5school1

Arabic Republic of Egypt Ministry of Education Book Sector

Mathematics Fifth Primary

2012-2013

‫غري مرصح بتداول هذا الكتاب‬ ‫خارج وزارة الرتبية والتعليم‬

5

Student's Book First Term

Authors Mr. Omar Fouad Gaballa Prof.Dr. Afaf Abo-ElFoutoh Salah Dr. ,ZZHT >HZÄ 9V\WOHLPS Mr. 4HOTV\K @HZZLY (S ROH[PLI Mr. :LYHÄLT ,SPHZ :RHUKLY

Revised by Mr. /\ZZLPU 4HOTV\K /\ZZLPU Counsellor of Mathematics M.O.E

Issued: 2009 D.N: 11224/2009 ISPN: 978-977-6294-15-8

‫ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ‬ Dear students:

It is extremely great pleasure to introduce the mathematics book for fifth primary.

We have been specially cautious to make learning mathematics enjoyable and useful since it has many practical applications in real life as well as in the other subjects. This gives you a chance to be aware of the importance of learning mathematics, to determine its value and to appreciate the mathematicians roles.

This book sheds new lights on the activities as a basic objective. Additionally, we have tried to introduce the subject simply and excitingly to help attaining mathematical knowledge as well as gaining the patterns of positive thinking skills which pave your way to creativity.

This book is divided into units, each unit contains lessons. Colors and pictures are effectively used to illustrate some mathematical concepts and the properties of figures. Lingual level of previous study has also been taken into consideration.

Our great interest here is to help you get the information by yourself in order to develop your self-learning skills.

Calculators and computer sets are used when there’s a need for. Exercises, practices, general exams, Activities, unit test, general tests, and final term tests attached with model answers have been included to help you review the curriculum completely.

Eventually, we hope getting on the right track for the benefits of our students as well as for our dearest Egypt hoping bright future to our dearest students.

Authors

Shorouk for Modern printing

Contents First Term: Unit one: Fractions Lesson One :

Approximating to the nearest hundredth and thousandth . .. . 4

Lesson Two:

Comparing fractions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10

Lesson Three:

Multiplication: Multiplying fractions and decimal numbers by 10, 100, 1000 15

Lesson Four:

Multiplying a fraction or a decimal number by an integer number .. 17

Lesson Five:

Multiplying common fractions . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19

Lesson Six:

Multiplying decimal fractions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 20

Lesson Seven: Division: (1) Dividing fractions .. .. .. .. .. .. .. .. .. .. .. .. .. .. 23 Lesson Eight: (2) Dividing fractions and decimal numbers by 10, 100, 1000 .. .. 25 Lesson Nine:

(3) Dividing an integer by a 3-digit number without having a remainder

27

Lesson Ten:

(4) Division by a decimal fraction and by a decimal number . .. 29

General Exercises .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36 Activity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39 Unit Test .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 40

Unit Two: Sets Lesson One :

What is a set? .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 42

Lesson Two:

Mathmatical expression of a set .. .. .. .. .. .. .. .. .. .. .. .. .. .. 44

Lesson Three: Belonging of an element to a set . .. .. .. .. .. .. .. .. .. .. .. .. .. 47 Lesson Four:

Types of sets .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 49

Lesson Five:

Equal sets . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 51

Lesson Six:

Inclusion and subsets .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 53

Lesson Seven: Intersection of two sets .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 56 Lesson Eight: Union of two sets .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 59

Lesson Nine:

The universal set .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 61

Lesson Ten:

The complement of a set .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 63

Lesson Eleven: The difference of two sets .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 65 General Exercises .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 67 Activity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 69 Unit Test .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 70

Unit Three: Geometry Lesson One

The Circle .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 72

Lesson Two:

Drawing a triangle given the lengths of its three sides .. .. .. 76

Lesson Three:

Drawing line segements from the vertices of a triangle perpendicular to its opposite sides .. .. .. .. .. .. .. .. .. .. .. .. .. 79

General Exercises .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 83 Activity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 85 Unit Test .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 86

Unit four: Probability Lesson One:

Experimental Probability .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 88

Lesson Two:

Theoretical Probability .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 91

General Exercises .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 94 Activity .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 95 Unit Test .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 96 General Exams .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 97 Answers

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 100

List of Symbols C

Set of counting numbers

G

Less than or equal to

E

Set of even numbers

=

Equal to

O

Set of odd numbers

≠

Not equal to

P

Set of prime numbers

Open curve

∅ or { }

Null set or empty set

Closed curve

∈

Belonging

Circle

∉

Not belonging

r

radius

⊂

Inclusion

r

Approximate ratio

⊄

Not inclusion

AB

Line segment A B

⋃

Union

AB

Ray A B

⋂

Intersection

AB

Straight line A B

U

Universal set

d

Angle

X`

Complement set

X-Y

X difference Y

m (d B) P (E)

Measure angle (B), m (dB) Probability of event P

N

Set of natural numbers

/

Congruent

>

More than

b

Triangle

H

More than or equal to

<

Less than

(X, Y)

Ordered pair X, Y

P (E)

(X, Y)

5HYLVLRQ

5HYLVLRQ

&RQYHUW WKH IROORZLQJ IUDFWLRQV WR IUDFWLRQV ZLWK GHQRPLQDWRU WKHQ FRQYHUW WR WKH GHFLPDO IRUP ([DPSOH

$

&

(

% ' )

:ULWH GRZQ WKH IROORZLQJ QXPEHUV LQ WKH IRUP RI GHFLPDO QXPEHUV DV VKRZQ

([DPSOH

$

&

%

'

:ULWH GRZQ WKH IROORZLQJ QXPEHUV LQ WKH IRUP RI DQ LQWHJHU QXPEHU DQG D IUDFWLRQ DV VKRZQ

([DPSOH

$

%

&

'

3XW WKH IROORZLQJ QXPEHUV LQ WKHLU DSSURSULDWH SODFHV LQVLGH WKH UHFWDQJOHV EHORZ

7KHQ FRPSOHWH

WR WKH QHDUHVW KXQGUHG

WR WKH QHDUHVW KXQGUHG

5I WR WKH QHDUHVW KXQGUHG \PM UI\ QK[ .QN\P 8ZQUI Za

0LGGOH

5HYLVLRQ &RPSOHWH WKH IROORZLQJ WDEOH 1XPEHU WR WKH QHDUHVW

1XPEHU

1XPEHU WR WKH QHDUHVW

1XPEHU WR WKH QHDUHVW

1XPEHU WR WKH QHDUHVW XQLW

0DWFK HDFK QXPEHU ZLWK LWV DSSUR[LPDWHG YDOXH WR WKH QHDUHVW XQLW

$SSUR[LPDWH WKH IROORZLQJ QXPEHUV WR WKH QHDUHVW WHQWK

$

'

%

(

&

)

)LQG WKH VXP RI WKH IROOZRLQJ RSHUDWLRQV WKHQ DSSUR[LPDWH WKH VXP DFFRUGLQJ WR WKH DSSUR[LPDWLRQ UHTXLUHG $

WR WKH QHDUHVW WHQWK

%

WR WKH QHDUHVW XQLW

&

WR WKH QHDUHVW WHQ

'

WR WKH QHDUHVW WHQWK

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

%TTVS\MQEXMRK XS XLI RIEVIWX LYRHVIHXL ERH XLSYWERHXL

)LUVW $SSUR[LPDWLQJ WR WKH QHDUHVW KXQGUHGWK

<RX ZLOO OHDUQ

- 7R DSSUR[LPDWH WR WKH QHDUHVW KXQGUHGWK - 7R DSSUR[LPDWH WR WKH QHDUHVW WKRXVDQGWK

7KLQN DQG GLVFXVV

7KH WHDFKHU DVNHG )DUHG DQG +XGD WR DSSUR[LPDWH WR WKH QHDUHVW KXQGUHGWK )DUHG DQVZHUHG

DQ

UHG

GWK

WK

WK

XV

QG

WKR

KX

WHQ

LWV

GV

V

UH

WHQ

QG

.H\ 7HUPV - $SSUR[LPDWLQJ - +XQGUHGWK - 7KRXVDQGWK

XQ

KX

7KH QXPEHU LV EHWZHHQ DQG

+RZHYHU LW LV FORVHU WR WKDQ 7KHUHIRUH DSSUR[LPDWHG WR WKH QHDUHVW KXQGUHGWK +RGD VXJJHVWHG 8VLQJ WKH RSSRVLWH IORZ FKDUW WR DSSUR[LPDWH WKH QXPEHU WR WKH QHDUHVW KXQGUHGWK

6WDUW

DSSUR[LPDWH WKH QXPEHU WR WKH QHDUHVW KXQGUHGWK

1R

,V WKH GLJLW ZULWWHQ DW WKH WKRXVDQGWK V SODFH OHVV WKDQ "

OHDYH RXW WKH WKRXVDQGWK V SODFH DQG WKH RWKHU GLJLWV WR WKH ULJKW

,QFUHDVH WKH GLJLW DW WKH KXQGUHGWK V SODFH E\ RQH DQG OHDYH RXW RWKHU GLJLWV WR LWV ULJKW

ZULWH WKH UHVXOW

5I

\PM UI\ QK[

VWRS

.QN\P 8ZQUI Za

<HV

3UDFWLFH $SSUR[LPDWH WKH IROORZLQJ QXPEHUV WR WKH QHDUHVW KXQGUHGWK

$

%

&

'

(

)

7KH FDSDFLW\ RI D FROD ERWWOH OLWHUV

OLWHUV WR WKH QHDUHVW KXQGUHGWK

$ PLFURPHWHU LV D GHYLFH XVHG IRU SUHFLVH PHDVXULQJ ,W LV XVHG WR PHDVXUH WKH WKLFNQHVV RI D SDSHU DQG LW ZDV PP &RPSOHWH WKH WKLFNQHVV RI WKH SDSHU PP WR WKH QHDUHVW KXQGUHGWK

WKLQN 7KH UHDGLQJ RI WKH RSSRVLWH JDV PHWHU FXELF PHWHU DSSUR[LPDWH WR WKH QHDUHVW

KXQGUHGWK

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

6HFRQG $SSUR[LPDWLQJ WR WKH QHDUHVW WKRXVDQGWK

7KLQN DQG GLVFXVV

7KH WHDFKHU DVNHG +HQG WR DSSUR[LPDWH WKH QXPEHU WR WKH QHDUHVW WKRXVDQGWK

RX

QG VD RX

VD

QG KX

WK

UH

QG

GW

WHQ

WK

WHQ

XQ

WK

LWV

V WHQ

WK

K

WK

+HQG DQVZHUHG

7KH QXPEHU LV EHWZHHQ and +RZHYHU LW LV FORVHU WR WKDQ 7KHUHIRUH

3UDFWLFH

6WDUW

,Q WKH RSSRVLWH ILJXUH &RPSOHWH WKH IORZ FKDUW WR UHSUHVHQW WKH VWHSV IROORZHG WR DSSUR[LPDWH WR WKH QHDUHVW WKRXVDQGWK )\EQTPI

DSSUR[LPDWH WKH QXPEHU WR WKH QHDUHVW WKRXVDQGWK

1R

$SSUR[LPDWH WR WKH QHDUHVW WKRXVDQGWK

6IWYPX 7KH GLJLW DW WKH WKRXVDQGWK SODFH LV WKH GLJLW DW WKH WHQ WKRXVDQGWK V SODFH LV ZKLFK LV JUHDWHU WKDQ 7KHUHIRUH ZH LQFUHDVH WKH GLJLW DW WKH WKRXVDQGWK V SODFH E\ RQH 7KHQ LW ZLOO EH

,V WKH GLJLW ZULWWHQ DW WKH SODFH

<HV

OHVV WKDQ " /HDYH RXW WKH GLJLW DW WKH SODFH DQG WKH RWKHU GLJLWV WR LWV ULJKW

,QFUHDVH WKH GLJLW DW WKH SODFH E\ RQH DQG OHDYH RXW RWKHU GLJLWV WR LWV ULJKW

ZULWH WKH UHVXOW

VWRS

2SXI XLEX

5I

:KHQ DSSUR[LPDWLQJ WR WKH QHDUHVW WKRXVDQGWK WKH UHVXOW RI DSSUR[LPDWLQJ PXVW LQFOXGH GHFLPDO GLJLWV HYHQ LI WKH GLJLW DW WKH WKRXVDQGWK V SODFH LV HTXDO WR ]HUR \PM UI\ QK[ .QN\P 8ZQUI Za

6

3UDFWLFH &RPSOHWH WKH IROORZLQJ WDEOH 1XPEHU

DSSUR[LPDWH WR WKH QHDUHVW KXQGUHGWK

DSSUR[LPDWH WR WKH QHDUHVW WKRXVDQGWK

&RPSOHWH

(DFK WDEOHW FRQWDLQV VRPH LQJUHGLHQWV DV VKRZQ LQ WKH IROORZLQJ WDEOH 7KH OHQJWK RI D FHOO PHDVXUHG E\ D PLFURVFRSH PP

FRPSRXQG

ZHLJKW LQ JP

ZHLJKW DSSUR[LPDWHG WR WKH QHDUHVW WKRXVDQGWK

$

%

&

PP

WR WKH QHDUHVW WKRXVDQGWK )\EQTPI

*LYHQ WKDW / 0 (VWLPDWH WKH VXP RI / 0 WKHQ FRPSDUH \RXU HVWLPDWLRQ ZLWK WKH VXP WR WKH QHDUHVW KXQGUHGWK

6IWYPX

/ 0 / 0

Shorouk for Modern printing

(VWLPDWH WKH UHVXOW

(VWLPDWH RI / (VWLPDWH RI 0 (VWLPDWH RI / 0

6LQFH WKH YDOXH LV FORVHU WR WKH HVWLPDWH WKH YDOXH LV DFFHSWDEOH

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

7

7KLQN $KPHG ERXJKW VRPH VWXII IURP D VKRSSLQJ FHQWHU &DQ \RX HVWLPDWH WKH WRWDO RI ZKDW KH SDLG LQ /( DSSUR[LPDWLQJ LW WR WKH QHDUHVW WHQ SRXQGV" 0DNH VXUH WKDW \RXU HVWLPDWLRQ LV DFFHSWDEOH IRU WKH DFWXDO VXP

5HFLHSW SULFH /(

VRDS

ZDVKLQJ SRZGHU

SHUIXPHV

PHDW

FORWKHV

YHJHWDEOHV

)\IVGMWIW $SSUR[LPDWH WR WKH QHDUHVW $

%

KXQGUHGWK

WKRXVDQGWK

&KRRVH WKH FRUUHFW DQVZHU IURP SDUDQWKHVHV $

WR WKH QHDUHVW

%

WHQWK KXQGUHGWK WKRXVDQGWK

WR WKH QHDUHVW

WHQWK KXQGUHGWK WKRXVDQGWK

& WR WKH QHDUHVW KXQGUHGWK

'

WR WKH QHDUHVW KXQGUHGWK

5I

\PM UI\ QK[

8

.QN\P 8ZQUI Za

LWHP

(

WR WKH QHDUHVW KXQGUHGWK

)

PHWHUV WR WKH QHDUHVW FHQWLPHWHU

:ULWH GRZQ WKH VPDOOHVW GHFLPDO IUDFWLRQ WKDW LQFOXGHV WKH GLJLWV WKHQ DSSUR[LPDWH WKDW QXPEHU WR WKH QHDUHVW KXQGUHGWK DQG QHDUHVW WKRXVDQGWK &RPSOHWH $

7KH QXPEHU WR WKH QHDUHVW

%

7KH GLIIHUHQFH EHWZHHQ

&

'

WR WKH QHDUHVW WHQWK

WR WKH QHDUHVW KXQGUHGWK

· WR WKH QHDUHVW KXQGUHGWK

$ URDG H[WHQGV IRU PHWHUV )LQG

LWV OHQJWK LQ NLORPHWHUV DSSUR[LPDWLQJ

WKH UHVXOW WR WKH QHDUHVW KXQGUHGWK &RPSOHWH $

GD\V

ZHHNV

%

KRXUV

GD\V

&

NLORPHWHUV

NLORPHWHUV

'

PRQWKV

\HDUV

*LYHQ WKDW

;

<

)LQG ; < DSSUR[LPDWLQJ WKH VXP WR WKH QHDUHVW KXQGUHGWK (VWLPDWH WKH VXP RI ; < ,V \RXU HVWLPDWLRQ DFFHSWDEOH" ([SODLQ

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

9

'SQTEVMRK JVEGXMSRW

7KLQN DQG GLVFXVV

<RX ZLOO OHDUQ - 7R FRPSDUH IUDFWLRQV

7KH IUDFWLRQ JUDSKHG E\ WKH FRORUHG VHFWLRQ

.H\ 7HUPV - JUHDWHU WKDQ !

- OHVV WKDQ - (TXDO WR

SXW ! RU

WKH FRORUHG VHFWLRQ SXW ! RU

7KH IUDFWLRQ JUDSKHG E\

:KHQ WKH QXPHUDWRU LV WKH VDPH LQ WKH WZR IUDFWLRQV

:KHQ WKH GHQRPLQDWRU LV WKH VDPH LQ WKH WZR IUDFWLRQV

!

EHFDXVH !

EHFDXVH !

3UDFWLFH 5I

SXW ! RU WR KDYH D WUXH VHQWHQFH

\PM UI\ QK[

.QN\P 8ZQUI Za

0HQWDO 0DWK

5HDUUDQJH WKH IROORZLQJ IUDFWLRQV DVFHQGLQJO\ IURP WKH VPDOOHVW WR WKH JUHDWHVW

5HDUUDQJH WKH IROORZLQJ IUDFWLRQV GHVFHQGLQJO\ IURP WKH JUHDWHVW WR WKH VPDOOHVW

)LQG WKH SRVVLEOH YDOXHV RI ; ZKLFK VDWLVI\ WKH IROORZLQJ UHODWLRQV ZKHUH ; LV DQ LQWHJHU $

[

%

[

5HDUUDQJH WKH IROORZLQJ IUDFWLRQV RQFH DVFHQGLQJO\ DQG WKH RWKHU GHVFHQGLQJO\ <RX FDQ XVH WKH QXPEHU OLQH

&RPSDULQJ WZR IUDFWLRQV ZLWK GLIIHUHQW GHQRPLQDWRUV :KLFK LV JUHDWHU

RU "

JUDSKHG E\ WKH

FRORUHG VHFWLRQ

JUDSKHG E\ WKH

FRORUHG VHFWLRQ

WKH IUDFWLRQ

WKH IUDFWLRQ

*

*

!

EHFDXVH !

L H !

2SXI XLEX

*

*

L H 7R FRPSDUH WZR IUDFWLRQV ZLWK GLIIHUHQW GHQRPLQDWRUV ZH ILQG WKHLU FRPPRQ GHQRPLQDWRU ,Q RWKHU ZRUGV ZH ILQG WKH OHDVW FRPPRQ GHQRPLQDWRU /&'

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

:KLFK LV JUHDWHU

RU "

/&' RI WKH WZR GHQRPLQDWRUV DQG *

*

*

*

*

7KHQ

!

L H

!

3UDFWLFH &RPSDUH WKH IROORZLQJ IUDFWLRQV $

%

'

(

&

)

&RPSDULQJ IUDFWLRQV DQG GHFLPDOV <RX FDQ FRQYHUW FRPPRQ IUDFWLRQV LQWR GHFLPDOV DQG \RX FDQ FRPSDUH WKHP DV \RX KDYH OHDUQHG LQ WKH DSSUR[LPDWLQJ OHVVRQ XVLQJ \RX FDOFXODWRU

)\EQTPI

2SXI XLEX 7KH VPDOOHVW QXPEHU LV ZKLOH WKH JUHDWHVW QXPEHU LV

7R FRPSDUH

5HDUUDQJH WKH IROORZLQJ QXPEHUV DVFHQGLQJO\

L H ZH FRPSDUH

! !

:H FRPSDUH

M

5I

\PM UI\ QK[

12

.QN\P 8ZQUI Za

7R FRPSDUH SURFHHG DV IROORZV

WKHQ !

7KHUHIRUH WKH DVFHQGLQJ 5HDUUDQJH LV 7KDW 5HDUUDQJH FDQ EH VKRZQ RQ WKH QXPEHU OLQH

5HDUUDQJH WKH IROORZLQJ IUDFWLRQV ILUVW

UHDUUDQJH

VHFRQG UHDUUDQJH

GHVFHQGLQJO\

DVFHQGLQJO\

)\IVGMWIW 6WDWH ZKDW WKH FRORUHG VHFWLRQ UHSUHVHQWV LQ HDFK ILJXUH WKHQ UHDUUDQJH WKH IUDFWLRQV DVFHQGLQJO\ $

$VFHQGLQJ 5HDUUDQJH

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

13

%

$VFHQGLQJ 5HDUUDQJH SXW RU

$ &

(

%

!

!

'

!

)

)LQG WKH YDOXHV RI D E DQG F LI

D

$

%

E

5HDUUDQJH WKH IROORZLQJ QXPEHUV DVFHQGLQJO\

5I

$

%

\PM UI\ QK[

.QN\P 8ZQUI Za

&

F

1YPXMTPMGEXMSR

1YPXMTP]MRK JVEGXMSRW ERH HIGMQEP RYQFIVW F] /HW V ZRUN WRJHWKHU

<RX ZLOO OHDUQ - 7R PXOWLSO\ IUDFWLRQV

:RUN ZLWK \RXU FODVVPDWH XVLQJ WKH FDOFXODWRU

DQG GHFLPDO QXPEHUV E\

(QWHU RQ \RXU FDOFXODWRU DV LOOXVWUDWHG RQ WKH RSSRVLWH ILJXUH WKHQ PXOWLSO\ WKDW QXPEHU E\ 1RWLFH WKH FKDQJH RI WKH SRVLWLRQ RI WKH GHFLPDO SRLQW ZLWKLQ WKH UHVXOW LQSXWV

RXWSXWV

+RZ PDQ\ SODFHV KDV WKH GHFLPDO SRLQW PRYHG WR WKH ULJKW GLUHFWLRQ" (QWHU RQ \RXU FODFXODWRU WKHQ PXOWLSO\ WKDW QXPEHU E\ 1RWLFH WKH WKH FKDQJH RI SRVLWLRQ RI WKH GHFLPDO SRLQW ZLWKLQ WKH UHVXOW LQSXWV

.H\ 7HUPV - 'HFLPDO IUDFWLRQ - 'HFLPDO QXPEHU

RXWSXWV

+RZ PDQ\ SODFHV KDV WKH GHFLPDO SRLQW PRYHG WR WKH ULJKW GLUHFWLRQ" 6WDUW

7KLQN

1XPEHU

,I ZH PXOWLSO\ WKDW QXPEHU E\ +RZ PDQ\ SODFHV ZLOO WKH GHFLPDO SRLQW PRYH WR WKH ULJKW GLUHFWLRQ" 1RWLFH WKH IORZ FKDUW LOOXVWUDWLQJ WKH RSHUDWLRQV ZLWKLQ WKDW OHVVRQ

Shorouk for Modern printing

* PRYH WKH GHFLPDO SRLQW RQH SODFH WR WKH ULJKW

* PRYH WKH GHFLPDO SRLQW SODFHV WR WKH ULJKW

* PRYH WKH GHFLPDO SRQW SODFHV WR WKH ULJKW

ZKHQ WKH QXPEHU RI SODFHV LV QRW HQRXJK SXW ]HURV DW WKH RWKHU SODFHV ZULWH GRZQ WKH QXPEHU

VWRS

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

3UDFWLFH &RPSOHWH

¯

¯

¯

¯

)\IVGMWIW &RPSOHWH $

¯

%

¯

&

¯

'

¯

(

¯

)

¯

*

¯

+

¯

,

¯

-

¯

.

¯

/

¯

&KRRVH WKH FRUUHFW DQVZHU IURP WKH SDUHQWKHVHV $

%

¯

&

¯

¯

3XW ! RU LQ WKH HPSW\ VSDFHV $

¯

¯

%

¯

¯

&

¯

¯

&RPSOHWH

5I

$

.LORJUDPV

JUDPV

%

/(

SLDVWHUV

&

GHFLPHWHUV

FHQWLPHWHUV

'

.LORPHWHUV

PHWHUV

\PM UI\ QK[

.QN\P 8ZQUI Za

1YPXMTP]MRK E JVEGXMSR SV E HIGMQEP RYQFIV F] ER MRXIKIV /HW V ZRUN WRJHWKHU 7KH PDWK WUHDFKHU DVNHG WKH JURXSV LQ WKH FODVV WR ILQG WKH DUHD RI D UHFWDQJOH LQ ZKLFK WKH OHQJWKV RI LWV VLGHV DUH FP DQG FP (DFK JURXS GUHZ WKH UHFWDQJOH DQG FDOFXODWHG LWV DUHD LQ D GLIIHUHQW PHWKRG )LOO LQ WKH EODQNV WR ILQG WKH DUHD

<RX ZLOO OHDUQ - 7R PXOWLSO\ D IUDFWLRQ RU D GHFLPDO QXPEHU E\ DQ LQWHJHU

7KH ILUVW JURXS

FP

¯

$UHD

¯

FP

FP

7KH VHFRQG JURXS $UHD

¯

¯

FP

.H\ 7HUPV

- 'HFLPDO IUDFWLRQ - 'HFLPDO QXPEHU

FP FP

7KH WKLUG JURXS $UHD

¯

¯ ¯

¯

FP FP FP

FP

7KH WHDFKHU VDLG 'HVSLWH WKH GLIIHUHQFH RI WKH VROXWLRQV DOO WKH DQVZHUV DUH FRUUHFW ([SODLQ WKDW E\ FRPSOHWLQJ WKH IROORZLQJ VHQWHQFHV ¯ ¯ ZKDW GR \RX REVHUYH"

¯ ¯ ¯ ZKDW GR \RX REVHUYH" Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

3UDFWLFH )LQG WKH SHULPHWHU RI HDFK RI WKH IROORZLQJ VKDSHV

P

FP

)\IVGMWIW , I * ZLWKRXW PXOWLSO\LQJ

* WKHQ FRPSOHWH WKH IROORZLQJ

$

¯

%

¯

&

¯

'

¯

(

¯

)

¯

*

¯

+

¯

,

¯

-

¯ ¯ ¯ ¯

)LQG WKH UHVXOW RI HDFK RI WKH IROORZLQJ

¯

$

¯

%

¯

&

'

¯

(

¯

)

¯ ¯

7 KH SULFH RI D EDU RI FKRFRODWH LV /( ZKDW LV WKH FRVW RI EDUV RI WKH VDPH NLQG" $ KPHG ERXJKW FDQV RI MXLFH 7KH SULFH RI HDFK FDQ ZDV /( :KDW LV WKH WRWDO FRVW RI WKH MXLFH" +RZ PXFK ZRXOG WKH VHOOHU SD\ EDFN WR $KPHG LI KH SDLG KLP /( "

5I

\PM UI\ QK[

.QN\P 8ZQUI Za

1YPXMTP]MRK GSQQSR JVEGXMSRW

7KLQN DQG GLVFXVV

6RDG ZDQWHG WR GHGXFH WKH SURGXFW RI ¯ 6KH XVHG

D SLHFH RI SDSHU DQG GLYLGHG LW DFFRUGLQJ WR WKH IROORZLQJ

<RX ZLOO OHDUQ - 0XOWLSOLFDWLRQ RI FRPPRQ IUDFWLRQV

VWHSV

L H

RI

¯

)\EQTPI )LQG WKH SURGXFW RI

¯

RI

L H

*

¯

*

,Q JHQHUDO

¯

¯

*

*

DQG VR RQ

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

1YPXMTP]MRK HIGMQEP JVEGXMSRW )LUVW 0XOWLSO\LQJ D GHFLPDO IUDFWLRQ E\ D GHFLPDO QXPEHU

<RX ZLOO OHDUQ - 7R PXOWLSO\ D GHFLPDO IUDFWLRQ E\ D GHFLPDO QXPEHU

1RWH WKDW

WHQWKV

*

*

* * * *

*

7KLQN DQG GLVFXVV

,Q WKH RSSRVLWH ILJXUH 7KH OHQJWK RI WKH UHFWDQJOH $%&' LV WHQWKV DQG LWV ZLGWK LV WHQWKV 7KH DUHD RI WKH UHFWDQJOH DW WKH ILJXUH 7KH QXPEHU RI VTXDUH XQLWV LQVLGH WKH VKDSH KXQGUHGWKV

'

$

&RPSOHWH XVLQJ WKH VKDSH

&

%

7KH UHFWDQJOH

OHQJWK

ZLGWK

DUHD

$%&'

