, asmaller denominator means the ONE has fewer parts and eachpart 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

�

NoteFractions with likedenominators have thesame denominator.

�14� and �

34� have like

denominators.

Fractions with likenumerators have thesame numerator.

�23� and �

25� have

like numerators.

Use of the symbol � for“equal to” dates back to1571. Use of the symbols� and � for “greaterthan” and “less than”dates back to 1631.

Compare. Write �, �, or � in each box.

1. �35� �

37� 2. �

23� �

49� 3. �

38� �

58� 4. �

26� �

25�


Unlike Numerators and Unlike DenominatorsSeveral 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, tocompare �

58� and �

35�, look at the table of equivalent

fractions on page 51. The table shows that bothfifths 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: 0.375Since 0.4 � 0.375, you know that �

25� � �

38�.

54 fifty-four

Fractions

NoteRemember that fractionscan be used to showdivision problems.

� a bab

�38� � �

18� � �

48� � � �

12�

4 48 4

�170� � �1

30� � �1

40� � � �

25�

4 210 2

Adding and Subtracting Fractions

Like DenominatorsAdding or subtracting fractions that have the samedenominator 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 DenominatorsWhen you are adding and subtracting fractions that have unlikedenominators, you must be especially careful. One way is tomodel the problem with pattern blocks. Remember thatdifferent denominators mean the ONE is divided into differentnumbers (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 wouldneed 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�

Clock FractionsA 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

NoteThousands of years ago,the ancient Babyloniansdivided the day into 24 hours, the hour into60 minutes, and theminute into 60 seconds.This system for keeping time is a good modelfor working with many fractions.

A clock face can help in solving simple fraction addition andsubtraction problems.

�13� � �

16� � ?

�13

� hour � 20 minutes

�16


�13� � �

16� � �

12�

�34� � �

13� � ?

�34


�13


�34� � �

13� � �1

52�

Using a CalculatorSome 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. �1

72� � �

14� 4. �

12� � �

23�


56 fifty-six

A furlong is a unit ofdistance, equal to �

18� mile.

It is often used tomeasure distances inhorse and dog races.

To add a distance infurlongs (eighths of amile) and a distancegiven in tenths of a mile, you could renamethe 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� � � �1

32�

�23� � � �1

82�

So, �14� � �

23� � �1

32� � �1

82� � �

11

12�.

�34� � �

25� � ?

A like denominator for these fractions is 4 * 5 � 20.Rename �

34

� and �25

� as 20ths:

�34� � � �

12

50�

�25� � � �2

80�

So, �34� � �

25� � �

12

50� � �2

80� � �2

70�.

Sometimes tools like pattern blocks or clock faces are nothelpful for solving a fraction addition or subtraction problem.Here is a method that always works.

Using a Like DenominatorTo add or subtract fractions that have different denominators,first rename them as fractions with a like denominator. A quicklike denominator to use is the product of the denominators.

If two fractions are renamed so that they have the samedenominator, that denominator is called a commondenominator.

fifty-seven 57

Fractions

Add or subtract.

1. �24� � �

15� 2. �

58� � �

12� 3. �1

72� � �

14� 4. �

12� � �

13� � �

14�


1 * 34 * 32 * 43 * 4

3 * 54 * 52 * 45 * 4

Use any method to solve these problems.

1. 6 * �23� 2. 3 * �

45� 3. �

34� * 6 4. 4 * �

34� 5. �

45� * 5


Multiplying Fractions and Whole Numbers

There are several ways to think about multiplying a wholenumber and a fraction.

Using a Number LineOne way to multiply a whole number and a fraction is to thinkabout “hops” on a number line. The whole number tells howmany hops to make, and the fraction tells how long each hopshould be. For example, to solve 4 * �

23�, imagine taking 4 hops on

a number line, each �23� unit long.

Using AdditionYou 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 AreaYou can think of multiplying with a fraction as finding thefraction 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�

Finding a Fraction of a Set

You can think of multiplication with fractions as finding afraction 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?


Rename as fractions: 0, 12, 15.3, 3.75, and 25%.