¯

*

6+$3(

¯

6+$3(

¯

3UDFWLFH

¯ ¯

)LQG WKH UHVXOW RI

¯

P RQH GHFLPDO QXPEHU P RQH GHFLPDO QXPEHU P 7ZR GHFLPDO QXPEHUV

\PM UI\ QK[

.QN\P 8ZQUI Za

5I

2EVHUYH DQG &RPSOHWH *

¯ ¯ RU ¯

*

6HFRQG (VWLPDWLQJ WKH SURGXFWV RI PXOWLSO\LQJ D IUDFWLRQ RU D GHFLPDO QXPEHU E\ D IUDFWLRQ RU D GHFLPDO QXPEHU

)\EQTPI (VWLPDWLQJ

)LQG WKH UHVXOW ¯ $

*

¯

%

&

LV HVWLPDWHG WR

LV HVWLPDWHG WR

¯

7KH HVWLPDWLRQ RI WKH

UHVXOW RI ¯

WKH UHVXOW

B 1XPEHUV FDQ EH PXOWLSOLHG DV LQWHJHUV DQG ZH ILQG WKH SRVWLWLRQ RI WKH GHFLPDO SRLQW LQ WKH UHVXOW

3UDFWLFH )LUVW HVWLPDWH WKH IROORZLQJ SURGXFWV WKHQ FRPSDUH \RXU HVWLPDWLRQ WR WKH DFWXDO UHVXOW $

¯

(VWLPDWHG UHVXOW

DFWXDO UHVXOW

%

¯

(VWLPDWHG UHVXOW

DFWXDO UHVXOW

&

¯ (VWLPDWHG UHVXOW

DFWXDO UHVXOW

)\IVGMWIW )LQG WKH UHVXOW $ '

¯

%

¯

(

¯

&

¯

¯

)

¯

&RPSDUH WKH SURGXFWV RI WKH IROORZLQJ E\ SXWWLQJ RU ! RU $

¯

¯

%

¯

¯

&

¯

¯

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

)LQG WKH UHVXOW

*

¯

¯

3XW ! RU RU WR PDNH WKH IROORZLQJ VHQWHQFHV WUXH $

¯

¯

%

¯

¯

&

¯

¯

'

¯

¯

¯

)LQG WKH UHVXOW $

*

%

*

8VH WKH UHVXOWHG SURGXFWV WR ILQG WKH YDOXH RI )LUVW ¯ ¯ 6HFRQG ¯ ¯ ZKDW GR \RX REVHUYH" )LQG WKH UHVXOWV RI $

¯

%

¯

&

¯

'

¯

,I WKH SULFH RI RQH PHWHU RI FORWK LV / ( ZKDW LV WKH FRVW RI PHWHUV RI FORWK" $ FDU FRYHUV HTXDO GLVWDQFHV LQ HTXDO WLPHV +RZ

PDQ\ NLORPHWHUV GRHV LW FRYHU LQ KRXUV DQG

PLQXWHV LI LWV VSHHG LV .LORPHWHUV SHU KRXU" (VWLPDWH WKH SURGXFWV RI WKH IROORZLQJ RSHUDWLRQV

WKHQ FRPSDUH \RXU HVWLPDWLRQ WR WKH DFWXDO

UHVXOW $

¯

%

¯

&

¯

6DOZD ERXJKW D SLHFH RI FORWK ZLWK PHWHUV LQ OHQJWK ,I WKH SULFH RI RQH PHWHU

5I

RI FORWK ZDV / ( ILQG WKH FRVW RI FORWK DSSUR[LPDWLQJ LW WR WKH QHDUHVW

\PM SRXQG UI\ QK[ .QN\P 8ZQUI

Za

(MZMWMSR

(MZMHMRK *VEGXMSRW

7KLQN DQG GLVFXVV

<RX ZLOO OHDUQ

- 7R GLYLGH DQ LQWHJHU

6DP\ ZDQWHG WR ILQG WKH

TXRWLHQW RI GLYLGLQJ E\

+H XVHG D UHFWDQJXODU SLHFH RI SDSHU DQG GLYLGHG LW LQWR HTXDO SDUWV WKHQ

E\ D FRPPRQ IUDFWLRQ - 7R GLYLGH D FRPPRQ IUDFWLRQ E\ DQ LQWHJHU - 7R GLYLGH D FRPPRQ IUDFWLRQ E\ DQRWKHU FRPPRQ IUDFWLRQ

GLYLGHG WKH SLHFH RI SDSHU LQWR HTXDO SDUWV $IWHU GUDZLQJ VDP\ QRWLFHG WKDW

.H\ 7HUPV - )UDFWLRQ - 'LYLVRQ

·

:RXOG \RX DJUHH ZLWK 6DP\" ZK\"

2SXI XLEX

¯

Ȧ · ¯

)\EQTPI 'LYLGH ·

2QH XQLW

2QH XQLW

'LYLGH HDFK XQLW LQWR HTXDO SDUWV WKHQ FRQVWUXFW SDUWV

ZKHUH HYHU\ SDUW LV HTXDO WR

XQLW

·

L H · *

Shorouk for Modern printing

XQLW

XQLW

XQLW

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

)\EQTPI

·

'LYLGH

XQLW

5HVXOW

·

*

:KDW GR \RX REHUYH"

7KLQN &DQ \RX ILQG WKH TXRWLHQW RI ·

3UDFWLFH )LQG WKH TXRWLHQW RI $

·

%

·

&

·

'

·

&RPSOHWH $

· * · ·

%

· · *

'LYLGH

5I

$

·

%

·

&

·

'

·

(

·

)

·

\PM UI\ QK[

.QN\P 8ZQUI Za

(MZMHMRK JVEGXMSRW ERH HIGMQEP RYQFIVW F]

<RX ZLOO OHDUQ

7KLQN DQG GLVFXVV

- 7R GLYLGH D GHFLPDO

$ FKDULW\ GRQRU ZDQWHG WR GLYLGH / ( E\ IDPLOLHV HTXDOO\ :KDW LV WKH VKDUH RI HDFK IDPLO\" 7KH VKDUH RI HDFK IDPLO\ ·

/(

IUDFWLRQ RU D GHFLPDO QXPEHU E\

*

:KDW GR \RX REHUYH UHJDUGLQJ WKH QXPEHU RI SODFHV PRYHG E\ WKH GHFLPDO SRLQW DQG LQ ZKLFK GLUHFWLRQ" <RX FDQ DOVR GLVFRYHU WKH SDWWHUQ RI FKDQJLQJ WKH SRVLWLRQ RI WKH GHFLPDO SRLQW WR WKH OHIW ZKLOH GLYLGLQJ E\ XVLQJ WKH FDOFXODWRU 8VLQJ WKH FDOFXODWRU

([SODQDWLRQ

·

*

·

*

·

*

+DYH \RX REVHUYHG D VSHFLILF SDWWHUQ" ZKDW LV WKDW SDWWHUQ" 6DHHG VDLG :KHQ GLYLGLQJ E\ WKH GHFLPDO SRLQW PRYHV RQH SODFH WR WKH OHIW :KHQ GLYLGLQJ E\ WKH GHFLPDO SRLQW PRYHV WZR SODFHV WR WKH OHIW :KHQ GLYLGLQJ E\ WKH GHFLPDO SRLQW PRYHV WKUHH SODFHV WR WKH OHIW

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

3UDFWLFH &RPSOHWH

·

·

·

·

)\IVGMWIW &RPSOHWH $

·

%

·

&

·

'

·

(

·

)

·

&KRRVH WKH FRUUHFW DQVZHU IURP WKH SDUHQWKHVHV $

·

%

·

&

·

3XW WKH VXWLDEOH VLJQ RU ! RU LQ HDFK RI WKH IROORZLQJ $

· ·

%

· *

&RPSOHWH $ & (

JUDPV PHWHUV NLORJUDPV

NJV

NLORPHWHUV WRQV

%

PHWHUV FP

'

PP FP

)

FP OLWHUV

$ FDU FRQVXPHV RQH OLWHU RI JDVROLQH WR WUDYHO .LORPHWHUV +RZ PDQ\ OLWHUV

5I

RI JDVROLQH GRHV LW QHHG WR WUDYHO D GLVWDQFH RI .LORPHWHUV"

\PM UI\ QK[

.QN\P 8ZQUI Za

(MZMHMRK ER MRXIKIV F] E HMKMX RYQFIV [MXLSYX LEZMRK E VIQEMRHIV 7KLQN DQG GLVFXVV

<RX ZLOO OHDUQ - WR GLYLGH DQ LQWHJHU

7KH OLEUDULDQ DW WKH VFKRRO DVNHG WKH VWXGHQWV WR KHOS KLP DUUDQJH WKH OLEUDU\ +H DVNHG WKHP WR SXW ERRNV RQ VKHOYHV HTXDOO\

E\ D GLJLW QXPEHU ZLWKRXW KDYLQJ D UHPDLQGHU

$KPHG WKRXJKW +RZ PDQ\ ERRNV VKRXOG ZH SXW RQ HYHU\

VKHOI" 6DPHLU DQVZHUHG :H GLYLGH E\ WKHQ WKH TXRWLHQW LV

ERRNV DQG WKH UHPDLQGHU LV ERRNV $KPHG VDLG L H *

GLYLGHQHG

GLYLVRU

TXRWLDQW

UHPDLQGHU

.H\ 7HUPV - 5HPDLQGHU

2SXI XLEX 7KH GLYLGHQHG WKH GLYLVRU * WKH TXRWLDQW WKH UHPDLQGHU 7KH UHPDLQGHU LV DOZD\V OHVV WKDQ WKH GLYLVRU

3UDFWLFH QXPEHU RI GLYLVLRQ GLYLGHQHG GLYLVRU TXRWLHQW UHPDQLGHU RSHUDWLRQ

5HODWLRQ DPRQJ WKH HOHPHQWV RI GLYLVRQ

([DPSOH

·

¯

·

·

·

¯

·

]HUR

·

¯

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

2SXI XLEX ,I WKH UHPDLQGHU LV ]HUR WKHQ WKH GLYLVRQ LV ZLWKRXW D UHPDLQGHU )\EQTPI )LQG WKH TXRWLHQW RI ·

5HVXOW

(VWLPDWLQJ WKH TXRWLHQW WR FKHFN WKH UHDVRQDELOLW\ RI WKH DQVZHU HVWLPDWLRQ

GLYLGHQHG

P

GLYLVRU

P

7KH UHDVRQDEOH HVWLPDWLRQ WR WKH TXRWLHQW LV

'R WKH GLYLVLRQ 7KH WHQV GLJLW

¯ ¯

¯ ¯

7KH XQLWV GLJLW

¯ ¯

¯

· 7KH TXRWLHQW LV FORVH WR WKH HVWLPDWLRQ 7KH DQVZHU LV UHDVRQDEOH

)\IVGMWIW :LWKRXW GRLQJ WKH GLYLVRQ FKRRVH WKH FRUUHFW DQVZHU IURP WKH SDUHQWKHVHV $

·

%

·

&

·

)LQG WKH TXRWLHQW RI WKH IROORZLQJ $

·

%

·

&

·

'

·

7KH UHVXOW RI PXOWLSO\LQJ QXPEHUV LV ,I RQH RI WKHP LV ILQG WKH RWKHU QXPEHU

5I

$Q RZQHU RI SDFNLQJ IRRG IDFWRULHV ZDQWHG WR SDFN NLORJUDPV RI VXJHU

\PM HTXDOO\ LQ SDFNV :KDW LV WKH ZHLJKW RI HDFK SDFN" UI\ QK[ .QN\P 8ZQUI

Za

(MZMWMSR F] E HIGMQEP JVEGXMSR ERH F] E HIGMQEP RYQFIV )LUVW 'LYLVRQ E\ D GHFLPDO IUDFWLRQ ZLWKRXW D UHPDLQGHU

<RX ZLOO OHDUQ - 7R GLYLGH D GHFLPDO

IUDFWLRQ E\ D GHFLPDO IUDFWLRQ

7KLQN DQG GLVFXVV

- 7R GLYLGH D GHFLPDO )LQG WKH TXRWLHQW RI WKH IROORZLQJ )LUVW · 5HVXOW

· ·

¯

6HFRQG · 5HVXOW

· ·

¯

QXPEHU E\ D GHFLPDO IUDFWLRQ

- 7R ILQG WKH TXRWLHQW RI LQILQLWH GLYLVRQ WR WKH QHDUHVW WHQWK RU QHDUHVW KXQGUHGWK

.H\ 7HUPV - ,QILQLWH GLYLVLRQ

'UDZ WKH KXQGUHGV FKDUW DQG LOOXVWUDWH WKH RSHUDWLRQ RI GLYLVRQ RQ LW

· %RWK WKH GLYLGHQHG DQG WKH GLYLVRU FDQ EH PXOWLSO\HG E\ WR PDNH WKH GHQRPLQDWRU DQ LQWHJHU

*

*

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

)\EQTPI )LQG WKH TXRWLHQW LQ HDFK RI WKH IROORZLQJ $

%

·

·

5HVXOW $

7R ILQG WKH TXRWLHQW RI · PDNH WKH GLYLVRU DQ

LQWHJHU E\ PXOWLSO\LQJ ERWK RI WKH GLYLGHQHG DQG WKH

GLYLVRU E\

*

7KH TXRWLHQW

*

)LQG WKH TXRWLHQW RI · (VWLPDWLRQ

%

P

7KH HVWLPDWLRQ LV

P

7KH DQVZHU LV DFFHSODEOH

7R FRQYHUW WKH GLYLVRU LQWR DQ LQWHJHU PXOWLSO\ ERWK WKH GLYLGHQHG DQG WKH

GLYLVRU E\ 7KH TXRWLHQW (VWLPDWLRQ

*

*

7KH GLYLGHQHG

P

7KH GLYLVRU

P

7KH HVWLPDWHG TXRWLHQW

* P

7KHUHIRUH WKH DQVZHU LV DFFHSWDEOH

3UDFWLFH 'LYLGH · DQG FKHFN UHDVRQDELOLW\ RI WKH TXRWLHQW

5HVXOW

·

* *

5I

\PM UI\ QK[ .QN\P 8ZQUI Za

30

(VWLPDWLRQ 7KH GLYLGHQHG

P

7KH GLYLVRU

P

7KH HVWLPDWHG TXRWLHQW

$QVZHU

:LWKRXW GRLQJ WKH GLYLVLRQ HVWLPDWH WKH TXRWLHQW LQ HDFK RI WKH

IROORZLQJ $

%

·

·

&KHFN WKH UHDVRQDELOLW\ RI \RXU HVWLPDWLRQ XVLQJ \RXU FDOFXODWRU

3UDFWLFH )LQG WKH TXRWLHQW LQ HDFK RI WKH IROORZLQJ $

·

%

·

·

·

&

·

'

(

·

)

)LQG WKH TXRWLHQW LQ HDFK RI WKH IROORZLQJ $

·

%

·

)LQG D QXPEHU ZKHQ PXOWLSOLHG E\ WKHQ WKH UHVXOW LV :KLFK RI WKH IROORZLQJ UHODWLRQV LV WUXH DQG ZKLFK LV IDOVH" ZKDW GR

\RX FRQFOXGH" $

¯ ¯

%

· ·

$ EXQGOH RI SDSHU KDV D KHLJKW RI FP ,I DOO LWV SDSHUV ZHUH RI HTXDO WKLFNQHVV ZKHUH WKH WKLFNQHVV RI HDFK SDSHU ZDV PLOLPHWHUV ILQG KRZ PDQ\ SDSHUV GRHV WKH EXQGOH LQFOXGH"

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

31

6HFRQG GLYLVLRQ E\ D GHFLPDO QXPEHU ZLWKRXW KDYLQJ D UHPDLQGHU

)\EQTPI )LQG WKH TXRWLHQW RI HDFK RI WKH IROORZLQJ $

%

·

·

7KHQ GLVFXVV WKH UHDVRQDELOLW\ RI \RXU DQVZHU

5HVXOW $

· ·

P P

7KH HVWLPDWHG TXRWLHQW LV 7KH TXRWLHQW LV

WKH TXRWLHQW LV FORVH WR WKH HVWLPDWLRQ 7KHUHIRUH WKH DQVZHU LV UHDVRQDEOH %

· ·

(VWLPDWLRQ

P

P

P P

7KH TXRWLHQW LV FORVH WR WKH HVWLPDWLRQ 7KHUHIRUH WKH DQVZHU LV UHDVRQDEOH

3UDFWLFH &RQYHUW WKH IROORZLQJ WR WKH GHFLPDO IRUPV $

&

5I

\PM UI\ QK[

32

%

'

.QN\P 8ZQUI Za

&RPSOHWH WKH IROORZLQJ WR HVWLPDWH WKH TXRWLHQW LQ HDFK RI WKH IROORZLQJ RSHUDWLRQV $

*

7KH HVWLPDWLRQ

%

*

*

*

7KH HVWLPDWLRQ

* *

)LQG WKH TXRWLHQW RI HDFK RI WKH IROORZLQJ $

·

%

·

&

·

'

·

(

·

)

·

)LQG WKH UHVXOW $

·

%

·

7KH OHQJWK RI D UROO RI FORWK LV PHWHUV ,W ZDV GLYLGHG LQWR HTXDO SDUWV ZKHUH WKH OHQJWK RI HDFK SDUW LV PHWHUV )LQG WKH QXPEHU RI WKHVH SDUWV :LWKRXW GRLQJ WKH PDWKHPDWLFDO RSHUDWLRQV HVWLPDWH WKH UHVXOW RI HDFK RI WKH IROORZLQJ $

* ·

%

· *

)LQG WKH TXRWLHQW $

·

%

·

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

33

7KLUG )LQGLQJ WKH TXRWLHQW RI LQILQLWH GLYLVLRQ WR WKH QHDUHVW WHQWK DQG QHDUHVW KXQGUHGWK

)\EQTPI :ULWH GRZQ HDFK IUDFWLRQ LQ WKH GHFLPDO IRUP $

%

WR WKH QHDUHVW KXQGUHGWK

5HVXOW $

7R FRQYHUW IURP D FRPPRQ IUDFWLRQ WR D GHFLPDO IUDFWLRQ

GLYLGH ·

1RWH WKDW WKH GLYLVRQ KDV QR UHPDLQGHU 7KHUHIRUH ZH VD\ WKH GLYLVRQ LV ILQLWH %

7R ILQG

LQ WKH IRUP RI D GHFLPDO IUDFWLRQ

GLYLGH ·

1RWH WKDW LQ WKDW FDVH WKH GLYLVRQ KDV D UHPDLQGHU 7KHUHIRUH ZH FDOO LW LQILQLWH GLYLVRQ :H FDQ FRQWLQXH RQ GLYLVRQ +RZHYHU WKH UHTXLUHG LV WR ILQG WKH TXRWLHQW DSSUR[LPDWHG WR WKH QHDUHVW KXQGUHGWK 7KHUHIRUH LW LV HQRXJK WR GLYLGH WLOO ZH KDYH WKUHH GHFLPDO GLJLWV WKHQ ZH DSSO\ WKH UXOHV RI DSSUR[LPDWLRQ

WR WKH QHDUHVW KXQGUHGWK

3UDFWLFH &RPSOHWH

$

WR WKH QHDUHVW

%

WR WKH QHDUHVW

'

&

5I

\PM UI\ QK[

34

.QN\P 8ZQUI Za

WR WKH QHDUHVW

WR WKH QHDUHVW

)\EQTPI )LQG WKH TXRWLHQW RI · WR WKH QHDUHVW WHQWK

5HVXOW HVWLPDWHG TXRWLHQW

P

P

P

-

)LQG WKH WHQV GLJLW ¯ ¯

7KHQ ZULWH LW DERYH WKH WHQV GLJLW

-

)LQG WKH XQLWV GLJLW

1RWH WKDW WKH GLYLVRU 7KHUHIRUH WKH XQLWV GLJLW ]HUR

- -

3ODFH WKH GHFLPDO SRLQW LQ LWV RULJLQDO SRVLWLRQ

7KHQ ZULWH LW DERYH WKH WHQWK GLJLW

-

)LQG WKH KXQGUHGWK GLJLW ¯ ¯

7KHQ ZULWH LW DERYH WKH KXQGUHGWK GLJLW

)LQG WKH WHQWK GLJLW ¯ ¯

,W LV HQRXJK WR GLYLGH XQWLO \RX JHW GHFLPDO GLJLWV EHFDXVH LW LV UHTXLUHG WR DSSUR[LPDWH WKH TXRWLHQW WR WKH QHDUHVW WHQWK

Ȧ · WR WKH QHDUHVW WHQWK 1RWH WKDW WKH HVWLPDWHG TXRWLHQW LV FORVH WR WKH DFWXDO TXRWLHQW 7KHUHIRUH WKH DQVZHU LV UHDVRQDEOH

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

35

([HUFLVHV

+IRIVEP )\IVGMWIW 'LYLGH WKH IROORZLQJ WKHQ DSSUR[LPDWH WKH TXRWLHQW WR WKH QHDUHVW

$

·

%

·

&

·

'

·

)LQG WKH UHVXOWV DQG DSSUR[LPDWH WKHP WR WKH QHDUHVW KXQGUHGWK $

·

%

*

&

·

'

·

3ODFH D VXWLEOH VLJQ ! RU

$

·

%

¯

&

·

·

'

· *

·

)LQG WKH TXRWLHQW LQ HDFK RI WKH IROORZLQJ

$

·

&

·

%

'

·

·

7KH DUHD RI D UHFWDQJOH LV FP DQG LWV ZLGWK LV FP )LQG LWV OHQJWK DQG DSSUR[LPDWH LW WR WKH QHDUHVW KXQGUHGWK RI FHQWLPHWHU

)LOO LQ WKH EODQNV

5I

$

·

\PM UI\ QK[

36

.QN\P 8ZQUI Za

%

·

([HUFLVHV

)LQG WKH TXRWLHQW RI · WR WKH QHDUHVW KXQGUHGWK 'LYLGH E\ WKHQ DGG

WR WKH TXRWLHQW

7KH DUHD RI D UHFWDQJOH LV VTXDUH PHWHUV DQG LWV OHQJWK LV PHWHUV )LQG LWV ZLGWK DQG SHULPHWHU

7KH VLGH OHQJWK RI D VTXDUH LV PHWHUV )LQG LWV DUHD DSSUR[LPDWLQJ LW WR WKH QHDUHVW KXQGUHGWK

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

37

7HFKQRORJ\

7HFKQRORJ\ +RZ GR \RX GHDO ZLWK ([FHO" &OLFN WKH VWDUW PHQX WKHQ FKRRVH SURJUDPV 2Q SURJUDPV FOLFN RQ RIILFH WKHQ FKRRVH ([FHO WKH VSUHDG VKHHWV $ WDEOH ZLOO DSSHDU RQ WKH PRQLWRU LW LV GLYLGHG LQWR URZV DQG FROXPQV (DFK VSDFH ZLWKLQ WKH WDEOH LV FDOOHG D FHOO )RU H[DPSOH % GHQRWHV WKH HOHPHQW ORFDWHG LQ URZ QXPEHU DQG FROXPQ % ZKLOH FHOO ' GHQRWHV WKH HOHPHQW ORFDWHG LQ WKH URZ QXPEHU DQG WKH FROXPQ ' )\EQTPI 8VH WKH VSHDG VKHHWV SURJUDP ([FHO WR ILQG WKH TXRWLHQW RI ·

(QWHU WKH QXPEHU LQ FHOO $ WKHQ HQWHU WKH QXPEHU LQ FHOO & DQG LQ FHOO ' 'R D PRXVH FOLFN RQ FHOO * DQG ZULWH WKHQ FOLFN RQ & ' DQG SUHVV (QWHU 'R D PRXVH FOLFN RQ FHOO - DQG ZULWH WKHQ FOLFN RQ $ * DQG SUHVV (QWHU 7KH TXRWLHQW DSSHDUV )LQG E\ \RXUVHOI *

1RWH WKDW WKH PXOWLSOLFDWLRQ V\PERO LV

DQG GLYLVLRQ V\PERO LV

5I

\PM UI\ QK[

38

.QN\P 8ZQUI Za

7R ILQG WKH VXP RI

'R D PRXVH FOLFN RQ FHOO . DQG ZULWH WKHQ ZULWH $ * DQG SUHVV (QWHU 7KH TXRWLHQW DSSHDUV

$FWLYLW\

$FWLYLW\ ,MWXSV] 8MT

*L]D S\UDPLGV ZHUH EXLOW PRUH WKDQ \HDUV %& .KXIX S\UDPLG LV WKH ODUJHVW RQH DV LW KDV D KHLJKW RI PHWHUV ZKHUH WKH EDVH DUHD LV DV HTXDO DV WKH DUHD RI IRRWEDOO SOD\JURXQGV (DFK VWRQH XVHG LQ EXOLGLQJ WKH S\UDPLG ZHLJKV DERXW WRQV :KDW LV WKH ZHLJKW RI VWRQHV LQ NLORJUDPV"

6RPH KXJH VWRQHV XVHG LQ EXOLGLQJ WKH S\UDPLG ZHLJK WRQV HDFK ,I DQ HOHSKDQW ZHLJKV WRQV ZKDW LV WKH QXPEHU RI HOHSKDQWV ZKLFK WKH WRWDO RI WKHLU ZHLJKW LV HTXDO WR WKH ZHLJKW RI RQH VWRQH" ,I WKH SURFHVV RI JOD]LQJ HDFK VWRQH LQ D S\UDPLG WDNHV PLQXWHV :LOO WKH SURFHVV RI JOD]LQJ VWRQHV WDNH KRXUV" ([SODLQ

8VH WKH NQRZOHGJH UHVRXUFHV L H OLEUDU\ ,QWHUQHW WR ZULWH D WHQ OLQH SDUDJUDSK DERXW *L]D S\UDPLGV

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

39

7HVW

8QLW 7HVW

)LQG WKH UHVXOWV RI HDFK RI WKH IROORZLQJ RSHUDWLRQV DSSUR[LPDWLQJ WKHP WR WKH QHDUHVW KXQGUHGWK

$

%

&

*

'

·

$ WUXFN FDQ KROG ER[HV RI RUDQJHV DW D WLPH +RZ PDQ\ WLPHV DUH QHHGHG WR GHOLYHU ER[HV E\ WKDW WUXFN"

$

:KLFK LV JUHDWHU IUDFWLRQV

RU " )LQG WKH GLIIHUHQFH EHWZHHQ WKH WZR

%

)LQG D QXPEHU WKDW LI PXOWLSO\HG E\ WKHQ WKH UHVXOW LV

$

5HDUUDQJH WKH IROORZLQJ IUDFWLRQV GHVFHQGLQJO\

%

7KH OHQJWK RI D UHFWDQJOH LV FP ZKLOH LWV ZLGWK LV FP )LQG LWV SHULPHWHU DQG LWV DUHD

5I

\PM UI\ QK[

40

.QN\P 8ZQUI Za

Shorouk for Modern printing

;LEX MW E WIX# 7KLQN DQG GLVFXVV

\RX ZLOO OHDUQ

- 7R GHILQH WKH

:KDW DUH WKH GD\V RI WKH ZHHN" 7KH

GD\V RI WKH ZHHN DUH 6DWXUGD\ 6XQ GD\ 0RQGD\ 7XHVGD\ :HGQHVGD\ 7KXUVGD\ DQG )ULGD\

PDWKHPDWLFDO FRQFHSW RI D VHW

?]\ROg

$XJ

5DPDGDQ

:KDW DUH WKH OHWWHUV RI WKH ZRUG +RVDP " 7KH OHWWHUV RI WKH ZRUG +RVDP DUH + 2 6 $ 0

:KDW DUH WKH GLJLWV RI WKH QXPEHU " 7KH GLJLWV RI WKH QXPEHU DUH $OO RI WKH DERYH FROOHFWLRQV DUH FDOOHG $ VHW 6R ZH VD\

.H\ 7HUPV - 7KH VHW - 7KH HOHPHQWV RI D

7KH VHW RI WKH GD\V RI WKH ZHHN WKH VHW RI WKH OHWWHUV RI WKH ZRUG +RVDP DQG WKH VHW RI WKH GLJLWV RI WKH QXPEHU 7KH VHW LW LV D FROOHFWLRQ RI NQRZQ REMHFWV WKDW DUH FOHDUO\

VHW

GH¿QHG DQG WKH\ KDYH D FHUWDLQ SURSHUW\ LQ FRPPRQ

2SXI XLEX 7KH EHDXWLIXO Ã&#x20AC;RZHUV LQ \RXU VFKRRO JDUGHQ GR QRW IRUP D VHW VLQFH WKH SURSHUW\ RI EHDXW\ GLIIHUV IURP D SHUVRQ WR DQRWKHU 7KH HOHPHQWV RI D VHW

7KH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG 0DJG\ DUH 0 $ * ' < (DFK OHWWHU LV FDOOHG DQ HOHPHQW RI WKH HOHPHQWV LQ WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG 0DJG\

3UDFWLFH 5I

\PM UI\ QK[

7KH VHW RI WKH FRORUV WKDW DSSHDU LQ WKH WUDI¿F VLJQDO KDV WKH IROORZLQJ HOHPHQWV 5HG DQG

.QN\P 8ZQUI Za

&RPSOHWH ,Q WKH IROORZLQJ ¿JXUH WKH VHW RI WKH PHDQV RI WUDQVSRUWDWLRQ KDV WKH

IROORZLQJ HOHPHQWV WKH SODQH WKH WUDLQ DQG

)\IVGMWIW &RPSOHWH WKH IROORZLQJ WDEOH DV LOOXVWUDWHG LQ WKH H[DPSOH 7KH H[SUHVVLRQ

( ZL[ UV[ H ZL[

7KH PRQWKV RI WKH +HJUL \HDU

D VHW

7KH WDOO VWXGHQWV LQ \RXU FODVV

1RW D VHW

7KH VHDVRQV RI WKH \HDU

7KH OHWWHUV RI WKH ZRUG (J\SW

7KH EHDXWLIXO VWRULHV

7KH SULPH QXPEHUV EHWZHHQ DQG

ZULWH GRZQ DOO WKH HOHPHQWV LQ WKH IROORZLQJ VHWV 7+( 6(7

7+( (/(0(176

7KH VHW RI WKH GLJLWV LQ WKH QXPEHU

7KH VHW RI WKH FRORUV LQ (J\SW V Ã&#x20AC;DJ

7KH VHW RI WKH GD\V LQ WKH ZHHN

7KH VHW RI WKH \HDU V PRQWKV WKDW KDYH OHVV WKDQ GD\V

7KH VHW RI GLJLW QXPEHUV DQG HDFK LV OLNH WKH RWKHU

7KH VHW RI WKH PRQWKV LQ WKH +HJUL \HDU

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

1EXLIQEXMGEP I\TVIWWMSR SJ E WIX )LUVW 7KH OLVWLQJ PHWKRG

\RX ZLOO OHDUQ - 7R ZULWH D VHW LI \RX NQRZ LWV HOHPDQWV

- 7R H[SUHVV D VHW

7R H[SUHVV D VHW ZULWH DOO WKH HOHPHQWV RI WKH VHW EHWZHHQ WZR EUDFHV ^ ` WKHQ SODFH WKH PDUN EHWZHHQ HYHU\ WZR HOHPHQWV 7KH VHW LV V\PREOL]HG E\ RQH RI WKH DOSKDEHWLF OHWWHUV ZULWWHQ LQ FDSLWDO IRQW OLNH ; < =

XVLQJ WKH OLVWLQJ PHWKRG - 7R H[SUHVV D VHW XVLQJ WKH GHVFULSWLRQ PHWKRG - 7R UHSUHVHQW WKH VHWV XVLQJ D 9HQQ GLDJUDP

)\EQTPIW

:ULWH GRZQ WKH VHW ; ZKHUH ; LV WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG $KPHG

6ROXWLRQ ; ^D K P H G` ,W FDQ EH ZULWWHQ DOVR DV ; ^K G D H P`

:ULWH GRZQ WKH VHW < ZKHUH < LV WKH VHW RI WKH GLJLWV LQ WKH QXPEHU

6ROXWLRQ

.H\ 7HUPV

< ^ ` ,W FDQ EH ZULWWHQ DOVR DV < ^ `

- 7KH OLVWLQJ PHWKRG

- 7KH GHVFULSWLRQ PHWKRG

2SXI XLEX

,W LV QRW LPSRUWDQW WR SD\ DWWHQWLRQ WR WKH RUGHU RI WKH HOHPHQWV ZKHQ ZULWLQJ D VHW

$Q\ VHW GRHV QRW KDYH D UHSHDWHG HOHPHQW

RUGHUHG SDLU LV GLIIHUHQW

ZKLOH WKH VHW ^ ` LV WKH

\PM UI\ QK[

VDPH DV WKH VHW ^ `

.QN\P 8ZQUI Za

,Q WKH RSSRVLWH ILJXUH WKH IURP WKH RUGHUHG SDLU

5I

ȶ

6HFRQG 7KH GHVFULSWLRQ PHWKRG

,Q WKDW PHWKRG ZH GH¿QH WKH SURSHUW\ ZKLFK GLVWLQJXLVKHV DQG GHWHUPLQHV WKH HOHPHQWV RI D VHW )RU H[DPSOH 7KH VHW ^F D U H` FDQ EH H[SUHVVHG DV IROORZV 7KH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG FDUH RU WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG UDFH RU WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG DFUH

,W FDQ EH ZULWWHQ LQ WKH IROORZLQJ IRUP ^[ [ LV RQH RI WKH OHWWHUV IRUPLQJ WKH ZRUG FDUH` DQG LW LV UHDG DV WKH VHW RI DOO [ ZKHUH [ LV RQH RI WKH OHWWHUV IRUPLQJ WKH ZRUG FDUH

7KLQN :ULWH GRZQ WKH VHW ; ZKHUH [ ^ ` 8VH WKH GHVFULSWLRQ PHWKRG

3UDFWLFH &RPSOHWH WKH WDEOH WR H[SUHVV WKH IROORZLQJ VHWV 7+( /,67,1* 0(7+2'

7+( '(6&5,37,21 0(7+2'

^F D U`

7KH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG FDU

^HDVW ZHVW QRUWK VRXWK`

^ `

7KH VHW RI WKH FRORUV IRUPLQJ (J\SW V Ã&#x20AC;DJ

^ `

7KH VHW RI WKH GLJLWV LQ WKH QXPEHU

^$ER %DNU 2PDU 2WKPDQ $OL`

7KH VHW RI WKH OHWWHUV RI WKH ZRUG 6HULHV

^ `

^ `

^ `

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

5HSUHVHQWLQJ VHWV XVLQJ 9HQQ GLDJUDPV

6FLHQWLVW -DQ 9HQQ FRXOG UHSUHVHQW HYHU\ HOHPHQW LQ D VHW E\ SODFLQJ D SRLQW RU DQ

PDUN LQ DQ\ FORVHG JHRPHWULF VKDSH VXFK DV D WULDQJOH D FLUFOH D UHFWDQJOH HWF

)RU H[DPSOH 7KH VHW ; ^ ` FDQ EH UHSUHVHQWHG E\ XVLQJ D 9HQQ GLDJUDP DV IROORZV ; ;

;

RU

RU

3UDFWLFH &RPSOHWH WKH IROORZLQJ WDEOH 7KH VHW

9HQQ GLDJUDP ;

; ^ `

<

7KH OLVWLQJ PHWKRG < ^ `

7KH GHVFULSWLRQ PHWKRG LV

= WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG

=

6WDLUV

; ^ `

;

<

<

7KH VHW RI WKH HOHPHQWV IRXQG LQ ; DQG <

5I

LV

\PM UI\ QK[

.QN\P 8ZQUI Za

&IPSRKMRK SJ ER IPIQIRX XS E WIX 7KLQN DQG GLVFXVV

\RX ZLOO OHDUQ

,I WKH IRRWEDOO WHDP RI \RXU FODVV FRQVLVWV RI 6DP\ +DWHP .KDOHG <DVVHU +DQ\ DQG 0DKHU :ULWH GRZQ WKH VHW ; ZKLFK

- %HORQJLQJ RI DQ

UHSUHVHQWV WKH IRRWEDOO WHDP RI \RXU FODVV

HOHPHQW WR D VHW

HOHPHQW WR D VHW

- 1RW EHORQJLQJ RI DQ

; ^ `

,V .KDOHG RQH RI WKH SOD\HUV RI WKH IRRWEDOO WHDP RI \RXU FODVV"

,V .KDOHG DQ HOHPHQW RI WKH HOHPHQWV RI WKH IRRWEDOO WHDP LQ \RXU FODVV" :H FDQ VD\ WKDW .KDOHG EHORQJV WR WKH VHW RI WKH IRRWEDOO WHDP LQ \RXU FODVV 7KLV FDQ EH V\PEOL]HG DV IROORZV .KDOHG Ǻ ;

.H\ 7HUPV

6LPLODUO\ 6DP\ Ǻ ; +DWHP Ǻ ; DQG VR RQ 7KH V\PERO Ǻ GHQRWHV WKH EHORQJLQJ RI DQ HOHPHQW WR D VHW

- %HORQJLQJ Ǻ - 1RW EHORQJLQJ ǻ

2SXI XLEX $KPHG LV QRW D SOD\HU LQ WKDW WHDP VR $KPHG GRHV QRW EHORQJ WR ; 7KLV LV V\PEROL]HG DV $KPHG Ç» ; 7KH V\PERO Ç» GHQRWHV 7KH QRW EHORQJLQJ RI DQ HOHPHQW WR D VHW

([DPSOH ,I < ^ `

7KHQ Ǻ < Ǻ < DQG Ǻ <

ZKLOH Ç» < DQG Ç» <

7KLQN ,I Ǻ ^ [` 7KHQ [ Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

)\IVGMWIW :ULWH HDFK RI WKH IROORZLQJ VHQWHQFHV XVLQJ RQH RI WKH V\PEROV Ǻ RU ǻ 7KH VHQWHQFH

7KH V\PERO Ǻ ;

LV DQ HOHPHQW RI WKH VHW ; EHORQJV WR WKH VHW <

% GRHV QRW EHORQJ WR WKH VHW 0

GRHV QRW EHORQJ WR WKH VHW 1

% LV DQ HOHPHQW RI WKH VHW .

,I ; LV D VHW ZKHUH ; ^ `

3ODFH WKH VXLWDEOH V\PERO Ǻ RU ǻ LQ WKH EODQNV WR PDNH HDFK VHQWHQFH WUXH $

;

%

;

&

;

'

;

(

;

)

;

*

;

+

;

3ODFH WKH VXLWDEOH V\PERO Ǻ RU ǻ LQ WKH EODQNV WR PDNH HDFK VHQWHQFH WUXH $

%

^ `

< WKH VHW RI WKH OHWWHUV IRUPLQJ WKH OHWWHUV RI WKH ZRUG (J\SW

&

7KH VHW RI WKH RGG QXPEHUV

'

WKH VHW RI WKH GD\V RI WKH ZHHN

(

7KH PRQWK PDUFK WKH VHW RI WKH

)

^ `

VHDVRQV RI WKH \HDU

)LOO LQ XVLQJ D VXLWDEOH QXPEHU

5I

$

,I Ǻ ^ [ ` 7KHQ [

%

,I Ǻ ^ [ ` 7KHQ [

&

,I Ǻ ^ [ ` 7KHQ [

'

Ǻ ^ ` DQG EHORQJV DOVR WR WKH VHW RI WKH SULPH IDFWRUV RI WKH QXPEHU

\PM UI\ QK[

.QN\P 8ZQUI Za

8]TIW SJ WIXW

7KLQN DQG GLVFXVV

\RX ZLOO OHDUQ

+RZ PDQ\ HOHPHQWV DUH WKHUH LQ WKH VHW ; ^D K P H G`" 7KH QXPEHU RI WKH HOHPHQWV LQ WKH VHW ; 6R ZH FDOO WKDW VHW D ¿QLWH VHW 7KH Ã&#x20AC;QLWH VHW

- )LQLWH VHW - ,QILQLWH VHW - 1XOO VHW HPSW\ VHW

,W LV D VHW WKDW KDV D OLPLWHG QXPEHU RI HOHPHQWV L H WKH QXPEHU RI LWV HOHPHQWV FDQ EH OLVWHG 6LPLODUO\ 7KH VHW ; ^J E U` LV D ¿QLWH VHW WKDW KDV

WKUHH HOHPHQWV

7KH VHW < WKH VHW RI WKH GD\V RI WKH ZHHN LV DOVR D ¿QLWH VHW WKDW KDV HOHPHQWV

.H\ 7HUPV

7KH LQÃ&#x20AC;QLWH VHW

,W LV D VHW WKDW KDV DQ XQOLPLWHG QXPEHU RI HOHPHQWV L H WKH QXPEHU RI LWV HOHPHQWV FDQ QRW EH OLVWHG )RU H[DPSOH 7KH VHW RI WKH HYHQ QXPEHUV ^ `

LV DQ LQ¿QLWH VHW EHFDXVH WKH QXPEHU RI LWV HOHPHWQV FDQ QRW EH OLVWHG

6LPLODUO\ 7KH VHW RI WKH GHFLPDO QXPEHUV EHWZHHQ DQG LV DQ LQ¿QLWH VHW ZKHUH VRPH RI LWV HOHPHQWV DUH

- )LQLWH VHW - ,QILQLWH VHW - 1XOO VHW ¡

7KH QXOO VHW

,W LV D VHW WKDW KDV QR HOHPHQWV ,W LV GHQRWHG E\ WKH V\PEROV ^ ` RU ¡ DQG UHDG DV )DL

)RU H[DPSOH 7KH VHW RI WKH VWXGHQWV LQ \RXU FODVV WKDW DUH \HDUV ROG RU WKH VHW RI WKH PRQWKV LQ D \HDU WKDW KDYH GD\V Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

2SXI XLEX 7KH QXOO VHW LV D ¿QLWH VHW LQ ZKLFK WKH QXPEHU RI LWV HOHPHQWV ]HUR 7KH VHW ^ ` KDV D QXPEHU RI HOHPHQWV WKDW LV HTXDO WR ]HUR ZKLOH WKH VHW ^ ` KDV D QXPEHU RI HOHPHQWV WKDW LV HTXDO WR DQG LW LV QRW D QXOO VHW

)\IVGMWIW :KLFK RI WKHVH VHWV LV D ¿QLWH VHW DQG ZKLFK RI WKHP LV DQ LQ¿QLWH VHW :ULWH WKH HOHPHWQV RI HYHU\ ¿QLWH VHW 7KH VHW 7KH VHW RI WKH GD\V LQ D ZHHN

1XPEHU RI

¿QLWH

,Q¿QLWH

HOHPHQWV

7KH VHW RI WKH PRQWKV LQ D JUHJRULDQ \HDU 7KH VHW RI WKH RGG QXPEHUV 7KH VHW RI WKH SULPH QXPEHUV OHVV WKDQ 7KH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG VRQGRV

7KH VHW RI WKH IDFWRUV RI WKH QXPEHU 7KH VHW RI WKH DOSKDEHWV LQ WKH (QJOLVK ODQJXDJH

:KLFK RI WKHVH VHWV LV D QXOO VHW DQG ZKLFK RI WKHP LV QRW D QXOO VHW

5I

$

7KH VHW RI VWXGHQWV LQ \RXU FODVV ZKR PDGH D WULS WR WKH PRRQ

%

7KH VHW RI WKH (J\SWLDQ JRYHUQRUDWHV LQ $VLD

&

7KH VHW RI WKRVH QXPEHUV GLYLVLEOH E\ DQG DUH EHWZHHQ

'

7KH VHW RI WKH IDFWRUV RI ZKLFK DUH GLYLVLEOH E\

(

7KH VHW RI WKRVH QXPEHUV GLYLVLEOH E\ DQG DUH EHWZHHQ

)

7KH VHW RI WKH JRYHUQRUDWHV LQ XSSHU (J\SW WKDW DUH ORFDWHG RQ WKH 0HGLWHUUDQHDQ VHD

\PM UI\ QK[

.QN\P 8ZQUI Za

)UYEP WIXW

7KLQN DQG GLVFXVV

&RPSOHWH 7KH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG 0DJG DUH 7KH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG *PDG DUH

\RX ZLOO OHDUQ

- (TXDOLW\ RI WZR VHWV

:KDW GR \RX REVHUYH " ,I WKH VHW ; ^ ` DQG WKH VHW < ^ ` :KDW GR \RX REVHUYH LQ WKRVH WZR VHWV" 1RWH 'R QRW SD\ DWWHQWLRQ WR WKH RUGHU RI WKH HOHPHQWV LQ D VHW :H REVHUYH WKDW WKH HOHPHQWV RI WKH VHW ; DUH WKH VDPH HOHPHQWV LQ WKH VHW < 7KH VHW ; 7KH VHW < ,I WKH WZR VHWV KDYH WKH VDPH HOHPHQWV H[DFWO\

.H\ 7HUPV

3UDFWLFH

- (TXDOLW\ RI WZR VHWV

,I ; LV WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG PRQWK ZKLOH < LV WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG PRQWKV $UH WKH WZR VHWV HTXDO" 6WDWH WKH UHDVRQ )\EQTPI )LQG WKH YDOXHV RI D DQG E WKDW PDNH HDFK VHQWHQFH WUXH

^D ` ^E `

^ D ` ^E `

^ D` ^ `

6ROXWLRQ

^D ` ^E `

ZKHQ D E

^ D ` ^E `

ZKHQ D E

^ D` ^ `

WKHQ D

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

)\IVGMWIW 3XW IRU WKH WUXH VHQWHQFH DQG IRU WKH IDOVH RQH $

^ ` ^ `

%

^D U F` WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG FDU

&

^ ` WKH IDFWRUV RI WKH QXPEHU

'

^[ ` ^ ` ZKHUH [

,I ; WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG ODE < WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG EDOO LV ; <"

,I ^[ ` WKH VHW RI WKH GLJLWV LQ WKH QXPEHU ¿QG WKH YDOXH RI [ 0DWFK WKH HTXDO VHWV LQ WKH IROORZLQJ FROXPQV WKH VHW RI WKH OHWWHUV IRUPLQJ WKH

^ `

ZRUG ]LZHO

^ `

WKH VHW RI WKH GLJLWV RI

^ G`

^6XPPHU ZLQWHU VSULQJ DXWXPQ`

^] O H Z /`

WKH VHW RI WKH PRQWKV LQ D \HDU WKDW KDYH GD\V

WKH VHW RI WKH VHDVRQV RI WKH \HDU

^G `

Ç·

WKH VHW RI WKH HYHQ QXPEHUV WKDW KDYH GLJLWV

SODFH IRU WKH WUXH VHQWHQFH DQG IRU WKH IDOVH VHQWHQFH

5I

$

^ ` WKH VHW RI WKH HYHQ QXPEHUV OHVV WKDQ

%

^ ` WKH VHW RI WKH GLJLWV RI

&

^ ` WKH VHW RI WKH FRXQWLQJ QXPEHUV WKDW DUH GLYLVLEOH E\

\PM UI\ QK[

.QN\P 8ZQUI Za

-RGPYWMSR ERH WYFWIXW -RGPYWMSR ERH WYFWIXW 7KLQN DQG GLVFXVV 7KLQN DQG GLVFXVV

,I ; WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG SLH DQG < WKH ,I ; WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG SLH DQG < WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG SLHU

VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG SLHU

:ULWH HDFK RI WKH WZR VHWV XVLQJ WKH OLVWLQJ PHWKRG :ULWH HDFK RI WKH WZR VHWV XVLQJ WKH OLVWLQJ PHWKRG

; ^S L H` ZKLOH < ^S L H U` ; ^S L H` ZKLOH < ^S L H U`

\RX ZLOO OHDUQ

\RX ZLOO OHDUQ

- ,QFOXVLRQ - ,QFOXVLRQ - 1RQ LQFOXVLRQ - 1RQ LQFOXVLRQ - 6XEVHWV - 6XEVHWV

$UH DOO WKH HOHPHQWV RI WKH VHW [ IRXQG LQ WKH VHW <" 'UDZ D $UH DOO WKH HOHPHQWV RI WKH VHW [ IRXQG LQ WKH VHW <" 'UDZ D 9HQQ GLDJUDP ZKLFK UHSUHVHQWV WKH WZR VHWV ; DQG < 9HQQ GLDJUDP ZKLFK UHSUHVHQWV WKH WZR VHWV ; DQG < <

<

<HV DOO WKH HOHPHQWV RI WKH VHW [ DUH <HV DOO WKH HOHPHQWV RI WKH VHW [ DUH IRXQG LQ WKH VHW \ IRXQG LQ WKH VHW \

; ;

6R ZH VD\ 6R ZH VD\

3 3

LL

U

H

; LV D VXEVHW RI < ; LV D VXEVHW RI < RU ; LV LQFOXGHG LQ < RU ; LV LQFOXGHG LQ <

É´ É´

7KLV LV GHQRWHG E\ [ \ \ 7KLV LV GHQRWHG E\ [ 7KH V\PERO É´ GHQRWHV WKH LQFOXVLRQ RI RQH 7KH V\PERO É´ GHQRWHV WKH LQFOXVLRQ RI RQH VHW LQ DQRWKHU VHW

.H\ 7HUPV - ,QFOXVLRQ ɴ - 1RQ LQFOXVLRQ ɶ ɶ 1RQ LQFOXVLRQ - 6XEVHWV 6XEVHWV

VHW LQ DQRWKHU VHW

,I ; ^ ` < ^ ` ,V ; É´ <"

,I ; ^ ` < ^ ` ,V ; É´ <"

ZH QRWH WKDW Ǻ ; ZKLOH ǻ <

ZH QRWH WKDW Ǻ ; ZKLOH ǻ <

7KHUHIRUH ; LV QRW D VXEVHW RI < EHFDXVH DOO WKH HOHPHQWV 7KHUHIRUH ; LV QRW D VXEVHW RI < EHFDXVH DOO WKH HOHPHQWV LQ ; DUH QRW LQFOXGHG LQ < VR ZH VD\ WKDW ; LV QRW D VXEVHW LQ ; DUH QRW LQFOXGHG LQ < VR ZH VD\ WKDW ; LV QRW D VXEVHW RI <

RI <

; LV QRW LQFOXGHG LQ \

; LV QRW LQFOXGHG LQ \

ɶ

DQG WKDW LV ZULWWHQ DV ; \ DQG WKDW LV ZULWWHQ DV ; \ ZKHUH WKH V\PERO ɶ GHQRWHV WKH QRQ LQFOXVLRQ RI D VHW LQ DQRWKHU VHW ZKHUH WKH V\PERO ɶ GHQRWHV WKH QRQ LQFOXVLRQ

ɶ

RI D VHW LQ DQRWKHU VHW Shorouk for Modern printing

U <MZ

\ [ U QZ S . [\ <MZ W W Z V\ [ * .Q ;\]LM [ *WWS V\ ;\]LM

)\EQTPIW

$

)LOO LQ WKH EODQNV XVLQJ RQH RI WKH WZR V\PEROV ɴ RU ɶ WR PDNH HDFK VHQWHQFH WUXH %

^ ` ^ `

^ ` ^ `

&

'

^ ` ^ `

^ ` ^ `

6ROXWLRQ $

^ ` É´ ^ `

%

^ ` ɶ ^ ` DV WKH ¿UVW VHW KDV RQO\ RQH HOHPHQW ZKLFK LV ZKLOH WKH VHFRQG VHW KDV RQO\ RQH HOHPHQW ZKLFK LV

&

^ ` ɶ^ ` EHFDXVH Ǻ ^ ` ZKLOH ǻ ^ `

'

^ ` É´ ^ ` L H DQ\ VHW ; LV D VXEVHW RI LWVHOI

; É´ ;

,Q WKH RSSRVLWH 9HQQ GLDJUDP

ZULWH ; < DQG = XVLQJ WKH OLVWLQJ PHWKRG ZKDW GR \RX REVHUYH"

6ROXWLRQ

<

; ^ ` < ^ ` DQG = ^ `

1RWH WKDW

;

; ɴ = ZKLOH < ɶ =

=

7KH V\PEROV ɴ DQG ɶUHIHU WR WKH UHODWLRQ EHWZHHQ D VHW DQG DQRWKHU VHW ZKLOH WKH V\PEROV Ǻ ǻ UHIHU WR WKH UHODWLRQ EHWZHHQ DQ HOHPHQW DQG D VHW 7KH QXOO VHW LV D VXEVHW RI DQ\ VHW L H Ƿ ɴ ; Ƿ ɴ < DQG Ƿ ɴ =

:ULWH GRZQ DOO WKH VXEVHWV RI ; ^ `

6ROXWLRQ WKH VXEVHWV DUH 7KH QXOO VHW Ç· 7KH VHWV ZKLFK KDYH RQH HOHPHQW HDFK ^ ` ^ ` DQG ^ ` 7KH VHWV ZKLFK KDYH HOHPHQWV HDFK ^ ` ^ ` DQG ^ ` 7KH VHWV ZKLFK KDYH HOHPHQWV HDFK ^ ` ;

$

:ULWH GRZQ DOO WKH VXEVHWV LQ HDFK RI WKH IROORZLQJ VHWV ; ^ `

%

< ^ `

6ROXWLRQ

5I

$

7KH VXEVHWV RI WKH VHW ; DUH Ç· DQG ^ `

%

7KH VXEVHWV RI WKH VHW < DUH Ç· ^ ` ^ ` DQG ^ `

\PM UI\ QK[

54

.QN\P 8ZQUI Za

)\IVGMWIW &RPSOHWH WKH WDEOH 6HW ;

8VH ɴ RU ɶ

6HW <

^ `

^ `

; <

^D E F`

^D E G H`

; <

^ `

7KH VHW RI WKH SULPH QXPEHUV

; <

7KH OHWWHUV RI 5DJE

WKH OHWWHUV RI *DEU

; <

^-DQXDU\ 0DUFK`

7KH PRQWKV RI WKH JUHJRULDQ \HDU

; <

^/RQGRQ`

7KH VHW RI WKH FDSLWDOV RI DOO WKH ZRUOG V

; <

FRXQWULHV

/RRN DW WKH RSSRVLWH 9HQQ GLDJUDP WKHQ FRPSOHWH WKH IROORZLQJ XVLQJ RQH RI WKH V\PEROV ɴ ɶ Ǻ RU ǻ

$

&

(

\ ;

%

^ ` <

'

;

)

;

;

<

<

^ ` ;

)LQG WKH VXEVHWV IRU HDFK RI WKH IROORZLQJ VHWV

$

^ `

%

^Ç·`

'

^ `

(

7KH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG EOEO

&

^ `

6WDWH ZKHWKHU HDFK VHQWHQFH LV WUXH RU IDOVH

$

^ ` É´ ^ `

%

^ ` É´ ^ `

&

Ç· É´ ^ `

'

Ǻ ^ `

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

55

-RXIVWIGXMSR SJ X[S WIXW 7KLQN DQG GLVFXVV

\RX ZLOO OHDUQ /RRN DW WKH RSSRVLWH 9HQQ - 7R ILQG WKH LQWHUVHFWLRQ GLDJUDP DQG FRPSOHWH RI WZR VHWV

<

;

; ^ ` < ^ ` $UH WKHUH DQ\ FRPPRQ HOHPHQWV LQ WKH VHWV ;DQG <" :KDW DUH WKH\" <HV WKHUH DUH FRPPRQ HOHPHQWV LQ WKH WZR VHWV ; DQG < ZKHUH Ǻ ; Ǻ < VLPLODUO\ Ǻ ; DQG Ǻ < M I HDFK RI WKH WZR HOHPHQWV DQG EHORQJ WR WKH WZR VHWV ; DQG <

.H\ 7HUPV - ,QWHUVHFWLRQ RI WZR VHWV È&#x203A;

7KHUHIRUH ^ ` LV WKH VHW RI WKH LQWHUVHFWLRQ RI WKH WZR VHWV ; DQG < 7KLV UHODWLRQ LV ZULWWHQ DV ; È&#x203A; < ^ ` 7KH LQWHUVHFWLRQ RI WZR VHWV WKH VHW RI DOO &RPPRQ HOHPHQWV LQ WKH VHWV ; DQG <

,Q WKH DERYH 9HQQ GLDJUDP WKDW UHSUHVHQWV WKH WZR VHWV ; È&#x203A; < LV UHSUHVHQWHG E\ WKH FRORUHG VHFWLRQ )\EQTPIW

5I

\PM UI\ QK[

.QN\P 8ZQUI Za

,I ; ^ ` < ^WKH VHW RI WKH GLJLWV RI ` 5HSUHVHQW WKH WZR VHWV ; DQG < E\ D 9HQQ GLDJUDP WKHQ ¿QG ; È&#x203A; < DQG < È&#x203A; ; ZKDW GR \RX REVHUYH"

6ROXWLRQ

;

; ^ ` < ^ ` ; È&#x203A; < ^ `

< È&#x203A; ; ^ `

<

)URP WKH DERYH ZH QRWLFH WKDW

; È&#x203A; < < È&#x203A; ; 7KH FRPPXWDWLYH SURSHUW\

8VH D 9HQQ GLDJUDP WR UHSUHVHQW WKH WZR VHWV ; ^D E H` < WKH VHW RI WKH OHWWHUV IRUPLQJ WKH ZRUG

<

;

Ã&#x20AC;J WKHQ ¿QG ; È&#x203A; < DQG < È&#x203A; ;

6ROXWLRQ

:H QRWLFH WKDW WKHUH DUH QRW FRPPRQ HOHPHQWV LQ [ DQG < 6R ZH VD\ ; DQG < DUH WZR GLVMRLQW VHWV WKHQ ; È&#x203A; < Ç·

J

I

/

H

E

D

< È&#x203A; ; Ç·

7KLQN LI ; ^ ` ¿QG ; È&#x203A; Ç·

)\IVGMWIW ;

<

,Q WKH RSSRVLWH 9HQQ GLDJUDP $

&RORU ; È&#x203A; < LQ UHG

%

&RORU ; È&#x203A; = LQ JUHHQ

&

FRORU < È&#x203A; = LQ \HOORZ

'

)LQG ; È&#x203A; < È&#x203A; = DQG ; È&#x203A; < È&#x203A; = ZKDW GR \RX REVHUYH"

=

&RPSOHWH $

^ ` È&#x203A; ^ `

%

^ ` È&#x203A; ^ `

&

^ ` È&#x203A; ^ `

'

^ ` È&#x203A; ^ `

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

&RPSOHWH ;

<

;

<

? È&#x203A; @

? È&#x203A; @

/RRN DW WKH RSSRVLWH 9HQQ GLDJUDP DQG ZULWH GRZQ ; < DQG = XVLQJ WKH

OLVWLQJ PHWKRG WKHQ ¿QG WKH IROORZLQJ $

&

(

%

; È&#x203A; <

< È&#x203A; =

'

)

^ ` È&#x203A; ;

; È&#x203A; =

^ ` È&#x203A; =

3ODFH WKH VXLWDEOH V\PERO Ǻ ǻߺ ɴ RU ɶ WR PDNH HDFK RI

^ ` È&#x203A; <

;

<

=

WKH IROORZLQJ VHQWHQFHV WUXH

$

,I ;

^ ` È&#x203A; ^ ` WKHQ ;

%

,I <

^ ` È&#x203A; ^ ` WKHQ ^ ` <

&

,I =

^ ` È&#x203A; ^ ` WKHQ =

'