0 � �01� 12 � �

112� 15.3 � �

11503

� 3.75 � �31

70

50� 25% � �1

2050�

Negative Numbers and Rational Numbers

People have used counting numbers (1, 2, 3, and so on) forthousands of years. Long ago people found that the countingnumbers did not meet all of their needs. They needed numbersfor in-between measures such as 2�

12� inches and 6�

56� hours.

Fractions were invented to meet these needs. Fractions canalso be renamed as decimals and percents. Most of the numbersyou 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 waythey are defined, can never be less than 0.) This led to theinvention of negative numbers. Negative numbers are numbersthat 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 positivenumber. 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

NoteEvery whole number (0, 1, 2, and so on) canbe renamed as a fraction.For example, 0 can bewritten as �

01�. And 8 can

be written as �81�.

Numbers like �2.75 and�100 may not look likenegative fractions, butthey can be renamed asnegative fractions.

�2.75 � � �141�, and

�100 � ��1010

�

Note

This method will workfor most of the decimalnumbers you see. But itwill not work for everydecimal number. Forexample, 0.4444…cannot be written as a fraction with adenominator of 10, 100,1,000, or any otherpower of 10.

Note

�12� � � �1

5000� � 0.50

�45� � � �1

80� � 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… �1

31�: Key in: 3 11 Answer: 0.2727…

Fractions, Decimals, and Percents

Fractions, decimals, and percents are different ways to writenumbers. Sometimes it is easier to work with a fraction insteadof a decimal or a percent. Other times it is easier to work witha decimal or a percent.

Renaming a Fraction as a DecimalYou can rename a fraction as a decimal if you can find anequivalent 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 calculatorfor this division.

Renaming a Decimal as a FractionTo change a decimal to a fraction, write the decimal as afraction with a denominator of 10, 100, or 1,000. Then you canrename the fraction in simplest form.

sixty-one 61

Fractions

Remember that � a b is true for

any fraction .

Note

The U.S. Constitution didnot take effect until 9 ofthe 13 original states hadapproved it.

�193� � 0.69 and

�193� � 69%

(decimal and percentrounded 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 � �1

50�, or �

12�.

For 0.307, the rightmost digit is 7, which is in the 1,000thsplace. So, 0.307 � �1

3,00070�.

For 4.75, the rightmost digit is 5, which is in the 100ths place. So, 4.75 � �

41

70

50� (a fraction) or 4 �1

7050� or 4 �

34�

(mixed numbers).

1 * 502 * 50

4 * 25 * 2

ab a

b

Renaming a Decimal as a PercentTo rename a decimal as a percent, try to write the decimal as afraction with a denominator of 100. Then use the meaning ofpercent (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% � �1

7050� � �

34� 1% � �1

100� 200% � �

21

00

00� � 2

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 � �1

5000� � 50% �

35� � 0.60 � �1

6000� � 60% �

38� � 0.375 � �

3170.05

� � 37.5%

Renaming a Percent as a DecimalTo rename a percent as a decimal, try to rename it as a fractionwith a denominator of 100. Then rename the fraction as a decimal.

Renaming a Percent as a FractionTo rename a percent as a fraction, try to write it as a fractionwith a denominator of 100.

Renaming a Fraction as a PercentTo 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�


62 sixty-two

Musicians make patterns of sound to create music. Mathematicscan 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 forthmotion. For musical instruments to produce sound, something mustbe set in motion.

The rate at which a string, a drum head,or a column of air vibrates is called thefrequency. Higher frequency vibrationsproduce higher-pitched notes. Frequency is measured in Hertz (Hz), or “vibrationsper second.” The human ear can hearvibrations 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

➤

➤

64 sixty-four

Instrument Length and Pitch

Many instruments rely on a vibrating column of air to makesound. A longer column of air vibrates at a lower frequency andmakes a deeper- or lower-pitched note. Shorter vibrating aircolumns 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

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.

➤

➤

➤

66 sixty-six

Percussion Instruments

Drums are percussion instruments. The size of the instrumentaffects the pitch it can play. The size and tightness of the drumhead and the materials that the drum head is made from alsoaffect 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.

➤

sixty-seven 67

Stringed Instruments

The pitch of the notes that a stringed instrument can play isrelated 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.

➤

➤

➤

➤

The Piano

Looking closely at the way a piano works can help you see someof 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? Howhave 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.

➤

fractions - edison park elementary

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