,I 5

^ ` È&#x203A; ^ `

WKHQ 5 ^ `

(

,I 0 ^ ` È&#x203A; ^ `

WKHQ 0 ^ `

,I ; ^ ` < ^ ` DQG = ^ ` 5HSUHVHQW HDFK RI ; < DQG = XVLQJ D 9HQQ GLDJUDP WKHQ ¿QG WKH IROORZLQJ

5I

)LUVW ; È&#x203A; < È&#x203A; =

:KDW GR \RX REVHUYH"

\PM UI\ QK[

.QN\P 8ZQUI Za

6HFRQG ; È&#x203A; < È&#x203A; =

9RMSR SJ X[S WIXW 7KLQN DQG GLVFXVV

/RRN DW WKH RSSRVLWH 9HQQ GLDJUDP DQG FRPSOHWH ; ^ ` < ^ `

<

;

\RX ZLOO OHDUQ - 7R ILQG WKH 8QLRQ RI WZR VHWV

7KH VHW WKDW KDV DOO WKH HOHPHQWV LQ ; RU < RU LQ ; DQG LQ < LV ^ ` 7KH XQLRQ RI WKH WZR VHWV ; DQG < LV UHSUHVHQWHG E\ WKH FRORUHG VHFWLRQ LQ 9HQQ GLDJUDP 7KLV UHODWLRQ FDQ EH ZULWWHQ DV ; Ȝ < 7KH FRORUHG VHFWLRQ UHSUHVHQWV WKH VHW ZKLFK LQFOXGHV DOO WKH HOHPHQWV LQ ; RU < RU ERWK RI WKHP LV

7KH VHW RI HOHPHQWV ZKLFK EHORQJV WR [ RU WR < LV FDOOHG WKH XQLRQ RI WKH WZR VHWV ; DQG < ,W LV ZULWWHQ DV

.H\ 7HUPV - 8QLRQ RI WZR VHWV Ȝ

; Ȝ < ^ ` )\EQTPI )LQG HDFK RI ; Ȝ < DQG ; ț < LQ WKH IROORZLQJ FDVHV ;

<

;

<

; Ȝ < ^ `

; Ȝ < ^ `

; ț < ^ `

; ț < Ƿ ; Ȝ < ^ ` ;

;

<

Shorouk for Modern printing

; ț < ^ ` < 1RWLFH WKDW <ɴ;

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

3UDFWLFH *LYHQ WKDW ; ^ ` DQG < ^ ` )LQG ; Ȝ ; ; Ȝ Ƿ ; Ȝ < DQG < Ȝ ; :KDW GR \RX REVHUYH"

)\IVGMWIW &RPSOHWH

$

^ ` Ȝ ^ `

&

^ ` Ȝ ^ `

%

^ ` Ȝ ^ `

'

^ ` Ȝ Ƿ

/RRN DW WKH RSSRVLWH 9HQQ GLDJUDP WKHQ ¿QG

<

; Ȝ < DQG < Ȝ ; :KDW GR \RX REVHUYH"

*LYHQ WKDW ; ^ ` < ^ ` DQG = ^ `

;

)LQG HDFK RI ; Ȝ < Ȝ = DQG [ Ȝ <Ȝ = :KDW GR \RX REVHUYH"

,Q HDFK RI WKH IROORZLQJ 9HQQ GLDJUDPV ZULWH ZKDW WKH FRORUHG VHFWLRQ UHSUHVHQWV <

=

;

=

<

;

<

;

=

<

;

=

<

;

<

;

=

=

5I \PM UI\ QK[

.QN\P 8ZQUI Za

8LI 9RMZIVWEP WIX

7KLQN DQG GLVFXVV

,I [ 7KH VHW RI WKH IRRWEDOO WHDP LQ \RXU FODVV < 7KH VHW RI WKH EDVNHWEDOO WHDP LQ \RXU FODVV 7KHQ ZH FDQ FKRRVH D KXJH VHW WKDW UHSUHVHQWV DOO WKH JLYHQ VHWV LQ WKH SUREOHP ZKHUH WKH JLYHQ VHWV DUH VXEVHWV RI LW 7KDW KXJH VHW LV FDOOHG WKH XQLYHUVDO VHW DQG LW LV GHQRWHG E\ WKH V\PERO 8 )\EQTPI 7KH XQLYHUVDO VHW FDQ EH WKH VHW RI WKH VWXGHQWV LQ \RXU FODVV RU WKH VHW RI DOO WKH VWXGHQWV LQ WKH ┬┐IWK JUDGH LQ WKH VFKRRO RU VLPSO\ WKH VHW RI DOO WKH VWXGHQWV LQ WKH VFKRRO

\RX ZLOO OHDUQ

- 7KH XQLYHUVDO VHW

7KH XQLYHUDO VHW раг LV WKH PRWKHU VHW ZKLFK LQFOXGHV DOO WKH JLYHQ VXEVHWV

7KH XQLYHUVDO VHW PD\ EH UHSUHVHQWHG E\ D UHFWDQJOH RQ WKH

.H\ 7HUPV - 8QLYHUVDO VHW 8

9HQQ GLDJUDP ZKLOH LWV VXEVHWV DUH UHSUHVHQWHG E\ FORVHG &XUYHV LQVLGH WKH UHFWDQJOH )\EQTPIW

,I ; ^(J\SW /LE\D 6XGDQ`

< ^6XGDQ 6RPDOLD` 8 LV SRVVLEO\ WKH VHW RI $UDE FRXQWULHV 7KLQN RI DQRWKHU GHVFULSWLRQ IRU WKH XQLYHUVDO VHW 8

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

,I ; ^ ` 8 LV SRVVLEO\ WKH VHW RI WKH RGG QXPEHUV

7KLQN RI DQRWKHU GHVFULSWLRQ RI 8

)\IVGMWIW 7KH JLYHQ VHWV LQ HDFK RI WKH IROORZLQJ FDVHV UHSUHVHQW VXEVHWV ZULWH GRZQ D VXLWDEOH XQLYHUVDO VHW IRU HDFK FDVH

; ^&DLUR +HOZDQ WK RI 2FWREHU FLW\`

< ^6KDUT\D $OH[DQGULD`

8

; 7KH VHW RI 0DWK WHDFKHUV DW \RXU VFKRRO

< WKH VHW RI VFLHQFH WHDFKHUV DW \RXU VFKRRO

8

; ^ `

< ^ `

8

UHSUHVHQW 8 E\ 9HQQ GLDJUDP

; ^7DKD +RXVVHQ <RXVVHI ,GUHHV 7DZ¿N $O +DNHHP`

8

7KH RSSRVLWH 9HQQ GLDJUDP UHSUHVHQWV WKH WZR VHWV / 0 DQG WKH XQLYHUVDO VHW 8 ,I ZH JLYH HDFK GLIIHUHQW VHFWLRQ ZLWKLQ WKH 9HQQ GLDJUDP RQH RI WKH IROORZLQJ QXPEHUV &DQ \RX UHSUHVHQW WKH IROORZLQJ VHFWLRQV XVLQJ WKH WZR VHWV / 0 DQG WKH V\PEROV ț DQG Ȝ

5I

$

6HFWLRQ

&

6HFWLRQV DQG

\PM UI\ QK[

8 %

6HFWLRQV DQG

'

6HFWLRQV DQG

/

0

.QN\P 8ZQUI Za

2-10

The Complement of a set

Think and discuss

you will learn If the set of the music team at school is:

☺ To find the

U= {Magdy, Yasser, Fayez, Lila, Souad}

complement of a set.

U

Then, the set of boys in the team X. X

X= {Magdy, Yasser , Fayez}

✗ Yasser

✗ Lila

Note that X ⊂ U However, if the set of the girls is x , then:

✗ Magdy

✗ Fayez

✗

X = { ...................... } Therefore, the set x is the complement of the set x.

Complete X ∪ X = ... , X ∩ X = .....

If the universal set

Key Terms

U ✗

U = {1, 2, 3, 5, 6, 8, 9}

✗

A⊂U

✗

6

9

✗

A ✗

and the set A = {2, 5, 8} where

3

5

✗ ✗

1

☺ The complement of set X

2

8

Then, we call the set {1, 3, 6, 9} The complement of the set A with regard to the set U . That set is written as A = {1, 3, 6, 9} and it is defined as a set of the elements in U that does not belong to the set A. It is denoted by the symbol A and written as A = {1, 3, 6, 9}

The complement of a set A with regard to the set U is A where: A ∪ A = U , A ∩ A = ∅

Shorouk for Modern printing

Stude

oko B nt's

m Ter t s Fir

63

2-10 Exercises U

1 Look at the opposite Venn diagram then complete: U

= ............

A = ............

A = ............

2 Look at the opposite Venn diagram then complete: U

= ............

X

= ............

X

= ............

XțX

= ............

XȜX

A

1

4

C

X

3 Look at the opposite Venn diagram then complete: = ............

X

= ............

Y

= ............

X

= ............

= ............

YțX

= ............

3

U

Y Ȝ X = ............

(Y Ȝ X)

2

U

= ............

Y

5

D

F

A

B

E

U

Y

2

4 = ............

7

6

5

3

1

4 If U is the set of the even numbers less than 16, A = { 4, 6, 10, 12} and B = {2, 6, 8, 14}. Find each of the following: AȜ B, (A Ȝ B)` A ț B and (A ț B) .

5 ,I 8 LV WKH VHW RI WKH IDFWRUV RI DQG $ LV WKH VHW RI WKH IDFWRUV ¿QG $ .

6 If U = { 1, 2, 3, 4, 5, 6},

X = {3, 4, 5} and Y = { 1, 2, 3}.

Find each of the following sets:

Ma

A

X

B

Y

C

XțY

D

(X ț Y)

E

XȜY

F

(X Ȝ Y)

G

X ȜY

H

X țY

the mat ics -

64

Fifth Prima ry

8LI (MJJIVIRGI SJ X[S WIXW

7KLQN DQG GLVFXVV

$ VXUYH\ KDV EHHQ DSSOLHG RQ VWXGHQWV IURP FODVV VKRZHG WKDW RQO\ VWXGHQWV UHDG $O *RPKRULD QHZVSDSHU 7KRVH VWXGHQWV DUH $KPDG 6DPHK +DQD\ DQG +DQDD +RZHYHU RQO\ VWXGHQWV UHDG $O $NKEDU QHZVSDSHU 7KRVH VWXGHQWV DUH 5DQD +RVVDP DQG +DQ\ ZKLOH WZR VWXGHQWV UHDG ERWK WKH QHZVSDSHUV 0RQD DQG 0DKPRXG DQG RQO\ RQH VWXGHQW GRHV QRW UHDG DQ\ QHZVSDSHU 6DPLU 7KLV FDQ EH LOOXVWUDWHG XVLQJ WKH RSSRVLWH 9HQQ GLDJUDP ; ^ $ KPHG 6DPHK +DQDD +DQDQ 0RQD 0DKPRXG`

\RX ZLOO OHDUQ

- 7R ILQG WKH GLIIHUHQFH RI WZR VHWV

8 <

;

5DQD

+RVVDP

+DQ\

+DQDD

0RQD 0DKPRXG

$KPDG

6DPHK

+DQDQ

< ^ 0 RQD 0DKPRXG 5DQD +DQ\ +RVVDP`

.H\ 7HUPV

6DPLU

7KH VHW RI VWXGHQWV ZKR UHDG $O *RPKRULD QHZVSDSHU DQG GR QRW UHDG $O $NEDU QHZVSDSHU LV ^$KPHG 6DPHK +DQDD +DQDQ`

- 7KH GLIIHUHQFH RI WZR VHWV ߶߻߷

L H 7KH VHW RI HOHPHQWV WKDW EHORQJV WR ; DQG GRHV QRW EHORQJ WR < 7KLV VHW LV FDOOHG ; GLIIHUHQFH < DQG LW LV ZULWWHQ DV ; < &RPSOHWH ; < ^ `

< ; ^ ` :KDW GR \RX REVHUYH"

2SXI XLEX ; < LV QRW HTXDO WR < ;

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

7KLQN 8

8 ;

8 ;

<

8 <

2SXI XLEX ; ; Ƿ

ZKLOH ; Ƿ ;

)\IVGMWIW

;

<

;

<

; <

; <

< ;

< ;

;

<

<

;

&

=

$ '

(

%

)

; <

; <

< ;

< ; ; = = <

5I

\PM UI\ QK[

66

.QN\P 8ZQUI Za

Exercise General Exercises 1 Place the suitable symbol Ǻ , ǻ , ɴ or ɶ in the blanks: A

8 ..... {5, 7}

B

{3} ..... {1, 3, 2}

C

2 ..... {22, 44}

D

{1, 2} .....The set of prime numbers

E

Ƿ ..... {0}

F

(X ț Y) ..... X

2 Complete each of the following sentences to have a true sentence:

67

A

If X = {2, 3}, Y = {3, 5}, then X ț Y = ..........

B

If {1, X} = {2, Y}, then X = .........., Y = ..........

C

If X ɴ Y, then X Ȝ Y = .........., X ț Y = ..........

D

{1, 2, 4} - {2, 4, 6} = ..........

E

If 4 Ǻ {2, X, 7}, then X = ..........

3 Choose the true sentence from the parentheses: A

{1, 7} .......... {0, 1, 2, 3, 4, ...}

B

X - X = ..........

C

If {2, 5, 7} = {5, A, 2} then A = .......... (2 or 5 or 7 or 0)

D

{5} - {1, 2, 5} = ..........

E

The number of subsets for the set {5} is ......

(Ǻ or ǻ or ɴ or ɶ)

Ƿ or (zero or {0} or {1}) ({5} or {1, 2} or Ƿ or {1, 2, 5}) (0 or 1 or 2 or 3)

4 If U ={1, 2, 3, 4, 5, 6 }, X = {2, 3, 5} and Y= { 3, 4, 5}. Represent the sets by Venn diagram, then write each of the following by the listing method.

X Ȝ Y, X ț Y, X - Y and X

5 Describe the colored section in each of the following shapes: X

X

U

Y

X X

....................

Shorouk for Modern printing

....................

.................... ....................

Stude

oko B nt's

m Ter t s Fir

67

([HUFLVH

/RRN DW WKH RSSRVLWH 9HQQ GLDJUDP DQG ¿QG WKH IROORZLQJ VHWV XVLQJ WKH OLVWLQJ PHWKRG

$

&

(

8

; Ȝ < ; <

%

'

; ț <

<

;

<C

; Ȝ < C

:ULWH GRZQ DOO WKH VXEVHWV IRU WKH VHW ; ^D E F` /RRN DW WKH RSSRVLWH 9HQQ GLDJUDPV WKHQ ¿QG WKH IROORZLQJ VHWV XVLQJ WKH OLVWLQJ

PHWKRG

$

8

; ț =

%

&

< =

'

; Ȝ =

(

= ;

)

;

; <

=

;

<

,I ; ^ ` < ^ ` SODFH WKH VXLWDEOH V\PERO Ǻ RU ǻ RU ɴ RU ɶ LQ WKH EODQNV

$

;

%

^ ` ; ț <

&

^ ` ; Ȝ <

'

; <

(

Ƿ <

)

^ ` ;

)LQG WKH YDOXH RI [ WR PDNH HDFK RI WKH IROORZLQJ VHQWHQFHV WUXH

$

Ǻ ^ [ `

&

^ ;` ț ^ ` ^ `

%

; Ǻ ^ ` ț ^ `

:ULWH GRZQ ZKDW HDFK FRORUHG VHFWLRQ UHSUHVHQWV LQ WKH IROORZLQJ 9HQQ

GLDJUDPV 8

8

<

;

<

8

8 ;

;

<

;

)LQG DOO WKH VXEVHWV IRU WKH VHW ; ^D E F G` ZKHUH HDFK VXEVHW KDV HOHPHQWV

)LQG WKH QXPEHU RI WKRVH VHWV

5I

\PM UI\ QK[

68

.QN\P 8ZQUI Za

$FWLYLW\

$FWLYLW\

8VH WKH PDS RI $5( ZLWK WKH KHOS RI \RXU WHDFKHU RI JHRJUDSK\ WKHQ ZULWH GRZQ WKH IROORZLQJ VHWV

; WKH VHW RI (J\SWLDQ FRDVWDO JRYHUQRUDWHV

< WKH VHW RI WKH JRYHUQRUDWHV LQ XSSHU (J\SW

= WKH VHW RI WKH JRYHUQRUDWHV LQ ORZHU (J\SW 'HOWD

)LQG ; ț < < Ȝ = DQG ; <

,V WKH VHW RI WKH FDSLWDOV RI WKH ZRUOG V FRXQWULHV D ¿QLWH VHW" ([SODLQ ZK\

)RUP D WHDP ZRUN ZLWK \RXU FODVVPDWHV WKHQ ¿QG

; 7KH VHW RI WKH VWXGHQWV LQ \RXU FODVV ZKR DUH ROGHU WKDQ \HDUV

< 7KH VHW RI WKH VWXGHQWV LQ \RXU FODVV ZKR DUH \RXQJHU WKDQ \HDUV

'RHV ; Ȝ < UHSUHVHQW WKH VHW RI WKH VWXGHQWV LQ \RXU FODVV"

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

69

Test

Unit Test 1 Place the suitable symbol Çş or Çť or É´ or Éś in the blanks to make each of the following sentences true: A

{52} ....... {2 , 5}.

B

{3} ....... {1, 3}.

C

5 ....... the set of the digits in the number 2513.

D

4 ....... {44}.

2 /RRN DW WKH RSSRVLWH 9HQQ GLDJUDP WKHQ ÂżQG HDFK RI WKH IROORZLQJ A

XČ&#x203A;Y

B

XČ&#x153;Y

C

X-Y

D

Y -X

E

Y

F

(X Č&#x153; Y)

U Y

X 5

1

4

2

3

3 Choose the correct answer from the parentheses

6

7

in each of the following sentences: A

If { 2, 3, 4} = { 3, 4, x} then x = ....... (2 or 3 or 4 or 5)

B

ǡ ....... {0}

(= or É´ or Éśor Çş)

C

If X É´ Y then X Č&#x203A; Y = .......

(X or Y or ǡ or U)

D

X - X = .......

(0 or {0} or ǡ or X)

E

If a Çş X then a ....... X

(Éś or Çť or Çş or É´)

4 Write down what the colored section in each of the following Venn diagram represents: U

U

Y

X

....................

U

Y

Y

X

....................

....................

5 IF U = { 1, 2,3, 4, 5, 6, 7 }, X = { 1, 2, 3, 4} and Y = {1, 2, 5, 6} Write down each of the following sets using the listing method:

Ma

A

XČ&#x153;Y

B

XČ&#x203A;Y

C

X-Y

D

X

E

Y

D

(X Č&#x203A; Y )

the mat ics -

70 70

Fifth Prima ry

X

8LI 'MVGPI +RVVDP KDV EHFRPH D QHZ PHPEHU LQ WKH VFRXWV WHDP +H

<RX ZLOO OHDUQ - 7R GH¿QH D FLUFOH - 7R GUDZ D FLUFOH

KDV IL[HG D VWDNH LQ WKH JURXQG DQG WLHG D URSH WR WKH VWDNH 7KHQ KH IL[HG DQRWKHU VWDNH DW WKH RWKHU HQG RI WKH URSH +H FRPSOHWHO\ SXOOHG WKH URSH DQG WXUQHG D IXOO WXUQ DURXQG WKH IL[HG VWDNH WR GUDZ D FXUYHG OLQH RQ WKH JURXQG

.H\ 7HUPV

$&7,9,7<

- &LUFOH - 7KH FHQWHU RI D FLUFOH - 7KH UDGLXV RI D FLUFOH - 7KH GLDPHWHU RI D

7DNH SDUW ZLWK \RXU SK\VLFDO HGXFDWLRQ WHDFKHU DW VFKRRO DQG GUDZ WKH FLUFOH RQ WKH IRRWEDOO SOD\JURXQG WKHQ PDUN LW XVLQJ OLPH

7KLV FORVHG FXUYH LV FDOOHG D FLUFOH 7KH SRLQW LQ ZKLFK WKH VWDNH LV IL[HG LV FDOOHG WKH

FLUFOH

- 7KH FKRUG RI D FLUFOH

FHQWHU RI WKH FLUFOH

+RZ WR GUDZ D FLUFOH 8VH WKH FRPSDVVHV DV LOOXVWUDWHG LQ WKH IROORZLQJ ¿JXUH WR GUDZ D FLUFOH

0

5I

\PM UI\ QK[

.QN\P 8ZQUI Za

0

7KH UHG FXUYHG OLQH UHSUHVHQWV WKH FLUFOH 0 7KH SRLQW 0 LV FDOOHG WKH FHQWHU RI WKH FLUFOH 7KH GLVWDQFH EHWZHHQ WKH VKDUS SRLQW RI WKH FRPSDVVHV DQG WKH SHQFLO

0

U

ZKLFK GUDZV WKH FLUFOH LV FDOOHG WKH OHQJWK RI WKH UDGLXV DQG LW LV GHQRWHG E\ WKH V\PERO U

)\EQTPI 'UDZ D FLUFOH RI D UDGLXV FP

WSPYXMSR

0

ว

DGMXVW WKH FRPSDVVHV WR WKH GLVWDQFH RI FP

ว

ว

ว

วก

ว ว

ว ว

ว ว

ว ว

ว ว

ว ว

ว ว

ว ว

ว ว

ว วก

'UDZ WKH FLUFOH XVLQJ WKH FRPSDVVHV

2SXI XLEX ,Q WKH RSSRVLWH ยฟJXUH D FLUFOH RI D UDGLXV U

I

H

D

*MVWX 7KH SRLQWV $ % ; DUH ORFDWHG RQ WKH FLUFOH

L H

D วบ &LUFOH WKHQ 0$ U

E วบ &LUFOH WKHQ 0 % U

F

&RPSOHWH [ วบ &LUFOH 0 WKHQ 7IGSRH 7KH SRLQW I LV ORFDWHG LQVLGH WKH FLUFOH 0

6R 0I LV VKRUWHU WKDQ WKH OHQJWK RI WKH UDGLXV

L H 0I U

0 G E [

&RPSOHWH G LV ORFDWHG LQVLGH WKH FLUFOH 0 WKHQ 8LMVH 7KH WZR SRLQWV F DQG H DUH ORFDWHG RXWVLGH WKH FLUFOH VR 0F ! U DQG VLPLOLDU\ 0H ! U

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

ว ว

/I] 8IVQW 7KH UDGLXV 7KH UDGLXV RI D FLUFOH LV D OLQH VHJPHQW ZKRVH HQGSRLQWV DUH WKH FHQWHU RI WKH FLUFOH DQG DQ\ SRLQW Ǻ WKH FLUFOH ([ 0$ 0%

0&

$

0

%

'

0'

&

6R 0$ 0% 0& 0' U $

7KH FKRUG RI D FLUFOH 7KH FKRUG RI D FLUFOH LV D OLQH VHJPHQW WKDW FRQQHFWV EHWZHHQ DQ\ WZR SRLQWV RQ WKH FLUFOH

([ 'UDZ &RPSOHWH

$%

$&

&'

'UDZ HDFK

LV FDOOHG

$&

0

%

&

$'

'

$' LV FDOOHG %

7KH GLDPHWHU RI D FLUFOH 7KH GLDPHWHU RI D FLUFOH LV D FKRUG WKDW FURVVHV WKH FHQWHU RI WKH FLUFOH GLDPHWHU [ 5DGLXV U 7KH GLDPHWHU LV WKH ORQJHVW FKRUG LQ D FLUFOH

0

$

3UDFWLFH

$ %

2Q WKH VDPH SDSHU RQ ZKLFK \RX GUHZ WKH FLUFOH ODEHO WKH IROORZLQJ SRLQWV $ % & ZKHUH 0$ FP 0% FP 0& FP &

'UDZ D FLUFOH RI D UDGLXV FP ZKHUH 0 LV WKH FHQWHU RI WKH FLUFOH

&RPSOHWH E\ FKRRVLQJ RQ RXWVLGH RU LQVLGH WKH FLUFOH WR PDNH HDFK VHQWHQFH WUXH

3RLQW $ LV ORFDWHG WKH FLUFOH

3RLQW % LV ORFDWHG WKH FLUFOH

5I 3RLQW & LV ORFDWHG WKH FLUFOH \PM UI\ QK[ .QN\P 8ZQUI Za

3RLQW 0 LV ORFDWHG WKH FLUFOH

74

)\GIVGMWIW ,Q WKH RSSRVLWH ¿JXUH WKHUH LV D FLUFOH ZKRVH FHQWHU LV 0 &RPSOHWH

7KH UDGLL RI WKH FLUFOH DUH

7KH GLDPHWHU RI WKH FLUFOH LV

7KH FKRUGV RI WKH FLUFOH DUH

$

'

0

%

&

,Q WKH RSSRVLWH ¿JXUH WKHUH LV D FLUFOH ZKRVH FHQWHU LV 1 &RPSOHWH )

7KH UDGLL RI WKH FLUFOH DUH

7KH GLDPHWHUV RI WKH FLUFOH DUH

7KH FKRUGV RI WKH FLUFOH DUH

(

$

1

%

&

'

,Q WKH RSSRVLWH ¿JXUH 0 1 DUH WZR FLUFOHV )LQG WKH OHQJWK RI 01 FP 1

0

FP

'UDZ D FLUFOH ZKRVH FHQWHU LV 0 DQG LWV GLDPHWHU LV FP WKHQ GUDZ D VWULDJKW OLQH WKDW SDVVHV WKURXJK WKH SRLQW 0 DQG LQWHUVHFWV WKH FLUFOH DW $ % 'UDZ DQRWKHU VWUDLJKW OLQH WKDW SDVVHV WKURXJK WKH SRLQW 0 DQG LQWHUVHFWV WKH FLUFOH DW & '

&RPSOHWH

$

$%

LV FDOOHG LQ WKH FLUFOH

%

&'

LV FDOOHG LQ WKH FLUFOH

&

0%

LV FDOOHG LQ WKH FLUFOH

'

3XW D VXLWDEOH VLJQ ! RU

$% &' 0& FP %' FP

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

75

(VE[MRK E XVMERKPI KMZIR XLI PIRKXLW SJ MXW XLVII WMHIW <RX KDYH OHDUQHG WR GUDZ D WULDQJOH LQ WKH IROORZLQJ WZR FDVHV

*LYHQ WKH OHQJWKV RI WZR RI LWV VLGHV DQG WKH PHDVXUH RI

<RX ZLOO OHDUQ - 7R GUDZ DQ

WKH LQFOXGHG DQJOH

HTXLODWHUDO WULDQJOH

*LYHQ WKH OHQJWK RI RQH RI LWV VLGHV DQG WKH PHDVXUHV

- 7R GUDZ DQ ,VRVFHOHV

RI WZR DQJOHV

WULDQJOH

- 7R GUDZ D VFDOHQH

1RZ \RX ZLOO OHDUQ WR GUDZ D WULDQJOH JLYHQ WKH OHQJWKV

WULDQJOH

RI LWV WKUHH VLGHV XVLQJ D UXOHU DQG D FRPSDVVHV )\EQTPI

.H\ 7HUPV - (TXLODWHUXO WULDQJOH - ,VRVFHOHV WULDQJOH - 6FDOHQH WULDQJOH

'UDZ WKH HTXLODWHUDO WULDQJOH $%& LQ ZKLFK WKH OHQJWK RI HYHU\ VLGH LV FP

7SPYXMSR

'UDZ WKH OLQH VHJPHQW $ % RI OHQJWK FP

$GMXVW WKH FRPSDVVHV WR D GLVWDQFH HTXDO WR FP WKHQ SODFH WKH VKDUS SRLQW DW $ DQG GUDZ DQ DUF

8VLQJ WKH VDPH GLVWDQFH SODFH

&

$

%

WKH FRPSDVVHV DW % WKHQ GUDZ DQRWKHU DUF WKDW LQWHUVHFWV WKH ¿UVW DUF DW &

'UDZ $& %& 6R \RX ZLOO KDYH WKH HTXLODWHUDO WULDQJOH $%&

3UDFWLFH 'UDZ WKH HTXLODWHUDO WULDQJOH /01 5I

ZKRVH SHULPHWHU LV FP

\PM UI\ QK[

.QN\P 8ZQUI Za

)\EQTPI 'UDZ WKH LVRVFHOHV WULDQJOH ;<= LQ ZKLFK WKH OHQJWK RI WKH EDVH FP DQG WKH OHQJWK RI HDFK RI LWV WZR RWKHU VLGHV FP =

7SPYXMSR

'UDZ WKH OLQH VHJPHQW ;< ZKHUH ;< FP

$GMXVW \RXU FRPSDVVHV WR D GLVWDQFH HTXDO WR FP DQG ZLWK WKH WLS RI WKH FRPSDVVHV DW ; DQG GUDZ DQ DUF

3ODFH WKH VKDUS SRLQW WR < 8VLQJ WKH VDPH GLVWDQFH WR GUDZ DQRWKHU DUF WKDW LQWHUVHFWV ZLWK WKH ¿UVW DUF DW =

'UDZ HDFK RI ; < < = VR \RX ZLOO KDYH WKH LVRVFHOHV WULDQJOH ;<=

FP

<

;

3UDFWLFH 'UDZ WKH LVRVFHOHV WULDQJOH LQ ZKLFK %& FP $% $& FP

)\EQTPI 'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP $& FP %

7SPYXMSR

'UDZ WKH OLQH VHJPHQW $& ZLWK OHQJWK FP

$GMXVW \RXU FRPSDVVHV WR D GLVWDQFH HTXDO WR FP WKHQ SODFH WKH VKDUS SRLQW & DW $ DQG GUDZ DQ DUF

FP

$

$GMXVW \RXU FRPSDVVHV WR D GLVWDQFH HTXDO WR FP WKHQ SODFH WKH VKDUS SRLQW DW & DQG GUDZ DQRWKHU DUF WKDW LQWHUVHFWV WKH ¿UVW DUF DW %

'UDZ $% %& VR \RX ZLOO JHW WKH WULDQJOH $%&

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

½"½ )\IVGMIWIW 'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP $& FP :KDW GR \RX REVHUYH"

'UDZ WKH WULDQJOH ;<= LQ ZKLFK ;< <= FP ;= FP 'UDZ WKH WULDQJOH /01 LQ ZKLFK /0 FP 01 FP 1/ FP 'UDZ D FLUFOH RI D UDGLXV FP DQG GUDZ $% DV LWV GLDPHWHU WKHQ FRPSOHWH GUDZLQJ WKH HTXLODWHUDO WULDQJOH $%& WKHQ FKRRVH WKH FRUUHFW DQVZHU

$

7KH SRLQW & LV ORFDWHG WKH FLUFOH LQVLGH RXWVLGH RQ

%

$& LV

D FKRUG D UDGLXV VRPHWKLQJ HOVH

'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP &$ FP 7KHQ GUDZ D FLUFOH ZKRVH FHQWHU LV % DQG LWV UDGLXV LV HTXDO WR FP WKHQ FRPSOHWH WKH IROORZLQJ

$

7KH SRLQW $ LV ORFDWHG WKH FLUFOH

%

7KH SRLQW & LV ORFDWHG WKH FLUFOH

&

7KH LV FDOOHG WKH UDGLXV RI WKH FLUFOH

'UDZ WKH HTXLODWHUDO WULDQJOH $%& ZKRVH VLGH LV HTXDO WR FP WKHQ GUDZ D FLUFOH ZKRVH FHQWHU LV $ DQG UDGLXV LV HTXDO WR FP WKHQ FRPSOHWH WKH IROORZLQJ

5I

$% LV FDOOHG LQ WKH FLUFOH

$& LV FDOOHG LQ WKH FLUFOH

%& LV FDOOHG LQ WKH FLUFOH

\PM UI\ QK[

.QN\P 8ZQUI Za

½"½

(VE[MRK PMRI WIKQIRXW JVSQ XLI ZIVXMGIW SJ E XVMERKPI TIVTIRHMGYPEV XS MXW STTSWMXI WMHIW ,QWURGXFWLRQ

'UDZLQJ D OLQH VHJPHQW IURP D SRLQW SHUSHQGLFXODU WR D VWUDLJKW OLQH

<RX ZLOO OHDUQ - 7R GUDZ D OLQH

,I $ % LV D VWUDLJKW OLQH & Ç» $ % 5HTXLUHG GUDZLQJ D OLQH VHJPHQW IURP SRLQW & SHUSHQGLFXODU WR $ %

VHJPHQW IURP D YHUWH[ RI D WULDQJOH SHUSHQGLFXODU WR WKH RSSRVLWH VLGH

- 7R GH¿QH WKH DOWLWXGHV RI D WULDQJOH

0DWHULDOV

7KH UXOHU WKH ULJKW WULDQJOH

.H\ 7HUPV 0HWKRG

- 7KH DOWLWXGHV RI D WULDQJOH $

&

'

%

3XW WKH UXOHU RQ $% 3XW RQH RI WKH ULJKW DQJOH VLGHV RI WKH VHW VTXDUH RQ WKH UXOHU DV LOOXVWUDWHG DERYH LQ WKH ¿JXUH

0RYH WKH VHW VTXDUH LQ WKH GLUHFWLRQ RI WKH DUURZ 7KH VHW VTXDUH ZLOO VOLGH RQ WKH UXOHU WLOO LW UHDFKHV WKH SRLQW &

'UDZ &' WKHQ &' $% Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

7KH DOWLWXGHV RI D WULDQJOH

)LUVW 7KH DOWLWXGHV RI WKH DFXWH WULDQJOH

,Q WKH RSSRVLWH ILJXUH $%& LV DQ

$

YLRXV VWHSV WR GUDZ

DFXWH WULDQJOH )ROORZ WKH VDPH SUH

$' %& DQG %( $&

)

(

$%

'

%

&

2SXI XLEX

&)

$' %( &) LQWHUVHFW DW RQH SRLQW ORFDWHG LQVLGH WKH WULDQJOH $%&

7KH OLQH VHJPHQWV $' %( &) DUH FDOOHG WKH DOWLWXGHV RI WKH WULDQJOH $%&

3UDFWLFH 'UDZ WKH HTXLODWHUDO WULDQJOH $%& ZKRVH VLGH LV HTXDO WR FP 7KHQ IURP LWV YHUWLFHV GUDZ WKH VHJPHQWV $' %( &) SHUSHQGLFXODU WR WKH RSSRVLWH VLGHV %& &$ $% UHVSHFWLYHO\ 7KHQ PHDVXUH WKH OHQJWKV RI $' %( &)

:KDW GR \RX REVHUYH"

'UDZ WKH WULDQJOH /01 LQ ZKLFK /0 FP 01 FP 1/ FP 7KHQ GUDZ D SHUSHQGLFXODU IURP / WR 01 WKDW LQWHUVHFWV LW DW ; $OVR GUDZ D SHUSHQGLFXODU IURP 0 WR / 1 WKDW LQWHUVHFWV LW DW < WKHQ PHDVXUH WKH OHQJWKV RI /; 0<

5I

\PM UI\ QK[

80

.QN\P 8ZQUI Za

6HFRQG 7KH DOWLWXGHV RI WKH ULJKW WULDQJOH

,Q WKH RSSRVLWH ¿JXUH $%& LV D ULJKW

$ '

WULDQJOH LQ & )ROORZ WKH VDPH VWHSV 'UDZ D OLQH VHJPHQW IURP $ SHUSHQGLFXODOU WR %& WKHQ WKH OLQH VHJPHQW ZLOO EH $ &

'UDZ D OLQH VHJPHQW IURP % SHUSHQGLFXODU WR $& %&

&

%

WKHQ WKH OLQH VHJPHQW ZLOO EH

'UDZ D SHUSHQGLFXODU OLQH VHJPHQW IURP & WR $% 7KH SHUSHQGLFXODU VHJPHQW LV &'

2SXI XLEX

$& %& &' LQWHUVHFW DW SRLQW & WKH ULJKW YHUWH[

7KH DOWLWXGHV RI WKH WULDQJOH DUH $& %& &'

3UDFWLFH 'UDZ WKH LVRVFHOHV WULDQJOH $%& ZKRVH ULJKW DQJOH LV % DQG LQ ZKLFK $% FP WKHQ GUDZ WKH OLQH VHJPHQW '% IURP SRLQW % SHUSHQGLFXODU WR $& DQG ¿QG WKH OHQJWK RI WKDW OLQH VHJPHQW

'UDZ D FLUFOH ZKRVH FHQWHU LV 0 DQG UDGLXV LV HTXDO WR FP 'UDZ WKH GLDPHWHU $% DQG ODEHO WKH SRLQW & Ǻ WKH FLUFOH 0 WKHQ GUDZ WKH WULDQJOH $%& DQG WKH

OLQH VHJPHQWV IURP LWV YHUWLFHV DQG SHUSHQGLFXODU WR WKH RSSRVLWH VLGHV RI WKH WULDQJOH $%& WKHQ ODEHO WKH SRLQW RI LQWHUVHFWLRQ IRU WKHVH OLQH VHJPHQWV

'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP P d% WKHQ PHDVXUH WKH DOWLWXGHV RI WKDW WULDQJOH

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

81

3-3 Third: The altitudes of the obtuse triangle:

In the opposite figure, ABC is a triangle whose angle C is obtuse . Follow the same previous steps: Draw

AD

9 BC .

and similarly

BE

9 AC ,

CF

9 AB

F

D

Note that

1

B C

The altitudes of the triangle are the segments: AD , AD , BE ,

BE

,

E

CF .

CF intersect at

one point which is located outside the triangle ABC.

Practice 1

Draw the triangle ABC in which AB = 5cm, BC = 6cm, m (dB) = 120° . Then, draw AD perpendicular to BC , and measure the length of AD . Draw also BE perpendicular to AC and measure the length of

BE .

Are AD and BE intersected at one point?

2

Draw the rectangle ABCD in which AB = 3cm, BC = 5cm, then label the point X Ǻ DA where AX = 2cm. How many locations can be labeled for the point X on the ray DA . Draw the traingle XBC then draw XY perpendicular from X to BC .

Ma

Can you know the length of XY without measuring it by a ruler?

the mat ics -

82 82

Fifth Prima ry

Exercises

General Exercises 1 Put ( ) for the true sentence and ( ) for the false one: A

The length of the diameter of a circle > the length of any chord which doesn't pass throught its center. (

)

The right triangle has only one altitude.

(

)

The line segments drawn from the vertices of the acute triangle perpendicular to the opposite sides intersect at one point inside the triangle.

(

)

D

Only one diameter can be drawn from any point on the circle.

(

)

E

The diameter of the circle divides it into two equal halves.

(

)

B C

2 Draw a circle whose center is N and diameter is 6cm. Then draw the diameter AB and the chord AC in the circle. Draw BC . Use the portractor to measure d ACB then draw CD 9 AB that intersects it at D and the circle at E, then choose the correct answer: A

The triangle ABC is ........

B

CE is .... in the circle (chord - diameter - radius).

C

The intersection point of the perpendicular line segments drawn from the

(right triangle - acute traingle - obtuse traingle)

vertices of the triangle ABC to the opposite sides is

(C - D - E)

3 Draw a circle whose center is M and radius 4cm then draw two radii

MX , MY

and the included angle between them measures 60° then draw X Y DQG ¿QG the length of X Y .

Shorouk for Modern printing

Stude

oko B nt's

m Ter t s Fir

83 83

([HUFLVHV

'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& &$ FP 7KHQ GUDZ WKH OLQH VHJPHQW IURP SRLQW & WKDW LV SHUSHQGLFXODU WR $ % DQG ¿QG LWV OHQJWK

'UDZ WKH WULDQJOH ;<= LQ ZKLFK ;< FP <= FP =; FP 'HWHUPLQH WKH W\SH RI WKH WULDQJOH DFFRUGLQJ WR WKH PHDVXUHV RI LWV DQJOHV WKHQ GUDZ WKH SHUSHQGLFXODU VHJPHQW IURP ; WR <= DQG PHDVXUH LWV OHQJWK '

$

,Q WKH RSSRVLWH ¿JXUH ¿QG WKH SHULPHWHU RI WKH VTXDUH $%&' JLYHQ WKH OHQJWK RI WKH FLUFOH V UDGLXV FP

&

%

,Q WKH RSSRVLWH ILJXUH 0 1 DUH WZR FLUFOHV ZKHUH WKHLU GLDPHWHUV DUH FP FP )LQG WKH OHQJWK RI 01

,Q WKH RSSRVLWH ¿JXUH WKHUH DUH WZR FLUFOHV ZLWK WKH VDPH FHQWHU 0 ,I WKHLU UDGLL ZHUH FP FP &RPSOHWH

$

%

0

1

$ '

& 0

7KH OHQJWK RI & ' FP

%

7KH OHQJWK RI $ % FP

'UDZ WKH UD\ ' & IURP ' ZKLFK LQWHUVHFWV WKH VPDOO FLUFOH DW ( DQG LQWHUVHFWV WKH ODUJH FLUFOH DW ) WKHQ ¿QG WKH OHQJWK RI ' )

'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP DQG $& FP 'UDZ WKH FLUFOH 0 LQ ZKLFK $ & LV WKH GLDPHWHU DQG ILQG WKH OHQJWK RI 0 %

5I

\PM UI\ QK[

84

.QN\P 8ZQUI Za

$FWLYLW\

$FWLYLW\ 3OD\ ZLWK WKH FRPSDVVHV

'UDZ WKH IROORZLQJ ILJXUH DQG LQQRYDWH RWKHU GHFRUDWLYH ILJXUHV

1RWLFH WKH SDWWHUQ E\ SODFLQJ SRLQWV RQ WKH FLUFOH 'UDZ WKH OLQH VHJPHQWV WKDW FRQQHFW HYHU\ WZR SRLQWV WKHQ \RX ZLOO EH DEOH WR GHWHUPLQH WKH QXPEHU RI ]RQHV ZKLFK DUH LQFOXGHG LQ WKH VXUIDFH RI D FLUFOH

RQH SRLQW RQH ]RQH

WZR SRLQWV WZR ]RQHV

WKUHH SRLQWV IRXU ]RQHV

IRXU SRLQWV ]RQHV

+RZ PDQ\ ]RQHV DUH WKHUH RQ WKH FLUFOH LQ FDVH RI KDYLQJ ¿YH SRLQWV"

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

85

7HVW 8QLW 7HVW &RPSOHWH WKH IROORZLQJ WR KDYH WUXH VHQWHQFHV $

7KH FKRUG RI D FLUFOH LV D OLQH VHJPHQW WKDW FRQQHFWV

%

7KH ORQJHVW FKRUG LQ D FLUFOH LV FDOOHG

&

7KH PLGSRLQW RI DQ\ GLDPHWHU LQ D FLUFOH LV RI WKH FLUFOH

'

LV XVHG LQ GUDZLQJ WKH FLUFOH

&KRRVH WKH FRUUHFW DQVZHU IURP WKH SDUHQWKHVHV $

,I 0 LV D FLUFOH ZKRVH GLDPHWHU LV FP ZKHUH 0$ FP WKHQ WKH SRLQW $ LV ORFDWHG LQVLGH RXWVLGH RQ WKH FLUFOH

%

,I $ % EHORQJ WR WKH FLUFOH 0 ZKHUH 0 Ǻ $% WKHQ $% LV FDOOHG D

FKRUG GLDPHWHU UDGLXV LQ WKH FLUFOH &

7KH QXPEHU RI DOWLWXGHV LQ DQ\ WULDQJOH

'

,I $% $& DUH WZR FKRUGV LQ D FLUFOH WKHQ %& LV D FKRUG GLDPHWHU UDGLXV LQ WKH VDPH FLUFOH

(

0

,Q WKH RSSRVLWH ¿JXUH LI WKH OHQJWK RI HDFK UDGLXV LQ WKH WKUHH FLUFOHV LV FP WKHQ WKH SHULPHWHU RI WKH WULDQJOH 0/1 FP

$

/

1

'UDZ D FLUFOH ZKRVH FHQWUH LV 0 DQG UDGLXV LV FP 7KHQ GUDZ LWV GLDPHWHU $% DQG GUDZ LWV FKRUG $& RI OHQJWK FP 'UDZ %& WKHQ ILQG LWV OHQJWK

%

'UDZ WKH LVRVFHOHV WULDQJOH $%& LQ ZKLFK %& FP DQG $% $& FP 7KHQ GUDZ SHUSHQGLFXODU VHJPHQWV IURP WKHLU YHUWLFHV WR WKHLU WKUHH VLGHV

5I

\PM UI\ QK[

86

.QN\P 8ZQUI Za

Shorouk for Modern printing

)\TIVMQIRXEP 4VSFEFMPMX] /HW V SOD\

<RX ZLOO OHDUQ

:KHQ \RX WRVV RQH FRLQ WKHUH DUH WZR SRVVLEOH ZD\V WKH

- 7R ILQG WKH SUREDELOLW\ FRLQ FDQ ODQG HLWKHU KHDG + IURP DQ H[SHULPHQW RU D VDPSOH

RU WDLO 7

7KH FODVV LV GLYLGHG LQWR JURXSV (DFK JURXS WRVVHV D FRLQ WLPHV WLPHV WLPHV DQG WLPHV WKHQ REVHUYH WKH

- 7R SUHGLFW XVLQJ D UHVXOWV DQG UHFRUG WKHP LQ WKH IROORZLQJ WDEOH JLYHQ SUREDELOLW\

QXPEHU RI WRVVLQJ QXPEHU RI D FRLQ RFFXUUHQFH RI KHDGV

.H\ 7HUPV : ([SHULPHQWDO SUREDELOLW\

: ([SHULPHQW : 6DPSOH : 3UHGLFWLRQ

QXPEHU RI RFFXUUHQFH RI WDLOV

WLPHV

WLPHV

WLPHV

WLPHV

:KDW GR \RX QRWLFH"

1RWLFH ,QFUHDVLQJ WKH QXPEHU RI WRVVLQJ D FRLQ WHOOV WKH IDFW WKDW WKH QXPEHU RI RFFXUUHQFH RI KHDGV LV QHDUO\ HTXDO WR WKH QXPEHU RI RFFXUUHQFH RI WDLOV )RU LQVWDQFH WRVVLQJ D FRLQ WLPHV WKH QXPEHUV RI RFFXUUHQFH RI KHDGV PD\ EH WLPHV ZKLOH WKH QXPEHU RI RFFXUUHQFH RI WDLOV PD\ EH ± WLPHV ,W LV VDLG WKH SUREDELOLW\ RI RFFXUUHQFH RI KHDGV LQ

5I

\PM UI\ QK[

WLPHV ZKLOH WKH SUREDELOLW\ RI RFFXUUHQFH RI

WDLOV LQ WLPHV

.QN\P 8ZQUI Za

7KLQN ,V LW SRVVLEOH WR SUHGLFW WKH QXPEHU RI RFFXUUHQFH RI KHDGV ZKHQ WRVVLQJ D FRLQ WLPHV" ([SODLQ

)\EQTPI 7KH RSSRVLWH WDEOH VKRZV WKH UHVXOW RI D VXUYH\ RI DVNLQJ VWXGHQWV DERXW WKHLU IDYRULWH EUHDNIDVW :KDW LV WKH SUREDELOLW\ RI FKRRVLQJ IRXO DQG WDPD\D"

%UHDNIDVW

:KDW LV WKH SUREDELOLW\ RI FKRRVLQJ SLHV"

)RXO DQG WDPD\LD

:KDW LV WKH SUREDELOLW\ RI FKRRVLQJ FKHHVH DQG GHVVHUW"

,I WKH QXPEHU RI VWXGHQW LV VWXGHQWV +RZ FDQ 3LH \RX SUHGLFW DERXW WKH QXPEHU RI VWXGHQWV FKRRVLQJ &KHHVH DQG GHVVHUW IRXO DQG WDPD\D"

7SPYXMSR

3UREDELOLW\ RI FKRRVLQJ IRXO DQG WDPD\D

3UREDELOLW\ RI FKRRVLQJ SLH

3UREDELOLW\ RI FKRRVLQJ FKHHVH DQG GHVVHUW

7KXV WKH SUHGLFWLRQ DERXW WKH QXPEHU RI VWXGHQWV FKRRVLQJ IRXO DQG WDPD\D

¯ VWXGHQWV

7KLQN ,I WKH QXPEHU RI VWXGHQWV LQ WKH VXUYH\ DSSOLHG ZDV VWXGHQWV +RZ FDQ \RX SUHGLFW $ 7KH QXPEHU RI VWXGHQWV FKRRVLQJ 3LH " % 7KH QXPEHU RI VWXGHQWV FKRRVLQJ &KHHVH DQG 'HVVHUW "

)\IVGMWIW (OHFWULF ODPSV PDQXIDFWXULQJ FRPSDQ\ NHHSV WUDFN RI ODPSV RI LWV SURGXFWLRQ WR NQRZ WKH PD[LPXP ZRUNLQJ KRXUV EHIRUH WHDULQJ GRZQ 7KH IROORZLQJ WDEOH OLVWV WKHVH GDWD 0D[LPXP ZRUNLQJ KRXUV

1XPEHU RI ODPSV

OHVV WKDQ K

± K

± K

PRUH WKDQ K

,I \RX ERXJKW D ODPS DW WKLV FRPSDQ\ ZKDW LV WKH SUREDELOLW\ WR WHDU GRZQ"

$ %HIRUH ZRUNLQJ KRXUV Shorouk for Modern printing

U <MZ

\ [ Z NQ WS W *

\ [

;\]LMV

% $IWHU ZRUNLQJ KRXUV

5ROOLQJ D QXPEHU FXEH QXPEHUHG IURP WR WLPHV

+RZ PDQ\ WLPHV DUH SUHGLFWHG WR JHW DQ HYHQ QXPEHU"

$ VXUYH\ ZDV DSSOLHG WR DVN VWXGHQWV DERXW WKH IRUHLJQ ODQJXDJH WKH\ SUHIHU WR VWXG\ VWXGHQWV SUHIHU (QJOLVK VWXGHQWV SUHIHU )UHQFK DQG VWXGHQWV SUHIHU *HUPDQ ,I WKH WRWDO QXPEHU RI VWXGHQWV LQ WKH VFKRRO LV VWXGHQWV +RZ PDQ\ VWXGHQWV DUH SUHGLFWHG WR SUHIHU VWXG\LQJ *HUPDQ"

(PDQ $PO DQG 0DKD FOHDQ WKHLU VFKRRO LQ WXUQV 7KH\ UROO D QXPEHU FXEH ZLWK WZR IDFHV QXPEHUHG WZR IDFHV QXPEHUHG DQG WZR IDFHV QXPEHUHG (PDQ GRHV WKH FOHDQLQJ LI WKH IDFHV QXPEHUHG DSSHDU $PO GRHV WKH FOHDQLQJ LI WKH IDFHV QXPEHUHG DSSHDU 0DKD GRHV WKH FOHDQLQJ LI WKH IDFHV QXPEHUHG DSSHDU +RZ RIWHQ GR \RX SUHGLFW HDFK RQH GRHV WKH FOHDQLQJ ZLWKLQ D GD\ PRQWK"

5I

\PM UI\ QK[

.QN\P 8ZQUI Za

8LISVIXMGEP 4VSFEFMPMX] 7KLQN DQG 'LVFXVV

'LVFXVV ZLWK \RXU WHDFKHU WKH IROORZLQJ H[SHULPHQWV WKHLU RXWFRPHV DQG WKH VDPSOH VSDFH RI HDFK

<RX ZLOO OHDUQ - 7R ILQG DQ HYHQW

([SHULPHQW 7RVVLQJ D UHJXODU FRLQ DQG REVHUYLQJ WKH RXWFRPHV 2XWFRPHV 7KHUH DUH SRVVLEOH ZD\V WKH FRLQ FDQ ODQG KHDGV + RU WDLOV 7

6DPSOH VSDFH 6 ^+ 7` ([SHULPHQW 5ROOLQJ D UHJXODU QXPEHU FXEH QXPEHUHG IURP WR 2XWFRPHV $OO SRVVLEOH RXWFRPHV DUH RU 6DPSOH VSDFH 6 ^ `

.H\ WHUPV : 7KHRUHWLFDO SUREDELOLW\

: 2XWFRPHV RI DQ H[SHULPHQW

: 6DPSOH VSDFH

([SHULPHQW +DYLQJ D EDE\ DQG GHWHUPLQLQJ WKH JHQGHU RI WKH QHZERUQ EDE\ 2XWFRPHV D ER\ % RU D JLUO *

6DPSOH 6SDFH 6 ^% *` ([SHULPHQW 3OD\LQJ D IRRWEDOO JDPH DQG GHWHUPLQLQJ WKH UHVXOW RI D WHDP 2XWFRPHV $OO SRVVLEOH RXWFRPHV DUH ZLQ RU « « RU « 6DPSOH 6SDFH 6 ^ « « « ` Shorouk for Modern printing

U <MZ

\ [ Z NQ WS W *

\ [

;\]LMV

([SHULPHQW

7R VSLQ D VSLQQHU GLYLGHG LQWR VHFWLRQV 2XWFRPHV DOO SRVVLEOH RXWFRPHV DUH

6DPSOH VSDFH 6 ^« `

7R VSLQ DQ RGG QXPEHU LV D VXEVHW RI WKH VDPSOH

VSDFH 7KLV LQGLYLGXDO RXWFRPH LV FDOOHG DQ HYHQW DQG WKH VXEVHW ^ `

$Q HYHQW LV D VXEVHW RI WKH VDPSOH VSDFH

,Q H[SHULPHQW 7KH SUREDELOLW\ RI VSLQQLQJ DQ RGG QXPEHU

QXPEHU RI RXWFRPHV RI DQ HYHQW 1XPEHU RI DOO SRVVLEOH RXWFRPHV

)\EQTPIW

$ EDJ FRQWDLQV ZKLWH PDUEOHV EODFN PDUEOHV DQG UHG PDUEOHV $OO PDUEOHV DUH HTXDOO\ OLNHO\ LQ VL]H 5DQGRPO\ D PDUEOH LV VHOHFWHG

&DOFXODWH WKH SUREDELOLW\ RI VHOHFWLQJ

$ $ EODFN PDUEOH

% $ \HOORZ PDUEOH

& $ ZKLWH RU UHG PDUEOH

ZKLWH

EODFN

UHG

WRWDO

7SPYXMSR $ 3 EODFN

% 3 \HOORZ

1XPEHU RI EODFN PDUEOHV 1XPEHU RI DOO SRVVLEOH RXWFRPHV

LPSRVVLEOH RXWFRPH

& 3 ZKLWH RU UHG

5I

\PM UI\ QK[

.QN\P 8ZQUI Za

$ ER[ FRQWDLQV FDUGV QXPEHUHG IURP WR 5DQGRPO\ D FDUG KDV EHHQ VHOHFWHG &DOFXODWH WKH

SUREDELOLW\ RI VHOHFWLQJ $ $ SULPH QXPEHU

% $ QXPEHU GLYLVLEOH E\ 7SPYXMSR 6 ^ `

1XPEHU RI HOHPHQWV LQ 6

$ 7KH HYHQW RI VHOHFWLQJ D SULPH QXPEHU ^ ` ZLWK QXPEHU RI HOHPHQWV

3 SULPH QXPEHU

% 7 KH HYHQW RI VHOHFWLQJ D QXPEHU GLYLVLEOH E\ ^ ` ZLWK QXPEHU RI HOHPHQWV

3 GLYLVLEOH E\

3UDFWLFH ,Q WKH RSSRVLWH ¿JXUH WKH VSLQQHU LV GLYLGHG LQWR FRORUHG VHFWLRQV :KDW LV WKH SUREDELOLW\ WKDW WKH VSLQQHU VWRSV DW $ 5HG

% %ODFN & *UHHQ

&LUFOH WKH WUXH DQVZHU $ 7RVVLQJ D UHJXODU FRLQ WKH SUREDELOLW\ RI ODQGLQJ KHDG «

% 5ROOLQJ D UHJXODU QXPEHU FXEH 7KH SUREDELOLW\ RI JHWWLQJ D QXPEHU GLYLVLEOH E\ LV

& $ FODVVURRP KROGV VWXGHQWV DUH ER\V DQG WKH UHVW DUH JLUOV $ VWXGHQW KDV EHHQ UDQGRPO\ VHOHFWHG WKH SUREDELOLW\ RI JHWWLQJ D JLUO LV

Shorouk for Modern printing

U <MZ

\ [ Z NQ WS W *

\ [

;\]LMV

+IRIVEP )\IVGMWIW &RPSOHWH $ $ ER[ FRQWDLQV ODPSV ODPSV DUH GHIHFWLYH $ ODPS KDV EHHQ UDQGRPO\ VHOHFWHG WKH SUREDELOLW\ RI JHWWLQJ D IXQFWLRQDO ODPS % 7KH SUREDELOLW\ RI IDLOLQJ D VWXGHQW LV 7KH SUREDELOLW\ RI VXFFHVV

& $ FDUG KDV EHHQ GUDZQ RXW RI FDUGV FRQWDLQLQJ WKH QXPEHUV

7KH SUREDELOLW\ RI VHOHFWLQJ D QXPEHU WKDW WKH VXP RI LWV WZR GLJLWV LV ' 5ROOLQJ D UHJXODU QXPEHU FXEH 7KH SUREDELOLW\ RI JHWWLQJ DQ HYHQ QXPEHU « 7KH SUREDELOLW\ RI JHWWLQJ DQ RGG QXPEHU 7KH SUREDELOLW\ RI JHWWLQJ D SULPH QXPEHU 7KH SUREDELOLW\ RI JHWWLQJ D QXPEHU JUHDWHU WKDQ 7KH IROORZLQJ WDEOH OLVWV WKH UHVXOWV RI D VXUYH\ DSSOLHG RQ VSHFWDWRUV RI 7 9 3URJUDP

DUDELF ¿OPV

IRUHLJQ ¿OPV

VHULHV

QHZV

IRRWEDOO PDWFKHV

1XPEHU RI VSHFWDWRUV

$ VSHFWDWRU KDV EHHQ UDQGRPO\ VHOHFWHG )LQG WKH SUREDELOLW\ RI VHOHFWLQJ D VSHFWDWRU SUHIHUV $ IRRWEDOO PDWFKHV

% IRUHLJQ ¿OPV

& VHULHV

' QHZV

$ FDUG KDV EHHQ UDQGRPO\ GUDZQ RXW RI FDUGV QXPEHUHG IURP WR

)LQG WKH SUREDELOLW\ RI JHWWLQJ $ $Q RGG QXPEHU % $ SULPH QXPEHU & $Q HYHQ QXPEHU JUHDWHU WKDQ $ VSLQQHU LV GLYLGHG LQWR HTXDO VHFWLRQV $ :KDW LV WKH SUREDELOLW\ RI VSLQQLQJ RQ DQ\ VHFWLRQ"

5I

% 6SLQQLQJ WKH VSLQQHU WLPHV +RZ PDQ\ WLPHV DUH

\PM SUHGLFWHG WR JHW WKH OHWWHU $ DV DQ RXWFRPH" UI\ QK[ .QN\P 8ZQUI Za

$

%

)

& (

'

%GXMZMX]

%GXMZMX] $ VXUYH\ KDV EHHQ DSSOLHG WR DVN VWXGHQWV DERXW WKHLU IDYRULWH JDPHV $ :KDW LV WKH SUREDELOLW\ WKDW IRRWEDOO LV WKH IDYRULWH JDPH" % ,I WKH QXPEHU RI \RXU VFKRRO VWXGHQWV LV VWXGHQWV WKHQ SUHGLFW WKH QXPEHU RI VWXGHQWV SUHIHUULQJ IRRWEDOO JDPHV & :KDW LV WKH SUREDELOLW\ WKDW EDVNHWEDOO LV WKH IDYRULWH JDPH" ' ,Q \RXU RSLQLRQ KRZ PDQ\ VWXGHQWV RXW RI VWXGHQWV ZRXOG SUHIHU D EDVNHWEDOO JDPH" 6WXGHQWV IDYRULWH *DPHV *DPH

1XPEHU RI VWXGHQWV

)RRWEDOO

%DVNHWEDOO

7DEOH WHQQLV

6ZLPPLQJ

Shorouk for Modern printing

U <MZ

\ [ Z NQ WS W *

\ [

;\]LMV

9RMX 8IWX

7HVW

7KH IROORZLQJ WDEOH OLVWV WKH QXPEHU RI YROXQWHHUV LQ JURXSV WR PDNH XQLIRUPV IRU WKH FOHDQHUV *URXS 1XPEHU RI YROXQWHHUV

'HVLJQ

3ULQWLQJ

'LVWULEXWLRQ

$ YROXQWHHU KDV EHHQ UDQGRPO\ VHOHFWHG :KDW LV WKH SUREDELOLW\ WR EH RQH RI WKH SULQWLQJ JURXS" $ 5ROOLQJ D UHJXODU QXPEHU FXEH :KDW LV WKH SUREDELOLW\ RI JHWWLQJ DQ HYHQ QXPEHU DQG QRW GLYLVLEOH E\ " % $ FDUG KDV EHHQ UDQGRPO\ VHOHFWHG RXW RI FDUGV QXPEHUHG IURP WR :KDW LV WKH SUREDELOLW\ RI JHWWLQJ D SULPH QXPEHU"

7RXULVWV VXFFHVVLYHO\ YLVLW (J\SW $ WRXULVW FRPSDQ\ KDV RUJDQL]HG D WULS IRU WRXULVWV WR YLVLW (J\SW IURP $UDE FRXQWULHV IURP (XURSH IURP $PHULFD DQG IURP $VLD 7KH QXPEHU RI WRXULVWV ZKR YLVLWHG (J\SW LQ WKLV PRQWK ZDV WRXULVWV :KDW LV WKH SUHGLFWHG QXPEHU RI WRXULVWV IURP (XURSH ZKR YLVLWHG (J\SW LQ WKLV PRQWK"

$ VSLQQHU LV GLYLGHG LQWR HTXDO VHFWLRQV $ :KDW LV WKH SUREDELOLW\ RI VSLQQLQJ WKH OHWWHU %"

%

$

% 6SLQ WKH VSLQQHU WLPHV ZKDW LV WKH SUHGLFWHG QXPEHU RI JHWWLQJ OHWWHU $"

5I

\PM UI\ QK[

.QN\P 8ZQUI Za

&

'

*HQHUDO ([DPV

([DP

&KRRVH WKH FRUUHFW DQVZHU IURP WKH SDUHQWKHVHV

Ǻ ǻ ɴ RU ɶ

$

^ ` ^ `

%

·

&

^ ` Ȝ ^ `

^ ` ^ ` ^ ` RU Ƿ

'

·

RU

(

7KH SUREDELOLW\ RI WKH SXSLO V VXFFHVV LQ DQ H[DP LV WKHUHIRUH WKH SUREDELOLW\

RI IDLOLQJ LV

RU

&RPSOHWH WKH IROORZLQJ WR KDYH D WUXH VHQWHQFH

$

,I Ǻ ^ [`

WKHQ [

%

DSSUR[LPDWH WR WKH QHDUHVW KXQGUHGWK

&

:KHQ GUDZLQJ D SDSHU RXW RI ¿YH LGHQWLFDO SDSHUV QXPEHUHG DQG WKHUHIRUH WKH SUREDELOLW\ WKDW WKH GUDZQ SDSHU KDV D SULPH QXPEHU

'

; < DUH WZR VHWV ZKHUH ; ɴ < WKHQ ; ț <

(

7KH ORQJHVW FKRUG LQ D FLUFOH LV FDOOHG

$

,I WKH XQLYHUVDO VHW 8 ^[ [ LV DQ RGG QXPEHU OHVV WKDQ `

¯ ^ ` < ^ ` 'UDZ 9HQQ GLDJUDP ZKLFK UHSUHVHQWV WKH VHWV 8 ¯ < WKHQ ¿QG ¯ ț < ¯ < <C

%

)LQG WKH SURGXFW RI ¯ DQG DSSUR[LPDWH LW WR WKH QHDUHVW KXQGUHGWK

EDUUHO KDV NJV RI RLO ,I ZH ZDQW WR SDFN LW LQ ERWWOHV ZKHUH HYHU\ ERWWOH $ KROGV NJV )LQG WKH QXPEHU RI ERWWOHV

'UDZ D FLUFOH 0 ZKRVH UDGLXV LV HTXDO WR FP WKHQ GUDZ LWV GLDPHWHU $ %

DQG ODEHO DQ\ SRLQW &Ǻ WKH FLUFOH 'UDZ WKH WULDQJOH $ % & DQG GUDZ &' $ %

ZKHUH ' Ǻ $ % ¿QG WKH OHQJWK RI &' "

$

)LQG WKH DUHD RI WKH UHFWDQJOH RI PHWHUV OHQJWK DQG PHWHUV ZLGWK

%

:KHQ UROOLQJ D UHJXODU QXPEHU FXEH :KDW LV WKH SUEDELOLW\ RI JHWWLQJ

$

$ QXPEHU PRUH WKDQ "

%

$ QXPEHU OHVV WKDQ RU HTXDO WR "

:KDW LV WKH QDPH RI WKH HYHQW LQ HDFK FDVH" &

$ VDPSOH RI EDOOV DUH UHG DQG WKH UHVW LV LQ GLIIHUHQW FRORUV

:KDW LV WKH SUHGLFWHG QXPEHU RI UHG EDOOV ZKHQ WKH VDPSOH FRQWDLQV EDOOV"

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

97

*HQHUDO ([DPV

([DP

)LUVW 3XW IRU WKH WUXH VHQWHQFH DQG IRU WKH IDOVH RQH $ % & '

7KH TXRWLHQW RI GLYLGLQJ E\ Ǻ ^ `

¯

7KH SUREDELOLW\ RI D FHUWDLQ HYHQW =HUR

6HFRQG )LQG WKH WKH UHVXOW LQ HDFK RI WKH IROORZLQJ $ ¯ DSSUR[LPDWH WR WKH QHDUHVW WKRXVDQGWK

% · DSSUR[LPDWH WR WKH QHDUHVW KXQGUHGWK

)LUVW )LOO LQ WKH IROORZLQJ WR KDYH D WUXH VHQWHQFH $

^ ` ț WKH VHW RI IDFWRUV IRU

%

,I ^ ` ^ [ ` WKHQ [

&

7R GUDZ D FLUFOH ZKRVH GLDPHWHU FP VHW WKH FRPSDVVHV WR D OHQJWK HTXDO WR FP

6HFRQG ILQG WKH UHVXOW LQ HDFK RI WKH IROORZLQJ $

·

%

^ ` ^ `

7KLUG :KDW LV WKH QXPEHU ZKLFK LI PXOWLSOLHG E\ WKH SURGXFW ZLOO EH 8

/RRN DW WKH RSSRVLWH 9HQQ GLDJUDP DQG ¿QG

$

¯ ț <

%

¯ Ȝ <

&

¯ \

'

\ȼ

;

<

'UDZ WKH WULDQJOH $ % & LQ ZKLFK $ % % & FP DQG

P d % WKHQ GUDZ $ ' % & ZKLFK LQWHUVHFWV LW DW ' WKHQ ILQG WKH OHQJWK RI $'

$ EDJ FRQWDLQV ZKLWH EDOOV UHG EDOOV DQG \HOORZ EDOOV $OO WKH EDOOV DUH HTXDO LQ VL]H ,I D EDOO LV UDQGRPHO\ GUDZQ

5I

)LUVW :KDW LV WKH SUREDELOLW\ WKDW WKH GUDZQ EDOO LV ZKLWH

6HFRQG :KDW LV WKH SUREDELOLW\ WKDW WKH GUDZQ EDOO LV QRW UHG

\PM UI\ QK[

98

.QN\P 8ZQUI Za

*HQHUDO ([DPV

([DP

&KRRVH WKH FRUUHFW DQVZHU IURP WKH SDUHQWKHVH $ ^ ` ^ `

Ǻ ǻ ɴ ɶ

%

· ·

&

7KH SUREDELOLW\ RI DQ LPSRVLEOH HYHQW

'

7KH VPDOOHVW IUDFWLRQ IURP WKH IROORZLQJ LV

(

¯ ¯

Ƿ

!

)LUVW &RPSOHWH WKH IROORZLQJ WR KDYH D WUXH VHQWHQFH

$

%

^ ` ț 7KH VHW RI WKH SULPH QXPEHUV

&

7KH SUREDELOLW\ WKDW .KDOHG ZLQV D JDPH LV WKH VDPH JDPH LV

DSSUR[LPDWH WR WKH QHDUHVW KXQGUHGWK

WKHQ WKH SUREDELOLW\ RI ORVLQJ

6HFRQG )LQG WKH IROORZLQJ $

¯

%

·

DSSUR[LPDWH WR WKH QHDUHVW WHQWK

)LUVW $ PDQ ERXJKW D 79 IRU /( +H SDLG /( RI LWV FRVW DQG SDLG WKH UHPDLQGHU RQ PRQWKO\ LQVWDOOPHQWV HDFK RI WKHP LV HTXDO WR / ( )LQG WKH 8 QXPEHU RI LQVWDOOPHQWV

$

$ Ȝ %

%

$ %

&

$

%

6HFRQG /RRN DW WKH RSSRVLWH 9HQQ GLDJUDP DQG ILQG $ Ȝ % ȼ

'UDZ D FLUFOH ZKRVH FHQWHU LV 0 DQG UDGLXV FP 'UDZ

GLDPHWHU $ % /DEHO WKH SRLQWV & ' ( ZKHUH 0& FP 0' FP 0( FP WKHQ FRPSOHWH WKH IROORZLQJ

$

0 ( LV FDOOHG

%

$ ( LV FDOOHG

&

' LV ORFDWHG WKH FLUFOH

$ IDPLO\ FRQVXPHV NJV RI PHDW PRQWKO\ ZKHUH WKH FRVW RI NJ RI PHDW LV /( )LQG ZKDW WKH IDPLO\ SD\V $SSUR[LPDWH WR WKH QHDUHVW SRXQG

$

'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP &$ FP WKHQ GUDZ SHUSHQGLFXODUV IURP LWV YHUWLFHV WR WKH RSSRVLWH VLGHV DQG ODEHO WKH SRLQW RI WKHLU LQWHUVHFWLRQ

%

$ VXUYH\ KDV EHHQ DSSOLHG RQ VWXGHQWV VSHQGLQJ WKHLU VXPPHU WLPH WKH UHVXOW ZDV DV IROORZV VWXGHQWV SUHIHU VXPPHU EHDFKHV VWXGHQWV SUHIHU JRLQJ WR FOXEV DQG WKH UHVW RI VWXGHQWV SUHIHU JRLQJ WR WKH FRXQWU\VLGH :KDW LV WKH SUREDELOLW\ WKDW RQH RI WKHP VSHQGV KLV VXPPHU LQ WKH FRXQWU\VLGH" :KDW LV \RXU SUHGLFWLRQ DERXW WKH QXPEHU RI VWXGHQWV SUHIHUULQJ WR VSHQG WKHLU VXPPHU WLPH LQ WKH FRXQWU\VLGH RXW RI VWXGHQWV" ZU \ <M

Shorouk for Modern printing

;\]L

.QZ[

S *WW MV\ [

99

$QVZHUV

:VS\[PVU VM [OL X\LZ[PVU PU [OPUR P[LT VU WHNL ! $ 3, ,_LYJPZLZ WHNL H I

K L

H /\UKYLK

M

I /\UKYLK J K L M

N O

;V [OL ULHYLZ[ O\UKYLK[O ;V [OL ULHYLZ[ [OV\ZHUK[O H [LU I J K RT

;OL VYKLY PZ! ɏ ɓ ɕ ɏɑ ɐɓ ɏɖ

ɏɖ

ɏɖ

ɏɖ

ɏɖ

H _ $ VY VY I _ $ VY ;OL HZJLUKPUN VYKLY PZ! ɏɐɓ ɐɑ ɑɒ ɓɔ ɏɏ ɏɐ ɓ ɑ ɐ ɓ ;OL KLZJLUKPUN VYKLY PZ! ɏɏ

ɏɐ ɔ ɒ ɑ ɏɐ ,_LYJPZLZ WHNLZ H I J K L M H H $ I I $ J J $ H ;OL HZJLUKPUN VYKLY PZ! ɏɐ ɏɐ ɏɐ ɏɐ ɏɐ ɏɕ ɏɓ ɏɑ ɕ ɓ I ;OL KLZJLUKPUN VYKLY PZ! ɑɐ ɑɖ ɑɓ ɔɖ ɏɖ ɐɏ

,_LYJPZLZ WHNL H I J K L M N O P Q 3 T H I J H $ I # J % H NYHTZ I WPHZ[LYZ J JT K TL[LYZ

\PM UI\ QK[

100

0 * $ * $ * * $ $ H

.QN\P 8ZQUI Za

.LULYHS L_LYJPZLZ WHNL ( ǻ ) ɴ * ǻ + ɶ , ɴ - ɴ ( b d ) _ $ ` $ * ` _ K b d , _ $ ( ɴ ) Ƿ * + Ƿ , _ ` _ ț ` ` _ _G ,_LYJPZLZ WHNL

J

9HKPP HYL! ) 4 4+ 4* 4( ;OL KPHTL[LY VM [OL JPYJSL PZ! ( * *OHYKZ VM [OL JPYJSL HYL! () (+ ) * *+

K

9HKPP HYL!(5 5- 5* 5) 5,

I

L M

,_LYJPZLZ WHNL

5I

,_LYJPZLZ WHNL H I J

3 , >OH[ (OTLK WH`Z $ * $ 3 , $ $ 3 , ,_LYJPZLZ WHNLZ H $ I $ J % H I J H % I $ J # K $ 7YHJ[PJL WHNL H I J K L M H I

WHWLYZ <UP[ [LZ[ WHNL ( ) * +

;OL KPHTL[LYZ VM [OL JPYJSL HYL! ), * - *OHYKZ VM [OL JPYJSL HYL ! ( ) )* 5 ) - + ,_LYJPZLZ WHNL ( ;OL WVPU[ * PZ SVJH[LK V\[ZPKL [OL JPYJSL ,_LYJPZLZ WHNL @LZ .LULYHS L_LYJPZLZ WHNL ( ) * + , <UP[ [LZ[ WHNL ( (U` [^V WVPU[Z VU [OL JPYJSL ) ;OL KPHTL[LY VM [OL JPYJSL * ;OL JLU[LY + ( JVTWHZZLZ ( 6\[ZPKL ) ( KPHTL[LY * + ( JOVYK , ,_LYJPZLZ WHNLZ ɖɎ $ -PYZ[! 7 ,]LY` OV\YZ $ ɏɎɎɎ :LJVUK! 7 (M[LY OV\YZ $ ɑɓɎ ɟ ɑɐɎ ɏɎɎɎ $ [PTLZ

$QVZHUV

.LULYHS L_LYJPZLZ WHNL ɕ ( 7 ]HSPK SHTWZ $ ɐɏ ɐɒ $ ɖ ) 7 :\JJLZZ $ ɏɑ ɏɓ * 7 Z\T VM [OL [^V KPNP[ PZ $ ɐɓ + 7 L]LU U\TILY $ ɑɔ $ ɏɐ 7 VKK U\TILY $ ɑɔ $ ɏɐ 7 WYPTL U\TILY $ ɑɔ $ ɏɐ 7 U\TILY NYLH[LY [OHU $ Ɏɔ $ ( 7 ^H[JOPUN MVV[IHSS TH[JOLZ ɑɔ $ $ ɏɎɎ ɐɎ $ ) 7 ^H[JOPUN MVYLPNU ÄSTZ $ ɏɎɎ ɏɓ $ * 7 ^H[JOPUN ZLYPLZ $ ɏɎɎ ɏɎ $ + 7 ^H[JOPUN UL^Z $ ɏɎɎ ( 7 VKK U\TILY $ ɏɎɓ $ ) 7 WYPTL U\TILY $ ɏɎɒ $ * 7 L]LU U\TILY NYLH[LY [OHU $ ɏɎɐ $ ( 7 [OL PUKPJH[VY Z[VWZ VU HU` ZLJ[PVU $ ɏɔ ) [PTLZ

H[DP

ZLJVUK! $

$ ? ʴ @ $ b d ? @ $ b d @G $ b d % $

IV[[SLZ

-PYZ[! ZX\HYLK TL[LY :LJVUK! $ ALYV PTWVZZPISL L]LU[

% H JLY[HPU L]LU[ ;OPYK! IHSSZ

<UP[ [LZ[ WHNL ɑɎ $ ɏ 7 WYPU[PUN ZL[ $ ɏɐɎ ɒ ( 7 VKK U\TILY PZ KP]PZPISL I` $ ɐɔ $ ɏɑ ) 7 WYPTL U\TILY $ ɐɓɗ ;OL WYVIHIPSP[` [OH[ [OL [V\YPZ[ PZ MYVT ɑɎ $ ɑ ,\YVWL $ ɏɎɎ ɏɎ 5\TILY VM ,\YVWLHU [V\YPZ[Z $ _ ɏɎɑ $ [V\YPZ[Z ( ɏɒ )

$ ɴ % & ALYV ' ( #

MPYZ[! $ % b d & ɏɑ

$ _ $ % & ɑ ɓ ' ? ( KPHTL[LY

H[DP

$ ɴ % & b d ' ( ɏɓ

%

MPYZ[! PUZ[HSSTLU[Z ZLJVUK! $ b d % b d

& b d $ YHKP\Z % JOVYK & V\[ZPKL

ZLJVUK! 3,

$ 7YVIHIPSP[` $ ɏɐ ɐɓ U\TILY $

H[DP

MPYZ[! $ % & '

ZLJVUK! $

%

MPYZ[! $ b d % & JT ZLJVUK! $

% b d ;OPYK! $ b d % b d & b d ' b d

MPYZ[! ;OL WYVIHIPSP[` [OH[ [OL IHSS PZ ^OP[L $ ɏɓ ZLJVUK! ;OL WYVIHIPSP[` [OH[ [OL IHSS PZ UV[ YLK $ ɏɓɖ

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

101

*HQHUDO H[HUFLVHV

*HQHUDO H[HUFLVHV WK 3ULP 8QLW RQH WKH IUDFWLRQV

&RPSOHWH

P NP

NLORPHWHUV PHWUHV

WR WKH QHDUHVW WHQWK

WR WKH QHDUHVW XQLW

WR WKH QHDUHVW

WR WKH QHDUHVW

WR WKH QHDUHVW KXQGUHGWK

WR WKH QHDUHVW

*

WKH QXPEHU

WR WKH QHDUHVW

± WR WKH QHDUHVW XQLW

*

GP FP

*

*

WR WKH QHDUHVW KXQGUHGWK

5I

\PM UI\ QK[

102

.QN\P 8ZQUI Za

*HQHUDO H[HUFLVHV

WR WKH QHDUHVW WHQWK

·

GD\V ZHHNV

WR WKH QHDUHVW ZHHN

WR WKH QHDUHVW ZKROH QXPEHU

·

D

WKHQ D

NP P FP

* ·

* WR WKH QHDUHVW ZKROH QXPEHU

*

·

± WR WKH QHDUHVW

WRQ NJ

KRXU PLQXWH VHFRQG

KRXU

&KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ

7KH TXRWLHQW RI GLYLGLQJ ·

$

%

&

'

'

GD\V WR WKH QHDUHVW ZHHN

$

Shorouk for Modern printing

%

&

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

103

*HQHUDO H[HUFLVHV

7KH QXPEHU WR WKH QHDUHVW KXQGUHGWK

$

%

&

'

&

'

&

'

&

'

'

·

$

·

$

%

%

· ·

$

%

(VWLPDWH WKH UHVXOW

$

%

&

7KH QXPEHU RI GD\V LQ KRXUV HTXDOV DSSUR[LPDWHO\

$

%

&

'

'

'

'

'

'

7KH QXPEHU RI PRQWKV LQ KDOI D \HDU

$

%

&

7KH QXPEHU RI \HDUV LQ PRQWKV

$

*

*

%

&

$ % &

$ % &

WR WKH QHDUHVW KXQGUHGWK

*

*

5I

\PM UI\ QK[

104

$

.QN\P 8ZQUI Za

%

&

*HQHUDO H[HUFLVHV

*

$

%

&

'

%

&

'

Â&#x2022;

$

!

7KH VPDOOHVW IUDFWLRQ LQ WKH IROORZLQJ LV

$

%

&

'

7KH JUHDWHVW QXPEHU LQ WKH IROORZLQJ LV

$

%

&

'

$QVZHU WKH IROORZLQJ TXHVWLRQV

$UUDQJH WKH IROORZLQJ QXPEHUV DVFHQGLQJO\

$UUDQJH WKH IROORZLQJ QXPEHUV GHVFHQGLQJO\

$UUDQJH WKH IROORZLQJ QXPEHUV GHVFHQGLQJO\

7KH OHQJWK RI D SLHFH RI FORWK LV P WRZHOV DUH PDGH RI LW WKH OHQJWK RI HDFK WRZHO LV P +RZ PDQ\ PHWUHV DUH UHPDLQGHU"

)LQG WKH UHVXOW RI WKH IROORZLQJ

$

± * WR WKH QHDUHVW

%

·

&

·

'

(

*

)

*

± ·

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

105

General exercises

G

45.334 * 100 = ..................

H

17 ÷ 0.85 = ..................

40

I

9375 ÷ 15 = ..................

J

25.25 ÷ 0.25 = ..................

B

3.2 * 10

D

2 dm

(6) Put the suitable symbol (> , = , <):

(7)

A

4.79 * 1000

C

2 m 5

E

140.44

47.9 * 100

5 m 2

0.32 * 1000 200 cm

34.044

If the price of one metre of cloth is 7.35 pounds , what is the price of 3.5 metres ?

(8)

A car consumes one litre of Benzine to cover a distance 10 km.

How many litres are needed so that the car covers a distance of 642.9 km.

(9)

Find the area of the rectangle if its dimensions are 2.4 and 4.5 cm. then approximate the result to the nearest tenth.

(10) If a = 18.24 , b = 8.354 Find the result of a + b to the nearest hundreadth.

Estimate the result of a + b. Is your estimate accurate or not?

(11) Mahmoud bought a computer for 2000 pounds. He paid 250 pounds cash money and divided the remainder into 50 equall monthly instalments. Calculate the value of each instalment.

(12) The product of two numbers is 625. If one of them is 25 then what is the other number ?

(13) Find the result of each of First : 2.4 * 4.7

Ma

WKHQ IURP WKH SUHYLRXV RSHUDWLRQV ¿QG WKH YDOXH RI A

the mat ics -

106

Second: 3.4 * 0.29

(2.4 * 4.7) * 0.29

Fifth Prima ry

B

2.4 * (3.4 * 0.29)

106

*HQHUDO H[HUFLVHV

$ PHGLFDO ¿UP SDFNHG OLWUH RI D PHGLFLQH LQ ERWWOHV HDFK RI WKHP LV RI FDSDFLW\ OLWUH +RZ PDQ\ ERWWOHV ZHUH XVHG "

6HFRQG 8QLW 6HWV

)LUVW FRPSOHWH

&RPSOHWH E\ SXWWLQJ WKH VXLWDEOH V\PERO Ǻ ǻ ɶ

$

^ `

%

^ ` ^ `

&

Â&#x2C6; ^ `

'

^ ` ^ `

(

^ `

)

^ ` ^ `

*

^ ` ^ `

+

^ ` ^ `

&RPSOHWH

$

^ ` ^ `

%

^ ` ^ `

&

^ ` ^ `

'

^ ` ^ `

(

^ ` ^ `

)

^D E F` ^E F D`

,I ; DQG < DUH WZR QRQ HPSW\ VHWV WKHQ

$

; Â&#x2C6;

&

,I ; < WKHQ ; <

%

; ;

'

,I ; < < WKHQ

&RPSOHWH E\ SXWWLQJ WKH VXLWDEOH V\PERO Ǻ ǻ ɶ

,I < ^ ` ^ ` WKHQ

$

^ ` <

&

<

Shorouk for Modern printing

%

^ ` <

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

107

*HQHUDO H[HUFLVHV

,I $ ^ ` ± ^ ` WKHQ

$

$

%

_ a $

'

$

(

_ a $

&

_ a $

,I ; ^ ` ^ ` WKHQ ; ^ ` ^ ` ,I ; < WKHQ ; ± < ,I ; < WKHQ ; È&#x203A; < ; ,I ; DQG < 8 WKHQ ; < < ; ^ ` ^ ` ^ ` ^ ` ]HUR ^ ` WKH VHW RI IDFWRUV RI WKH QXPEHU

6HFRQG &KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ ^ ` ^ ` $

Ǻ

%

Ç»

&

'

ɶ

%

Ç»

&

'

ɶ

'

Ç· ^ ` $

Ǻ

7KH QXPEHU RI VXEVHWV RI WKH VHW ^ ` HTXDOV $

%

&

^ ` WKH VHW RI IDFWRUV RI WKH QXPEHU LV $

^ `

%

^ `

&

^ `

'

^ `

&

Ç·

'

,I ; < WKHQ ; < $

;

%

<

,I 8 LV WKH VHW RI RGG QXPEHUV OHVV WKDQ WKHQ ^ ` 8

5I

$

Ǻ

\PM UI\ QK[

108

.QN\P 8ZQUI Za

%

Ç»

&

'

ɶ

*HQHUDO H[HUFLVHV

,I ^ ` ^ ; ` WKHQ ;

'

,I ^ D ` ɶ ^ ` WKHQ D $ % &

'

^ ` ^ ` $ Ǻ % ǻ

&

'

ɶ

,I ; < WKHQ ; ± < $ ; % <

&

Â&#x2C6;

'

8

,I ; < < WKHQ ; < $ Ǻ % ǻ

&

'

ɶ

,I ^ ` ^ [ ` WKHQ [ $ % &

'

'

ɶ

'

ɶ

$

%

&

,I 8 ^ ` WKHQ

)LUVW Â&#x2C6; 8 $ Ǻ % Ç» &

6HFRQG 8 8 $ Ǻ % ǻ &

7KLUG ^ ` 8 $ Ǻ % ǻ &

'

ɶ

7KLUG $QVZHU WKH IROORZLQJ TXHVWLRQV 5HSUHVHQW WKH WZR VHWV $ DQG % E\ 9HQQ GLDJUDP LQ HDFK RI WKH IROORZLQJ FDVHV WKHQ ¿QG $ % $

$ ^ ` % ^ `

&

$ ^ ` % ^ `

%

$ ^ ` % ^ `

5HSUHVHQW WKH WZR VHWV $ DQG % LQ HDFK RI WKH IROORZLQJ E\ 9HQQ GLDJUDP WKHQ ¿QG $ % $

$ ^ ` % ^ `

&

$ ^D P [` % ^D I [ P`

Shorouk for Modern printing

%

$ ^ ` % ^ `

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

109

General exercises

Using the operations of intersection and union, The difference and complement, express the shaded part in each of the follwing diagrams.

Fig. (1)

Fig. (2)

Fig. (3)

Fig. (4)

Fig. (5)

Fig. (6)

Use the opposite Venn diagram to write the following sets: U A X–Y B X Y C X D (X Y Y) E (X Y)

X

5

x

Y

3

x

2 x1

x

4

x

6

x

Use the opposite Venn diagram to write the following sets: Z X A X,Y,Z B Y–X,Z–Y C X – (Z ) D (Y – X) 2 5 1 4 3 (Z – Y) Y E (X – Z) (Z – Y) Use the symbols of intersection, union, difference and complement to write the shaded part in each Venn diagram of the following diagrams: x

Fig. (1)

Fig. (2)

Fig. (3)

x

x

x

x

Fig. (4)

Using the opposite diagram write the following sets: A C

X

Y

X–Y

B D

X (X

Y

U 6 x7 x

Y)

8

x

Ma

the mat ics -

110 110

Fifth Prima ry

Y

Z

5

3

x

x

4

x

1

X

x

2

x

9

x

*HQHUDO H[HUFLVHV

8QLW 7KUHH *HRPHWU\

)LUVW &RPSOHWH

$Q\ FKRUG ZKLFK SDVVHV WKURXJK WKH FHQWUH RI WKH FLUFOH LV FDOOHG LQ LW

7KH WULDQJOH ZKRVH PHXVXUHV RI DQJOHV DUH ┬Г ┬Г DQG ┬Г LV FDOOHG

7KH WULDQJOH LQ ZKLFK WKHUH DUH WZR HTXDO VLGHV LQ OHQJWK LV FDOOHG

7KH WULDQJOH ZKLFK WKH PHDVXUHV RI LWV DQJOHV DUH ┬Г ┬Г DQG ┬Г LV FDOOHG

7KH WULDQJOH ZKLFK WKH OHQJWKV RI LWV VLGHV DUH HTXDO LV FDOOHG

$ FLUFOH LV RI GLDPHWHU OHQJWK FP WKHQ LWV UDGLXV OHQJWK FP

,Q WKH RSSRVLWH ┬┐JXUH &RPSOHWH

$

P тАл ▄втАм$%&

%

7KH SHULPHWHU RI ( $%& FP

&

7KH QXPEHU RI DOWLWXGHV RI WULDQJOH $%&

7KH QXPEHU RI DOWLWXGHV RI WKH REWXVH DQJOHG WULDQJOH LV

,Q WKH RSSRVLWH ┬┐JXUH &RPSOHWH

$ LV FDOOHG D GLPHWHU LQ WKH FLUFOH ZKRVH FHQWUH LV 0

%

< = LV FDOOHG LQ WKH FLUFOH ZKRVH FHQWUH LV 0

&

(DFK RI ; 0 < 0 DQG = 0 LV FDOOHG LQ WKH FLUFOH 0

'

( <0= LV FDOOHG WULDQJOH GXH WR LWV VLGHV

7KH FKRUG RI WKH FLUFOH ZKLFK SDVVHV WKURXJK LWV FHQWUH LV

$Q\ OLQH VHJPHQW ZKLFK MRLQV WZR SRLQWV RQ WKH FLUFOH LV FDOOHG

,I WKH OHQJWKV RI VLGHV RI ( $%& DUH OHQJWK XQLW WKHQ LW LV FDOOHG WULDQJOH DQG WKH PHDVXUH RI HDFK RI LWV DQJOHV

7KH GLDPHWHU RI WKH FLUFOH RI UDGLXV FP HTXDOV FP

,W LV SRVVLEOH WR GUDZ D WULDQJOH LI WKH OHQJWKV RI DUH NQRZQ

:KDW DUH WKH NLQGV W\SHV RI WKH WULDQJOH GXH WR LWV DQJOHV Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

111

*HQHUDO H[HUFLVHV

'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP $& FP WKHQ GUDZ WKH FLUFOH 0 ZKRVH GLDPHWHU LV $ & WKHQ ¿QG

$

7KH SHULPHWHU RI WKH WULDQJOH $%&

%

8VH WKH SURWUDFWRU WR ¿QG WKH PHDVXUH RI

&

7KH OHQJWKV RI $ 0 % 0 & 0 ZKDW GR \RX GHGXFH "

'

7KH W\SH RI ( 0%& GXH LWV DQJOHV

(

0HQWLRQ WZR LVRVFHOHV WULDQJOHV

â&#x20AC;« Ü¢â&#x20AC;¬$%&

'UDZ WKH FLUFOH ZKRVH GLDPHWHU OHQJWK FP 'UDZ % & DV D GLDPHWHU RI LW WKHQ XVH % DV D FHQWUH DQG XVH WKH FRPSDVVHV ZLWK OHQJWK FP WR GUDZ DQ DUF WR LQWHUVHFW WKH FLUFOH DW ; DQG < MRLQ HDFK RI % ; % < & < & ; ; <

,I ( LV WKH SRLQW RI LQWHUVHFWLRQ RI % & DQG ; <

)LUVW &RPSOHWH XVLQJ WKH SURWUDFWRU

$

%

&

'

6HFRQG &KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ

$

â&#x20AC;« Ü¢â&#x20AC;¬%;&

P â&#x20AC;« Ü¢â&#x20AC;¬%<&

P â&#x20AC;« Ü¢â&#x20AC;¬%(&

P â&#x20AC;« <(& Ü¢â&#x20AC;P

( %;& LV WULDQJOH DFXWH DQJOHG REWXVH DQJOHG ULJKW DQJOHG

%

( %;< LV WULDQJOH DFXWH DQJOHG REWXVH DQJOHG ULJKW DQJOHG

&

( &;< LV WULDQJOH DFXWH DQJOHG REWXVH DQJOHG ULJKW DQJOHG

5I

\PM UI\ QK[

112

.QN\P 8ZQUI Za

*HQHUDO H[HUFLVHV

7KLUG &RPSOHWH

$

7KH SRLQW RI LQWHUVHFWLRQ RI DOWLWXGHV RI ( ;%< LV

%

7KH DOWLWXGHV RI ( ;%& LQWHUVHFW DW WKH SRLQW

,Q RSSRVLWH ¿JXUH

,I WKH UDGLXV OHQJWK RI WKH FLUFOH 0 HTXDOV FP

DQG WKH UDGLXV OQJWK RI WKH FLUFOH 1 FP

DQG WKH UDGLXV OHQJWK RI WKH FLUFOH ( FP

&RPSOHWH WKH IROORZLQJ

$

01 FP

&

(1 FP

'

7KH SHULPHWHU RI ( 0(1 FP

%

0( FP

,Q RSSRVLWH ¿JXUH

3XW WKH VXLWDEOH V\PERO ! RU RU

WR JHW D FRUUHFW VWDWHPHQW

$

$ 0 $ %

%

& % $ %

&

0 & 0 %

'

$ & 0 $

'UDZ WKH UHFWDQJOH $%&' ZKHUH $% FP %& FP WDNH WKH SRLQW

/ Ǻ $ ' ZKHUH $/ FP

'UDZ ( /%& WKHQ GUDZ / = SHUSHQGLFXODU WR % &

)LQG WKH OHQJWK RI / = ZLWKRXW PHDVXULQJ WKHQ ¿QG WKH SHULPHWHU RI WKH UHFWDQJOH '/=&

'UDZ WKH LVRVFHOHV WULDQJOH $%& ZKLFK LV ULJKW DQJOHG DW % ZKHUH $% FP IURP % GUDZ WKH OLQH VHJPHQW ZKLFK LV SHUSHQGLFXODU WR $ & VD\ % ' DQG PHDVXUH LWV OHQJWK Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

113

*HQHUDO H[HUFLVHV

'UDZ WKH FLUFOH 0 ZKRVH UDGLXV OHQJWK LV FP 'UDZ $ % DV D GLDPHWHU LQ LW ORFDWH WKH SRLQWV & ' DQG ( VXFK WKDW 0& FP 0' FP 0( FP WKHQ FRPSOHWH

$

0 ( LV FDOOHG

&

$ ( LV FDOOHG

%

7KH SRLQW ' OLHV WKH FLUFOH

'UDZ ( ;<= ZKLFK LV HTXLODWHUDO DQG LWV VLGH OHQJWK FP 'UDZ D FLUFOH RI FHQWUH ; DQG UDGLXV OHQJWK FP

&RPSOHWH WKH IROORZLQJ

$

; < LV FDOOHG LQ WKH FLUFOH ;

%

; = LV FDOOHG LQ WKH FLUFOH ;

&

< = LV FDOOHG LQ WKH FLUFOH ;

'

7KH SHULPHWHU RI ( ;<= FP

7KH IRXUWK XQLW WKH SUREDELOLW\ WK \HDU SULP

&KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ

$V WKURZLQJ D IDLU GLH RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ WKH QXPEHU HTXDOV

$

%

&

'

,I D IDLU GLH LV WKURZQ RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU JUHDWUH WKDQ LV

$

%

]HUR

&

'

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ DQ RGG QXPEHU LV

$

%

&

'

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU OHVV WKDQ LV

5I

\PM UI\ QK[

114

$

.QN\P 8ZQUI Za

%

&

'

*HQHUDO H[HUFLVHV

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU JUHDWU WKDQ RQH LV $ % & '

$V WRVVLQJ D PHWDOLF FRLQ RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D KHDG LV $ % & '

7KH SUREDELOLW\ WKDW WKH HOHSDKDQW Ã&#x20AC;LHV $ % ]HUR & Â&#x2C6; '

,W LV WKDW WKH VXQ ULVHV IURP ZHVW $ H[SHFWHG % SRVVLEOH & VXUH ' LPSRVVLEOH

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU GLYLVELOH E\ LV $ % & '

$V WKURZLQJ D PHWDOLF FRLQ RQFH WKHQ WKH SUREDELOLW\ WKDW D WDLO DSSHDUV LV $ % & '

$ FODVV KDV SXSLOV RI WKHP DUH ER\V DQG WKH UHPDLQGHU DUH JLUOV ,I D SXSLO LV FKRVHQ UDQGRPO\ WKHQ WKH SUREDELOLW\ WKDW WKH SXSLO LV D JLUO LV $ % & '

$V WKURZLQJ D IDLU GLH RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D SULPH QXPEHU LV $ % & '

$V WKURZLQJ D IDLU GLH RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU OHVV WKDQ RU HTXDO WR LV $ % & '

7KH SUREDELOLW\ RI WKH VXUH HYHQW LV $ ]HUR % & '

7KH SUREDELOLW\ WKDW D ELUG Ã&#x20AC;LHV

$

Shorouk for Modern printing

%

&

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [ '

115

*HQHUDO H[HUFLVHV

7KH IROORZLQJ WDEOH VKRZV WKH UHVXOW RI D VXUYH\ KDV EHHQ DSSOLHG WR NQRZ WKH YLHZV RI SXSLOV DERXW WKH IDYRULWH JDPH WR WKHP $

,I RQH SXSLO LV FKRVHQ DW UDQGRP DQVZHU IROORZLQJ TXHVWLRQV 7KH JDPH

IRRWEDOO

KDQGEDOO

EDVNHWEDOO

QXPEHU RI YLHZV

:KDW LV WKH SUREDELOLW\ WKDW RQH RI WKHP SUHIHUV IRRWEDOO

:KDW LV WKH SUREDELOLW\ WKDW RQH RI WKHP SUHIHUV KDQGEDOO

:KDW LV WKH SUREDELOLW\ WKDW RQH RI WKHP SUHIHUV EDVNHWEDOO

%

,I WKHUH DUH SXSLOV ZKDW LV WKH H[SHFWHG YDOXH RI WKH QXPEHU RI SXSLOV ZKR SUHIHU IRRWEDOO

&

,I WKHUH DUH SXSLOV ZKDW LV WKH H[SHFWHG YDOXH RI WKH QXPEHU RI SXSLOV ZKR SUHIHU EDVNHWEDOO

$ VSLQQHU LV GLYLGHG LQWR HTXDO FLUFXODU VHFWRUV LQ DUHD RI WKHP DUH UHG WZR VHFWRUV DUH JUHHQ RQH VHFWRU LV EURZQ ZKDW LV WKH SUREDELOLW\ WKDW WKH SRLQWHU $

:LOO VWDQG RQ WKH UHG FRORXU"

%

:LOO VWDQG RQ WKH EURZQ FRORXU"

&

:LOO VWDQG RQ WKH JUHHQ FRORXU "

$ IDLU GLH LV WKURZQ RQFH ZKDW LV WKH SUREDELOLW\ RI HDFK RI WKH IROORZLQJ HYHQWV"

5I

$

DSSHDULQJ DQ RGG QXPEHU

%

DSSHDULQJ DQ HYHQ QXPEHU

&

DSSHDULQJ D SULPH QXPEHU DQG QRW HYHQ

'

DSSHDULQJ DQ RGG SULPH QXPEHU

(

DSSHDULQJ D QXPEHU OHVV WKDQ RQH

DSSHDULQJ D QXPEHU PRUH WKDQ \PM UI\ QK[ .QN\P 8ZQUI Za

)

116

*HQHUDO H[HUFLVHV

*

DSSHDULQJ D QXPEHU OHVV WKDQ

+

DSSHDULQJ D QXPEHU JUHDWHU WKDQ ]HUR DQG OHVV WKDQ

,

:KDW LV WKH QDPH RI HDFK RI WKH SUHYLRXV HYHQWV "

$ EDJ FRQWDLQV UHG EDOOV EODFN EDOOV DQG ZKLWH EDOOV DOO RI WKHP DUH LGHQWLFDO DQG HTXDO LQ YROXPH $ EDOO LV GUDZQ UDQGRPO\ FDOFXODWH WKH SUREDELOLW\ WKDW $

7KH GUDZQ EDOO LV EODFN

%

WKH GUDZQ EDOO LV QRW JUHHQ

&

WKH GUDZQ EDOO LV UHG RU ZKLWH

'

WKH GUDZQ EDOO LV QHLWKHU UHG QRU ZKLWH

$ GLH LV WKURZQ WLPHV KRZ PDQ\ WLPHV WKDW WKH DSSHDULQJ QXPEHU LV RGG" $ ER[ FRQWDLQV FDUGV QXPEHUHG IURP WR ,I D FDUG LV GUDZQ UDQGRPO\ FDOFXODWH WKH SUREDELOLW\ WKDW WKH GUDZQ FDUG FDUULHV $

DQ RGG QXPEHU

%

D QXPEHU GLYLVELOH E\

&

$Q HYHQ SULPH QXPEHU

'

D QXPEHU GLYLVLEOH E\

(

D QXPEHU GLYLVLEOH E\

)

D QXPEHU GLYVLEOH E\ RU

*

D QXPEHU OHVV WKDQ

+

D QXPEHU LQ ZKLFK WKH XQLWV GLJLW LV

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

117

([DPV

7HVW 2QH &RPSOHWH HDFK RI WKH IROORZLQJ

·

WR WKH QHDUHVW

GD\V ZHHNV WR WKH QHDUHVW ZHHN

Ã&#x2014;

^ ` ^ `

&KRRVH WKH FRUUHFW DQVZHU

KDOYHV ¿IWK $ ! % & Â&#x201D;

^ ` ^ ` $ Ǻ % ǻ &

7KH QXPEHU RI DOWLWXGHV RI WKH WULDQJOH LV $ ]HUR % &

7KH SUREDELOLW\ RI WKH LPSRVVLEOH HYHQW $ Ç· % ]HUR &

FP WR WKH QHDUHVW PHWUH $ % &

$

)LQG E\ OLVWLQJ PHWKRG

; <

'

ɶ

'

'

'

; ± <

8 ;

<

[

[

; <

[

[

[

[

[

;C <C

%

$UUDQJH WKH IROORZLQJ QXPEHUV DVFHQGLQJO\

$

'UDZ WKH HTXLODWHUDO WULDQJOH $%& ZKRVH VLGH OHQJWK FP WKHQ

GUDZ $ ' B % & 7KHQ ¿QG

5I

'

,Q 9HQQ GLDJUDP LQ IURQW RI \RX

7KH SHULPHWHU RI ( $%&

\PM UI\ QK[

118

.QN\P 8ZQUI Za

â&#x20AC;«& Ü¢â&#x20AC;¬$' E\ PHDVXULQJ

P

([DPV

,Q WKH RSSRVLWH ¿JXUH &RPSOHWH

0 $

7KH ORQJHVW FKRUG LQ WKH FLUFOH LV DQG

LW LV FDOOHG %

0

$

%

; % LV FDOOHG LQ WKH FLUFOH ZKRVH FHQWUH LV ; $ $V WKURZLQJ D IDLU GLH RQFH FDOFXODWH WKH SUREDELOLW\ RI

)LUVW DSSHDULQJ D QXPEHU JUHDWHU WKDQ

6HFRQG DQ DSSHDULQJ HYHQ QXPEHU JUHDWHU WKDQ

7KLUG DSSHDULQJ DQ HYHQ SULPH QXPEHU % )LQG WKH SHULPHWHU RI WKH UHFWDQJOH ZKRVH OHQJWK LV FP DQG LWV ZLGWK LV WKHQ &DOFXODWH LWV DUHD

7HVW 7ZR $QVZHU WKH IROORZLQJ TXHVWLRQV

&KRRVH WKH FRUUHFW DQVZHU

· · $ % &

7KH ORQJHVW FKRUG LQ WKH FLUFOH LV FDOOHG $ D UDGLXV % D FKRUG & D GLDPHWHU

Ã&#x2014; Ã&#x2014; $ ! %

^ ` $ Ǻ % ǻ

KRXUV $ %

'

'

D WDQJHQW

&

&

'

ɶ

&

'

&RPSOHWH HDFK RI WKH IROORZLQJ

$%& LV DQ HTXLODWHUDO WULDQJOH RI VLGH OHQJWK FP WKHQ LWV SHULPHWHU FP

,I ; < WKHQ ; <

· ·

^ ` ± ^ `

± Ã&#x2014; WR WKH QHDUHVW WHQWK Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

119

([DPV

$

%

$ FDU FRYHUV HTXDO GLVWDQFHV LQ HTXDO WLPHV ,I WKLV FDU FRYHUHG N

LQ RQH KRXU KRZ PDQ\ NP DUH FRYHUHG LQ KRXUV :ULWH ZKDW LV UHSUHVHQWHG E\ VKDGHG SDUW LQ HDFK GLDJUDP %

$

,I WKH XQLYHUVDO VHW 8 WKH VHW RI DOO IDFWRUV RI WKH QXPEHU DQG ; ^ ` < ^ ` 'UDZ 9HQQ GLDJUDP ZKLFK UHSUHVHQWV WKH VHWV 8 [ \ WKHQ )LQG ; < ; ± < <ఴ ;ఴ ఴ % :ULWH DOO WKH VXEVHWV RI WKH VHW ; ^D E` ZKDW LV WKH QXPEHU RI VXEVHWV" $ 'UDZ ( $%& ZKHUH $% FP %& FP &$ FP WKHQ GUDZ D FLUFOH RI FHQWUH % DQG LWV UDGLXV OHQJWK FP )URP WKH GUDZLQJ FRPSOHWH

)LUVW WKH SRLQW $ OLHV WKH FLUFOH

6HFRQG WKH SRLQW & OLHV WKH FLUFOH

7KLUG $ % LV FDOOHG LQ WKH FLUFOH % $ PHWDOOLF FRLQ ZDV WKURZQ RQFH ¿QG WKH SUREDELOLW\ RI DSSHDULQJ D KHDG $

7HVW 7KUHH &RPSOHWH HDFK RI WKH IROORZLQJ $

^ ` WKH VHW RI DOO IDFWRUV RI WKH QXPEHU HTXDOV

%

·

&

7R GUDZ D FLUFOH RI GLDPHWHU OHQJWK FP WKHQ WKH RSHQLQJ GLVWDQFH

RI WKH FRPSDVVHV VKRXOG EH FP

5I

'

± WR WKH QHDUHVW

(

$V WKURZLQJ D PHWDOOLF FRLQ RQFH WKHQ WKH VDPSOH VSDFH DQG WKH QXPEHU RI HOHPHQWV RI WKH VDPSOH VSDFH

\PM UI\ QK[

120

.QN\P 8ZQUI Za

([DPV

&KRRVH WKH FRUUHFW DQVZHU

,I ^ ` ^ \ ` WKHQ \ $ % &

7KH GHFLPDO IRUP RI WKH FRPPRQ IUDFWLRQ LV

'

%

'

'

'

$

&

é ×

$

!

%

&

7KH TXRWLHQW RI GLYLGLQJ ·

$

%

&

WR WKH QHDUHVW

$ $

%

&

)LQG WKH DUHD RI WKH VTXDUH ZKRVH VLGH OHQJWK LV P DSSUR[LPDWLQJ WKH UHVXOW WR WKH QHDUHVW WHQWK

%

,Q WKH RSSRVLWH ¿JXUH

7KUHH FLUFOHV RI FHQWUHV 0 1 DQG ( RI UDGLXV OHQJWK

FP IRU HDFK ¿QG WKH SHULPHWHU RI 0(1

$

‫ & ܢ‬

'UDZ ( $%& ZKHUH $& FP %& FP P

; %

'UDZ $ ' B %& PHDVXUH WKH OHQJWK RI $ ' WKHQ GUDZ % ; B $ & 'R $' DQG %; LQWHUVHFW DW RQH SRLQW" PHDVXUH WKH OHQJWK RI % ; %

,I ; \ ¿QG ; < DSSUR[LPDWLQJ WKH UHVXOW WR WKH QHDUHVW RQH WKRXVDQGWK

$

,I WKH SULFH RI D SLHFH RI VZHHW LV SRXQGV ZKDW LV WKH SULFH RI

%

SLHFHV RI WKH VDPH NLQG $ EDJ FRQWDLQV ZKLWH EDOOV UHG EDOOV DQG EODFN EDOOV DOO WKH EDOO

DUH LGHQWLFDO DQG HTXDO LQ VL]H ,I D EDOO LV GUDZQ UDQGRPO\ :KDW LV WKH SUREDELOLW\ WKDW WKH GUDZQ EDOO LV

)LUVW ZKLWH

Shorouk for Modern printing

6HFRQG QRW ZKLWH

7KLUG ZKLWH RU UHG

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

121

Exams

Test Four Complete the following: (1) ∅ .................... {a , b} (2) 6.35 + 17.025 = .................... to the nearest

1 100

(3) It is .................... that the sun rises from the west.

(4) As throwing a fair die once , then the probability of appearing a number less than 3 is ....................

(5) The altitudes of the right angled triangle intersect at ....................

Choose the correct answer : (1) 63.598 A

(2) 3

63.60 to the nearest part of ....................

10

1 2

÷

A

6

7 12

B

100

C

1000

D

10 000

B

18 3

C

50 12

D

4

B

17.14

C

11.74

D

117.4

B

∉

C

D

⊄

= ....................

(3) 67.5 – 55.67 = ....................

A

117.4

(4) 3 .................... {13 , 303} A

∈

(5) The chord which passes through the centre of the circle is called ....................

A

a diameter

B

C

A

Arrange ascendingly: 0.6 ,

B

From the opposite Venn diagram, write the following sets:

a radius 3 8

,

3 4

Ma

Y

X ,Y

the mat ics -

122 122

a side

, 0.8 U

X,Y,Z X

D

a tangent

Y

Z

, ,

Fifth Prima ry

X

Y

X–Y

, ,

Y

Z

Y – Z , Z – X.

x

7

x

Z

X

4

6

x

2

x

3

x

5

x

1

x

([DPV

$

)LQG WKH DUHD RI WKH UHFWDQJOH ZKRVH OHQJWK LV P DQG LWV ZLGWK P WR WKH QHDUHVW KXQGUHGWK

%

$V WKURZLQJ D IDLU GLH RQFH FDOFXODWH WKH SUREDELOLW\ RI DSSHDULQJ

D QXPEHU JUHDWHU WKDQ

D QXPEHU JUHDWHU WKDQ RU HTXDO

DQ RGG SULPH QXPEHU

D QXPEHU GLYLVLEOH E\

$

'UDZ ( $%& ZKHUH $% FP %& FP &$ FP %LVHFW $ & DW 0 WKHQ GUDZ D FLUFOH RI UDGLXV OHQJWK FP DQG 0 LV LWV FHQWUH WKHQ ┬┐QG

7ZR HTXDO OLQH VHJPHQWV LQ OHQJWK LQ WKH FLUFOH 0

:KDW LV WKH QDPH RI $ &

%

,Q WKH RSSRVLWH ┬┐JXUH FRPSOHWH

P тАл ▄втАм$0& P тАл ▄втАмP тАл ▄втАм$&0

P тАл& ▄втАм0% ┬Г

$% "

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

123

‫تم الطبع بمطابع ال�شروق الحديثة‬ ‫بالموا�شفات الفنية الآتية‬ ‫عدد ال�صفحات بالغالف‪�134 :‬صفحة‬ ‫المقا�س‪� 114 * 82 :‬صم‬ ‫نـ ـ ـ ــوع ال ـ ـ ــورق‪ :‬اليقل الداخلى عن ‪ 80‬جرام والغالف ‪ 200‬جرام‬ ‫ال ـ ـ ــوان الطب ـ ــع‪ 4 :‬لون للداخلى و‪ 4‬لون للغالف‬

‫جميع حقوق الطبع محفوظة لوزارة الرتبية والتعليم داخل جمهورية مرص العربية‬

*HQHUDO H[HUFLVHV

*HQHUDO H[HUFLVHV WK 3ULP 8QLW RQH WKH IUDFWLRQV

&RPSOHWH

P NP

NLORPHWHUV PHWUHV

WR WKH QHDUHVW WHQWK

WR WKH QHDUHVW XQLW

WR WKH QHDUHVW

WR WKH QHDUHVW

WR WKH QHDUHVW KXQGUHGWK

WR WKH QHDUHVW

*

WKH QXPEHU

WR WKH QHDUHVW

± WR WKH QHDUHVW XQLW

*

GP FP

*

*

WR WKH QHDUHVW KXQGUHGWK

5I

\PM UI\ QK[

102

.QN\P 8ZQUI Za

*HQHUDO H[HUFLVHV

WR WKH QHDUHVW WHQWK

·

GD\V ZHHNV

WR WKH QHDUHVW ZHHN

WR WKH QHDUHVW ZKROH QXPEHU

·

D

WKHQ D

NP P FP

* ·

* WR WKH QHDUHVW ZKROH QXPEHU

*

·

± WR WKH QHDUHVW

WRQ NJ

KRXU PLQXWH VHFRQG

KRXU

&KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ

7KH TXRWLHQW RI GLYLGLQJ ·

$

%

&

'

'

GD\V WR WKH QHDUHVW ZHHN

$

Shorouk for Modern printing

%

&

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

103

*HQHUDO H[HUFLVHV

7KH QXPEHU WR WKH QHDUHVW KXQGUHGWK

$

%

&

'

&

'

&

'

&

'

'

·

$

·

$

%

%

· ·

$

%

(VWLPDWH WKH UHVXOW

$

%

&

7KH QXPEHU RI GD\V LQ KRXUV HTXDOV DSSUR[LPDWHO\

$

%

&

'

'

'

'

'

'

7KH QXPEHU RI PRQWKV LQ KDOI D \HDU

$

%

&

7KH QXPEHU RI \HDUV LQ PRQWKV

$

*

*

%

&

$ % &

$ % &

WR WKH QHDUHVW KXQGUHGWK

*

*

5I

\PM UI\ QK[

104

$

.QN\P 8ZQUI Za

%

&

*HQHUDO H[HUFLVHV

*

$

%

&

'

%

&

'

Â&#x2022;

$

!

7KH VPDOOHVW IUDFWLRQ LQ WKH IROORZLQJ LV

$

%

&

'

7KH JUHDWHVW QXPEHU LQ WKH IROORZLQJ LV

$

%

&

'

$QVZHU WKH IROORZLQJ TXHVWLRQV

$UUDQJH WKH IROORZLQJ QXPEHUV DVFHQGLQJO\

$UUDQJH WKH IROORZLQJ QXPEHUV GHVFHQGLQJO\

$UUDQJH WKH IROORZLQJ QXPEHUV GHVFHQGLQJO\

7KH OHQJWK RI D SLHFH RI FORWK LV P WRZHOV DUH PDGH RI LW WKH OHQJWK RI HDFK WRZHO LV P +RZ PDQ\ PHWUHV DUH UHPDLQGHU"

)LQG WKH UHVXOW RI WKH IROORZLQJ

$

± * WR WKH QHDUHVW

%

·

&

·

'

(

*

)

*

± ·

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

105

General exercises

G

45.334 * 100 = ..................

H

17 ÷ 0.85 = ..................

40

I

9375 ÷ 15 = ..................

J

25.25 ÷ 0.25 = ..................

B

3.2 * 10

D

2 dm

(6) Put the suitable symbol (> , = , <):

(7)

A

4.79 * 1000

C

2 m 5

E

140.44

47.9 * 100

5 m 2

0.32 * 1000 200 cm

34.044

If the price of one metre of cloth is 7.35 pounds , what is the price of 3.5 metres ?

(8)

A car consumes one litre of Benzine to cover a distance 10 km.

How many litres are needed so that the car covers a distance of 642.9 km.

(9)

Find the area of the rectangle if its dimensions are 2.4 and 4.5 cm. then approximate the result to the nearest tenth.

(10) If a = 18.24 , b = 8.354 Find the result of a + b to the nearest hundreadth.

Estimate the result of a + b. Is your estimate accurate or not?

(11) Mahmoud bought a computer for 2000 pounds. He paid 250 pounds cash money and divided the remainder into 50 equall monthly instalments. Calculate the value of each instalment.

(12) The product of two numbers is 625. If one of them is 25 then what is the other number ?

(13) Find the result of each of First : 2.4 * 4.7

Ma

WKHQ IURP WKH SUHYLRXV RSHUDWLRQV ¿QG WKH YDOXH RI A

the mat ics -

106

Second: 3.4 * 0.29

(2.4 * 4.7) * 0.29

Fifth Prima ry

B

2.4 * (3.4 * 0.29)

106

*HQHUDO H[HUFLVHV

$ PHGLFDO ¿UP SDFNHG OLWUH RI D PHGLFLQH LQ ERWWOHV HDFK RI WKHP LV RI FDSDFLW\ OLWUH +RZ PDQ\ ERWWOHV ZHUH XVHG "

6HFRQG 8QLW 6HWV

)LUVW FRPSOHWH

&RPSOHWH E\ SXWWLQJ WKH VXLWDEOH V\PERO Ǻ ǻ ɶ

$

^ `

%

^ ` ^ `

&

Â&#x2C6; ^ `

'

^ ` ^ `

(

^ `

)

^ ` ^ `

*

^ ` ^ `

+

^ ` ^ `

&RPSOHWH

$

^ ` ^ `

%

^ ` ^ `

&

^ ` ^ `

'

^ ` ^ `

(

^ ` ^ `

)

^D E F` ^E F D`

,I ; DQG < DUH WZR QRQ HPSW\ VHWV WKHQ

$

; Â&#x2C6;

&

,I ; < WKHQ ; <

%

; ;

'

,I ; < < WKHQ

&RPSOHWH E\ SXWWLQJ WKH VXLWDEOH V\PERO Ǻ ǻ ɶ

,I < ^ ` ^ ` WKHQ

$

^ ` <

&

<

Shorouk for Modern printing

%

^ ` <

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

107

*HQHUDO H[HUFLVHV

,I $ ^ ` ± ^ ` WKHQ

$

$

%

_ a $

'

$

(

_ a $

&

_ a $

,I ; ^ ` ^ ` WKHQ ; ^ ` ^ ` ,I ; < WKHQ ; ± < ,I ; < WKHQ ; È&#x203A; < ; ,I ; DQG < 8 WKHQ ; < < ; ^ ` ^ ` ^ ` ^ ` ]HUR ^ ` WKH VHW RI IDFWRUV RI WKH QXPEHU

6HFRQG &KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ ^ ` ^ ` $

Ǻ

%

Ç»

&

'

ɶ

%

Ç»

&

'

ɶ

'

Ç· ^ ` $

Ǻ

7KH QXPEHU RI VXEVHWV RI WKH VHW ^ ` HTXDOV $

%

&

^ ` WKH VHW RI IDFWRUV RI WKH QXPEHU LV $

^ `

%

^ `

&

^ `

'

^ `

&

Ç·

'

,I ; < WKHQ ; < $

;

%

<

,I 8 LV WKH VHW RI RGG QXPEHUV OHVV WKDQ WKHQ ^ ` 8

5I

$

Ǻ

\PM UI\ QK[

108

.QN\P 8ZQUI Za

%

Ç»

&

'

ɶ

*HQHUDO H[HUFLVHV

,I ^ ` ^ ; ` WKHQ ;

'

,I ^ D ` ɶ ^ ` WKHQ D $ % &

'

^ ` ^ ` $ Ǻ % ǻ

&

'

ɶ

,I ; < WKHQ ; ± < $ ; % <

&

Â&#x2C6;

'

8

,I ; < < WKHQ ; < $ Ǻ % ǻ

&

'

ɶ

,I ^ ` ^ [ ` WKHQ [ $ % &

'

'

ɶ

'

ɶ

$

%

&

,I 8 ^ ` WKHQ

)LUVW Â&#x2C6; 8 $ Ǻ % Ç» &

6HFRQG 8 8 $ Ǻ % ǻ &

7KLUG ^ ` 8 $ Ǻ % ǻ &

'

ɶ

7KLUG $QVZHU WKH IROORZLQJ TXHVWLRQV 5HSUHVHQW WKH WZR VHWV $ DQG % E\ 9HQQ GLDJUDP LQ HDFK RI WKH IROORZLQJ FDVHV WKHQ ¿QG $ % $

$ ^ ` % ^ `

&

$ ^ ` % ^ `

%

$ ^ ` % ^ `

5HSUHVHQW WKH WZR VHWV $ DQG % LQ HDFK RI WKH IROORZLQJ E\ 9HQQ GLDJUDP WKHQ ¿QG $ % $

$ ^ ` % ^ `

&

$ ^D P [` % ^D I [ P`

Shorouk for Modern printing

%

$ ^ ` % ^ `

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

109

General exercises

Using the operations of intersection and union, The difference and complement, express the shaded part in each of the follwing diagrams.

Fig. (1)

Fig. (2)

Fig. (3)

Fig. (4)

Fig. (5)

Fig. (6)

Use the opposite Venn diagram to write the following sets: U A X–Y B X Y C X D (X Y Y) E (X Y)

X

5

x

Y

3

x

2 x1

x

4

x

6

x

Use the opposite Venn diagram to write the following sets: Z X A X,Y,Z B Y–X,Z–Y C X – (Z ) D (Y – X) 2 5 1 4 3 (Z – Y) Y E (X – Z) (Z – Y) Use the symbols of intersection, union, difference and complement to write the shaded part in each Venn diagram of the following diagrams: x

Fig. (1)

Fig. (2)

Fig. (3)

x

x

x

x

Fig. (4)

Using the opposite diagram write the following sets: A C

X

Y

X–Y

B D

X (X

Y

U 6 x7 x

Y)

8

x

Ma

the mat ics -

110 110

Fifth Prima ry

Y

Z

5

3

x

x

4

x

1

X

x

2

x

9

x

*HQHUDO H[HUFLVHV

8QLW 7KUHH *HRPHWU\

)LUVW &RPSOHWH

$Q\ FKRUG ZKLFK SDVVHV WKURXJK WKH FHQWUH RI WKH FLUFOH LV FDOOHG LQ LW

7KH WULDQJOH ZKRVH PHXVXUHV RI DQJOHV DUH ┬Г ┬Г DQG ┬Г LV FDOOHG

7KH WULDQJOH LQ ZKLFK WKHUH DUH WZR HTXDO VLGHV LQ OHQJWK LV FDOOHG

7KH WULDQJOH ZKLFK WKH PHDVXUHV RI LWV DQJOHV DUH ┬Г ┬Г DQG ┬Г LV FDOOHG

7KH WULDQJOH ZKLFK WKH OHQJWKV RI LWV VLGHV DUH HTXDO LV FDOOHG

$ FLUFOH LV RI GLDPHWHU OHQJWK FP WKHQ LWV UDGLXV OHQJWK FP

,Q WKH RSSRVLWH ┬┐JXUH &RPSOHWH

$

P тАл ▄втАм$%&

%

7KH SHULPHWHU RI ( $%& FP

&

7KH QXPEHU RI DOWLWXGHV RI WULDQJOH $%&

7KH QXPEHU RI DOWLWXGHV RI WKH REWXVH DQJOHG WULDQJOH LV

,Q WKH RSSRVLWH ┬┐JXUH &RPSOHWH

$ LV FDOOHG D GLPHWHU LQ WKH FLUFOH ZKRVH FHQWUH LV 0

%

< = LV FDOOHG LQ WKH FLUFOH ZKRVH FHQWUH LV 0

&

(DFK RI ; 0 < 0 DQG = 0 LV FDOOHG LQ WKH FLUFOH 0

'

( <0= LV FDOOHG WULDQJOH GXH WR LWV VLGHV

7KH FKRUG RI WKH FLUFOH ZKLFK SDVVHV WKURXJK LWV FHQWUH LV

$Q\ OLQH VHJPHQW ZKLFK MRLQV WZR SRLQWV RQ WKH FLUFOH LV FDOOHG

,I WKH OHQJWKV RI VLGHV RI ( $%& DUH OHQJWK XQLW WKHQ LW LV FDOOHG WULDQJOH DQG WKH PHDVXUH RI HDFK RI LWV DQJOHV

7KH GLDPHWHU RI WKH FLUFOH RI UDGLXV FP HTXDOV FP

,W LV SRVVLEOH WR GUDZ D WULDQJOH LI WKH OHQJWKV RI DUH NQRZQ

:KDW DUH WKH NLQGV W\SHV RI WKH WULDQJOH GXH WR LWV DQJOHV Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

111

*HQHUDO H[HUFLVHV

'UDZ WKH WULDQJOH $%& LQ ZKLFK $% FP %& FP $& FP WKHQ GUDZ WKH FLUFOH 0 ZKRVH GLDPHWHU LV $ & WKHQ ¿QG

$

7KH SHULPHWHU RI WKH WULDQJOH $%&

%

8VH WKH SURWUDFWRU WR ¿QG WKH PHDVXUH RI

&

7KH OHQJWKV RI $ 0 % 0 & 0 ZKDW GR \RX GHGXFH "

'

7KH W\SH RI ( 0%& GXH LWV DQJOHV

(

0HQWLRQ WZR LVRVFHOHV WULDQJOHV

â&#x20AC;« Ü¢â&#x20AC;¬$%&

'UDZ WKH FLUFOH ZKRVH GLDPHWHU OHQJWK FP 'UDZ % & DV D GLDPHWHU RI LW WKHQ XVH % DV D FHQWUH DQG XVH WKH FRPSDVVHV ZLWK OHQJWK FP WR GUDZ DQ DUF WR LQWHUVHFW WKH FLUFOH DW ; DQG < MRLQ HDFK RI % ; % < & < & ; ; <

,I ( LV WKH SRLQW RI LQWHUVHFWLRQ RI % & DQG ; <

)LUVW &RPSOHWH XVLQJ WKH SURWUDFWRU

$

%

&

'

6HFRQG &KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ

$

â&#x20AC;« Ü¢â&#x20AC;¬%;&

P â&#x20AC;« Ü¢â&#x20AC;¬%<&

P â&#x20AC;« Ü¢â&#x20AC;¬%(&

P â&#x20AC;« <(& Ü¢â&#x20AC;P

( %;& LV WULDQJOH DFXWH DQJOHG REWXVH DQJOHG ULJKW DQJOHG

%

( %;< LV WULDQJOH DFXWH DQJOHG REWXVH DQJOHG ULJKW DQJOHG

&

( &;< LV WULDQJOH DFXWH DQJOHG REWXVH DQJOHG ULJKW DQJOHG

5I

\PM UI\ QK[

112

.QN\P 8ZQUI Za

*HQHUDO H[HUFLVHV

7KLUG &RPSOHWH

$

7KH SRLQW RI LQWHUVHFWLRQ RI DOWLWXGHV RI ( ;%< LV

%

7KH DOWLWXGHV RI ( ;%& LQWHUVHFW DW WKH SRLQW

,Q RSSRVLWH ¿JXUH

,I WKH UDGLXV OHQJWK RI WKH FLUFOH 0 HTXDOV FP

DQG WKH UDGLXV OQJWK RI WKH FLUFOH 1 FP

DQG WKH UDGLXV OHQJWK RI WKH FLUFOH ( FP

&RPSOHWH WKH IROORZLQJ

$

01 FP

&

(1 FP

'

7KH SHULPHWHU RI ( 0(1 FP

%

0( FP

,Q RSSRVLWH ¿JXUH

3XW WKH VXLWDEOH V\PERO ! RU RU

WR JHW D FRUUHFW VWDWHPHQW

$

$ 0 $ %

%

& % $ %

&

0 & 0 %

'

$ & 0 $

'UDZ WKH UHFWDQJOH $%&' ZKHUH $% FP %& FP WDNH WKH SRLQW

/ Ǻ $ ' ZKHUH $/ FP

'UDZ ( /%& WKHQ GUDZ / = SHUSHQGLFXODU WR % &

)LQG WKH OHQJWK RI / = ZLWKRXW PHDVXULQJ WKHQ ¿QG WKH SHULPHWHU RI WKH UHFWDQJOH '/=&

'UDZ WKH LVRVFHOHV WULDQJOH $%& ZKLFK LV ULJKW DQJOHG DW % ZKHUH $% FP IURP % GUDZ WKH OLQH VHJPHQW ZKLFK LV SHUSHQGLFXODU WR $ & VD\ % ' DQG PHDVXUH LWV OHQJWK Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

113

*HQHUDO H[HUFLVHV

'UDZ WKH FLUFOH 0 ZKRVH UDGLXV OHQJWK LV FP 'UDZ $ % DV D GLDPHWHU LQ LW ORFDWH WKH SRLQWV & ' DQG ( VXFK WKDW 0& FP 0' FP 0( FP WKHQ FRPSOHWH

$

0 ( LV FDOOHG

&

$ ( LV FDOOHG

%

7KH SRLQW ' OLHV WKH FLUFOH

'UDZ ( ;<= ZKLFK LV HTXLODWHUDO DQG LWV VLGH OHQJWK FP 'UDZ D FLUFOH RI FHQWUH ; DQG UDGLXV OHQJWK FP

&RPSOHWH WKH IROORZLQJ

$

; < LV FDOOHG LQ WKH FLUFOH ;

%

; = LV FDOOHG LQ WKH FLUFOH ;

&

< = LV FDOOHG LQ WKH FLUFOH ;

'

7KH SHULPHWHU RI ( ;<= FP

7KH IRXUWK XQLW WKH SUREDELOLW\ WK \HDU SULP

&KRRVH WKH FRUUHFW DQVZHU IURP WKRVH JLYHQ

$V WKURZLQJ D IDLU GLH RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ WKH QXPEHU HTXDOV

$

%

&

'

,I D IDLU GLH LV WKURZQ RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU JUHDWUH WKDQ LV

$

%

]HUR

&

'

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ DQ RGG QXPEHU LV

$

%

&

'

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU OHVV WKDQ LV

5I

\PM UI\ QK[

114

$

.QN\P 8ZQUI Za

%

&

'

*HQHUDO H[HUFLVHV

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU JUHDWU WKDQ RQH LV $ % & '

$V WRVVLQJ D PHWDOLF FRLQ RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D KHDG LV $ % & '

7KH SUREDELOLW\ WKDW WKH HOHSDKDQW Ã&#x20AC;LHV $ % ]HUR & Â&#x2C6; '

,W LV WKDW WKH VXQ ULVHV IURP ZHVW $ H[SHFWHG % SRVVLEOH & VXUH ' LPSRVVLEOH

$V WKURZLQJ D IDLU GLH RQFH DQG REVHUYLQJ WKH DSSHDULQJ QXPEHU RQ WKH XSSHU IDFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU GLYLVELOH E\ LV $ % & '

$V WKURZLQJ D PHWDOLF FRLQ RQFH WKHQ WKH SUREDELOLW\ WKDW D WDLO DSSHDUV LV $ % & '

$ FODVV KDV SXSLOV RI WKHP DUH ER\V DQG WKH UHPDLQGHU DUH JLUOV ,I D SXSLO LV FKRVHQ UDQGRPO\ WKHQ WKH SUREDELOLW\ WKDW WKH SXSLO LV D JLUO LV $ % & '

$V WKURZLQJ D IDLU GLH RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D SULPH QXPEHU LV $ % & '

$V WKURZLQJ D IDLU GLH RQFH WKHQ WKH SUREDELOLW\ RI DSSHDULQJ D QXPEHU OHVV WKDQ RU HTXDO WR LV $ % & '

7KH SUREDELOLW\ RI WKH VXUH HYHQW LV $ ]HUR % & '

7KH SUREDELOLW\ WKDW D ELUG Ã&#x20AC;LHV

$

Shorouk for Modern printing

%

&

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [ '

115

*HQHUDO H[HUFLVHV

7KH IROORZLQJ WDEOH VKRZV WKH UHVXOW RI D VXUYH\ KDV EHHQ DSSOLHG WR NQRZ WKH YLHZV RI SXSLOV DERXW WKH IDYRULWH JDPH WR WKHP $

,I RQH SXSLO LV FKRVHQ DW UDQGRP DQVZHU IROORZLQJ TXHVWLRQV 7KH JDPH

IRRWEDOO

KDQGEDOO

EDVNHWEDOO

QXPEHU RI YLHZV

:KDW LV WKH SUREDELOLW\ WKDW RQH RI WKHP SUHIHUV IRRWEDOO

:KDW LV WKH SUREDELOLW\ WKDW RQH RI WKHP SUHIHUV KDQGEDOO

:KDW LV WKH SUREDELOLW\ WKDW RQH RI WKHP SUHIHUV EDVNHWEDOO

%

,I WKHUH DUH SXSLOV ZKDW LV WKH H[SHFWHG YDOXH RI WKH QXPEHU RI SXSLOV ZKR SUHIHU IRRWEDOO

&

,I WKHUH DUH SXSLOV ZKDW LV WKH H[SHFWHG YDOXH RI WKH QXPEHU RI SXSLOV ZKR SUHIHU EDVNHWEDOO

$ VSLQQHU LV GLYLGHG LQWR HTXDO FLUFXODU VHFWRUV LQ DUHD RI WKHP DUH UHG WZR VHFWRUV DUH JUHHQ RQH VHFWRU LV EURZQ ZKDW LV WKH SUREDELOLW\ WKDW WKH SRLQWHU $

:LOO VWDQG RQ WKH UHG FRORXU"

%

:LOO VWDQG RQ WKH EURZQ FRORXU"

&

:LOO VWDQG RQ WKH JUHHQ FRORXU "

$ IDLU GLH LV WKURZQ RQFH ZKDW LV WKH SUREDELOLW\ RI HDFK RI WKH IROORZLQJ HYHQWV"

5I

$

DSSHDULQJ DQ RGG QXPEHU

%

DSSHDULQJ DQ HYHQ QXPEHU

&

DSSHDULQJ D SULPH QXPEHU DQG QRW HYHQ

'

DSSHDULQJ DQ RGG SULPH QXPEHU

(

DSSHDULQJ D QXPEHU OHVV WKDQ RQH

DSSHDULQJ D QXPEHU PRUH WKDQ \PM UI\ QK[ .QN\P 8ZQUI Za

)

116

*HQHUDO H[HUFLVHV

*

DSSHDULQJ D QXPEHU OHVV WKDQ

+

DSSHDULQJ D QXPEHU JUHDWHU WKDQ ]HUR DQG OHVV WKDQ

,

:KDW LV WKH QDPH RI HDFK RI WKH SUHYLRXV HYHQWV "

$ EDJ FRQWDLQV UHG EDOOV EODFN EDOOV DQG ZKLWH EDOOV DOO RI WKHP DUH LGHQWLFDO DQG HTXDO LQ YROXPH $ EDOO LV GUDZQ UDQGRPO\ FDOFXODWH WKH SUREDELOLW\ WKDW $

7KH GUDZQ EDOO LV EODFN

%

WKH GUDZQ EDOO LV QRW JUHHQ

&

WKH GUDZQ EDOO LV UHG RU ZKLWH

'

WKH GUDZQ EDOO LV QHLWKHU UHG QRU ZKLWH

$ GLH LV WKURZQ WLPHV KRZ PDQ\ WLPHV WKDW WKH DSSHDULQJ QXPEHU LV RGG" $ ER[ FRQWDLQV FDUGV QXPEHUHG IURP WR ,I D FDUG LV GUDZQ UDQGRPO\ FDOFXODWH WKH SUREDELOLW\ WKDW WKH GUDZQ FDUG FDUULHV $

DQ RGG QXPEHU

%

D QXPEHU GLYLVELOH E\

&

$Q HYHQ SULPH QXPEHU

'

D QXPEHU GLYLVLEOH E\

(

D QXPEHU GLYLVLEOH E\

)

D QXPEHU GLYVLEOH E\ RU

*

D QXPEHU OHVV WKDQ

+

D QXPEHU LQ ZKLFK WKH XQLWV GLJLW LV

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

117

([DPV

7HVW 2QH &RPSOHWH HDFK RI WKH IROORZLQJ

·

WR WKH QHDUHVW

GD\V ZHHNV WR WKH QHDUHVW ZHHN

Ã&#x2014;

^ ` ^ `

&KRRVH WKH FRUUHFW DQVZHU

KDOYHV ¿IWK $ ! % & Â&#x201D;

^ ` ^ ` $ Ǻ % ǻ &

7KH QXPEHU RI DOWLWXGHV RI WKH WULDQJOH LV $ ]HUR % &

7KH SUREDELOLW\ RI WKH LPSRVVLEOH HYHQW $ Ç· % ]HUR &

FP WR WKH QHDUHVW PHWUH $ % &

$

)LQG E\ OLVWLQJ PHWKRG

; <

'

ɶ

'

'

'

; ± <

8 ;

<

[

[

; <

[

[

[

[

[

;C <C

%

$UUDQJH WKH IROORZLQJ QXPEHUV DVFHQGLQJO\

$

'UDZ WKH HTXLODWHUDO WULDQJOH $%& ZKRVH VLGH OHQJWK FP WKHQ

GUDZ $ ' B % & 7KHQ ¿QG

5I

'

,Q 9HQQ GLDJUDP LQ IURQW RI \RX

7KH SHULPHWHU RI ( $%&

\PM UI\ QK[

118

.QN\P 8ZQUI Za

â&#x20AC;«& Ü¢â&#x20AC;¬$' E\ PHDVXULQJ

P

([DPV

,Q WKH RSSRVLWH ¿JXUH &RPSOHWH

0 $

7KH ORQJHVW FKRUG LQ WKH FLUFOH LV DQG

LW LV FDOOHG %

0

$

%

; % LV FDOOHG LQ WKH FLUFOH ZKRVH FHQWUH LV ; $ $V WKURZLQJ D IDLU GLH RQFH FDOFXODWH WKH SUREDELOLW\ RI

)LUVW DSSHDULQJ D QXPEHU JUHDWHU WKDQ

6HFRQG DQ DSSHDULQJ HYHQ QXPEHU JUHDWHU WKDQ

7KLUG DSSHDULQJ DQ HYHQ SULPH QXPEHU % )LQG WKH SHULPHWHU RI WKH UHFWDQJOH ZKRVH OHQJWK LV FP DQG LWV ZLGWK LV WKHQ &DOFXODWH LWV DUHD

7HVW 7ZR $QVZHU WKH IROORZLQJ TXHVWLRQV

&KRRVH WKH FRUUHFW DQVZHU

· · $ % &

7KH ORQJHVW FKRUG LQ WKH FLUFOH LV FDOOHG $ D UDGLXV % D FKRUG & D GLDPHWHU

Ã&#x2014; Ã&#x2014; $ ! %

^ ` $ Ǻ % ǻ

KRXUV $ %

'

'

D WDQJHQW

&

&

'

ɶ

&

'

&RPSOHWH HDFK RI WKH IROORZLQJ

$%& LV DQ HTXLODWHUDO WULDQJOH RI VLGH OHQJWK FP WKHQ LWV SHULPHWHU FP

,I ; < WKHQ ; <

· ·

^ ` ± ^ `

± Ã&#x2014; WR WKH QHDUHVW WHQWK Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

119

([DPV

$

%

$ FDU FRYHUV HTXDO GLVWDQFHV LQ HTXDO WLPHV ,I WKLV FDU FRYHUHG N

LQ RQH KRXU KRZ PDQ\ NP DUH FRYHUHG LQ KRXUV :ULWH ZKDW LV UHSUHVHQWHG E\ VKDGHG SDUW LQ HDFK GLDJUDP %

$

,I WKH XQLYHUVDO VHW 8 WKH VHW RI DOO IDFWRUV RI WKH QXPEHU DQG ; ^ ` < ^ ` 'UDZ 9HQQ GLDJUDP ZKLFK UHSUHVHQWV WKH VHWV 8 [ \ WKHQ )LQG ; < ; ± < <ఴ ;ఴ ఴ % :ULWH DOO WKH VXEVHWV RI WKH VHW ; ^D E` ZKDW LV WKH QXPEHU RI VXEVHWV" $ 'UDZ ( $%& ZKHUH $% FP %& FP &$ FP WKHQ GUDZ D FLUFOH RI FHQWUH % DQG LWV UDGLXV OHQJWK FP )URP WKH GUDZLQJ FRPSOHWH

)LUVW WKH SRLQW $ OLHV WKH FLUFOH

6HFRQG WKH SRLQW & OLHV WKH FLUFOH

7KLUG $ % LV FDOOHG LQ WKH FLUFOH % $ PHWDOOLF FRLQ ZDV WKURZQ RQFH ¿QG WKH SUREDELOLW\ RI DSSHDULQJ D KHDG $

7HVW 7KUHH &RPSOHWH HDFK RI WKH IROORZLQJ $

^ ` WKH VHW RI DOO IDFWRUV RI WKH QXPEHU HTXDOV

%

·

&

7R GUDZ D FLUFOH RI GLDPHWHU OHQJWK FP WKHQ WKH RSHQLQJ GLVWDQFH

RI WKH FRPSDVVHV VKRXOG EH FP

5I

'

± WR WKH QHDUHVW

(

$V WKURZLQJ D PHWDOOLF FRLQ RQFH WKHQ WKH VDPSOH VSDFH DQG WKH QXPEHU RI HOHPHQWV RI WKH VDPSOH VSDFH

\PM UI\ QK[

120

.QN\P 8ZQUI Za

([DPV

&KRRVH WKH FRUUHFW DQVZHU

,I ^ ` ^ \ ` WKHQ \ $ % &

7KH GHFLPDO IRUP RI WKH FRPPRQ IUDFWLRQ LV

'

%

'

'

'

$

&

é ×

$

!

%

&

7KH TXRWLHQW RI GLYLGLQJ ·

$

%

&

WR WKH QHDUHVW

$ $

%

&

)LQG WKH DUHD RI WKH VTXDUH ZKRVH VLGH OHQJWK LV P DSSUR[LPDWLQJ WKH UHVXOW WR WKH QHDUHVW WHQWK

%

,Q WKH RSSRVLWH ¿JXUH

7KUHH FLUFOHV RI FHQWUHV 0 1 DQG ( RI UDGLXV OHQJWK

FP IRU HDFK ¿QG WKH SHULPHWHU RI 0(1

$

‫ & ܢ‬

'UDZ ( $%& ZKHUH $& FP %& FP P

; %

'UDZ $ ' B %& PHDVXUH WKH OHQJWK RI $ ' WKHQ GUDZ % ; B $ & 'R $' DQG %; LQWHUVHFW DW RQH SRLQW" PHDVXUH WKH OHQJWK RI % ; %

,I ; \ ¿QG ; < DSSUR[LPDWLQJ WKH UHVXOW WR WKH QHDUHVW RQH WKRXVDQGWK

$

,I WKH SULFH RI D SLHFH RI VZHHW LV SRXQGV ZKDW LV WKH SULFH RI

%

SLHFHV RI WKH VDPH NLQG $ EDJ FRQWDLQV ZKLWH EDOOV UHG EDOOV DQG EODFN EDOOV DOO WKH EDOO

DUH LGHQWLFDO DQG HTXDO LQ VL]H ,I D EDOO LV GUDZQ UDQGRPO\ :KDW LV WKH SUREDELOLW\ WKDW WKH GUDZQ EDOO LV

)LUVW ZKLWH

Shorouk for Modern printing

6HFRQG QRW ZKLWH

7KLUG ZKLWH RU UHG

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

121

Exams

Test Four Complete the following: (1) ∅ .................... {a , b} (2) 6.35 + 17.025 = .................... to the nearest

1 100

(3) It is .................... that the sun rises from the west.

(4) As throwing a fair die once , then the probability of appearing a number less than 3 is ....................

(5) The altitudes of the right angled triangle intersect at ....................

Choose the correct answer : (1) 63.598 A

(2) 3

63.60 to the nearest part of ....................

10

1 2

÷

A

6

7 12

B

100

C

1000

D

10 000

B

18 3

C

50 12

D

4

B

17.14

C

11.74

D

117.4

B

∉

C

D

⊄

= ....................

(3) 67.5 – 55.67 = ....................

A

117.4

(4) 3 .................... {13 , 303} A

∈

(5) The chord which passes through the centre of the circle is called ....................

A

a diameter

B

C

A

Arrange ascendingly: 0.6 ,

B

From the opposite Venn diagram, write the following sets:

a radius 3 8

,

3 4

Ma

Y

X ,Y

the mat ics -

122 122

a side

, 0.8 U

X,Y,Z X

D

a tangent

Y

Z

, ,

Fifth Prima ry

X

Y

X–Y

, ,

Y

Z

Y – Z , Z – X.

x

7

x

Z

X

4

6

x

2

x

3

x

5

x

1

x

([DPV

$

)LQG WKH DUHD RI WKH UHFWDQJOH ZKRVH OHQJWK LV P DQG LWV ZLGWK P WR WKH QHDUHVW KXQGUHGWK

%

$V WKURZLQJ D IDLU GLH RQFH FDOFXODWH WKH SUREDELOLW\ RI DSSHDULQJ

D QXPEHU JUHDWHU WKDQ

D QXPEHU JUHDWHU WKDQ RU HTXDO

DQ RGG SULPH QXPEHU

D QXPEHU GLYLVLEOH E\

$

'UDZ ( $%& ZKHUH $% FP %& FP &$ FP %LVHFW $ & DW 0 WKHQ GUDZ D FLUFOH RI UDGLXV OHQJWK FP DQG 0 LV LWV FHQWUH WKHQ ┬┐QG

7ZR HTXDO OLQH VHJPHQWV LQ OHQJWK LQ WKH FLUFOH 0

:KDW LV WKH QDPH RI $ &

%

,Q WKH RSSRVLWH ┬┐JXUH FRPSOHWH

P тАл ▄втАм$0& P тАл ▄втАмP тАл ▄втАм$&0

P тАл& ▄втАм0% ┬Г

$% "

Shorouk for Modern printing

;\]L

U <MZ

\ [ QZ S .

W W * MV\ [

123

‫تم الطبع بمطابع ال�شروق الحديثة‬ ‫بالموا�شفات الفنية الآتية‬ ‫عدد ال�صفحات بالغالف‪�136 :‬صفحة‬ ‫المقا�س‪� 114 * 82 :‬صم‬ ‫نـ ـ ـ ــوع ال ـ ـ ــورق‪ :‬اليقل الداخلى عن ‪ 80‬جرام والغالف ‪ 200‬جرام‬ ‫ال ـ ـ ــوان الطب ـ ــع‪ 4 :‬لون للداخلى و‪ 4‬لون للغالف‬

‫جميع حقوق الطبع محفوظة لوزارة الرتبية والتعليم داخل جمهورية مرص العربية‬