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The set of Integers Notes: 1) The set of counting numbers C= {1, 2, 3, 4,.} 2) The set of natural numbers N = {0, 1, 2, 3, 4,.} 3) The set of integers Z Z= Z - U {0} U Z + OR Z= Z - U N Where Z - = {..,-3,-2, 1} Z + = {1, 2, 3, 4, } Then Z= {..-3, -2,-1,0,1,2,3,4, } Z - Z + 0 is non negative and non positive N 2
Remarks (study hard) 1) N Z 2) C N Z 3) Z + Z = φ 4) Z N = Z 5) Z = Z U N 6) N - Z = φ 7) Z + U Z =Z-{0} 8) Z + - Z = Z + 9) The set of non- negative = Z + U{0}= N i.e {0, 1, 2, 3, 4, 5,.} 10) The set of non positive = Z U {0} i.e {..-3,-2,-1, 0} 3
Exercises 1) Underline the integer numbers from the following numbers 8, 0, 14 100, 0.77, - 635, 32 8,3 1 2 2) Complete 1) N- {0} = 2) Z + Z = 3) Z N = 4) Z - Z + = 5) Z + U Z U {0} = 6) Z + - Z = 7) Z- Z = 8) Z- N= 9) Z + U {0} = 10) N - Z + = 11) N Z = 4
3) Complete using,,, 1) N..Z 2) Z +.Z 3) {0}. Z 4) Z +..N 5) 0 N 6) {0}..Z 7) 0.. Z + 4) Complete 1) Any positive integer is that any negative integer. 2) The smallest +ve integer is..and we can t determine the greatest + ve integer. 3) The greatest ve integers is we can t determine the smallest ve integer. 4) The set of integers which are less than 3 5) The set of integers which are greater than -2 6) The set of integers which are less than -5. 7) The set of integers which are less than 6 and greater than -2. 5
8) The complement of Z with respect to Z.. 9) The complement of Z with respect to Z. 10) The complement of N with respect to Z.. The Absolute value Note The absolute value of a number is its distance from 0 on number line The absolute value of any number x is denoted by x 6
Some Exercises 1) Find : 1) 5 = 2) 6 = 3) - 10 = 4) - 7 = 5) 7+ 3 = 6) 3 x 5 = 2) Complete 1) If x =7 Then x=..or. 2) If x =8 Then x=..or. 3) If 9 = x Then x=.. 4) The absolute value of -12 is. 5) - 5 {-1, 0, 3, x} Then x= 6){ 2, x } U { -4,0,4 } ={0,-2,2,-4,4} Then x =.. 3) Write the opposite (inverse) of each of the following integers : 5, -6, -3, 0, - 5 4) Compare using >, > or = : 1) 4-14 2) -4-1 7
3) -5-14 4) -8-2 5) -6 4 6) -2 2 7) 0-5 8) 0 8 9) -8 - -8 10) -5-5 5) Write the integers representing each of the following: a) X > -1 b) X < 7 c) - 4 > X > 4 d) The previous integer to -9 is and its next integer is e) The previous integer to 3 is.. and its next integer is.. 6) Arrange the following numbers in ascending order: a) -9, 17, -9, -15, 16, 0. b) -1, 3, 1, -5, 7, - -10. 8
o Arrange the following numbers in ascending order: a) 3, -30, - 8, 0, 11. b) 1, -11, 3, - -7, 5, -22. 7) Write the integers representing each of the following: a) A temperature of 3 degrees below zero b) A bank deposit of L.E. 100 c) A loss of 5 yards in a football. d) A withdrawal of L.E 25. 9
The addition of integers and its properties Study hard (+ Ve) + (+ve) = (+ve) (-ve) + (-ve) = (-ve) But (+ve) + (-ve) = (+ve) or (-ve) according to the sign of the greatest integer. Zero has no additive inverse and no absolute value. i.e the additive inverse of zero = zero. The properties of addition: 1) Closure property: The sum of two integers is an integer i.e is always possible. 2) Commutative property: a +b = b +a 3) Associative property: a + b + c = ( a + b ) + c = ( a + c ) + b = ( b + c ) + a 10
4) The additive identity (neutral) = zero, it means when we add zero to any integer it doesn't change its value. i.e a + 0 = a Revision 1) Use the property of addition in Z to find : a) -5 + 29 + 15 b) -7 + 2 + (-13) c) 2015 + 180 + (-1015) d) 55 + (-255) + 45 + 225 e) 5 + (-3) + 7 + (-9) f) 24 + (-19) + (-24) + 9 2) Complete: a) The additive inverse of zero is b) The additive inverse of ( a) is c) a + (-a) =.. d) a + b = b + a is called.. Property. e) The additive identity of integer is... 11
3)If X = { -2, 4, 2, -6 } then find : a) The relation between X and Z. b) Is X closed under addition? Why? Subtraction of integer Its properties 1) Z is closed under subtraction operation. i.e : the result of subtracting any two integers is an integer. 2) The subtraction operation in Z is not commutative. 3) The subtraction operation in Z is not associative. 12
1) Find each of the following: 1) 3 + 7 + 6= 2) -3 +6 + (-2) = 3) -3 + 7-5= 4) - 6 - (-3) 5 = 5) (-6) + (-2) + (-1) = 6) -4 7 5= 7) - 9 +7-3 = 8) -17-13 +10 = 9) -2+ 5 (-3) +1 = 10) -10-4 + (-3) (-6) = 11) -9 (4-7) = Some Exercises 2) Complete: 1) 4 + (-3) = -3 + 2) -8 +.. = (-8) 3) (-7) +.. = 0 4) 6 + (-6) =. 13
5) 6 + (-11) = 6) -8-3 = 7) -2 + 5 =. 8) [5 + (-8)] +7 = 5 + (.. + 7 ) 14
Revision about addition and subtraction 1) Find the result : 1) -12 7= 2) -77 + (-3+77) = 3) -6 +6 = 4) -7-4 = 5) 6-11 = 2) Complete using,,, 1) 1-9 + 3..Z 6) 6 11.. Z+ 2) {9} Z 7) -2 (-3).. Z+ 3) 35.. Z 8) -5 + 5.. Z 4) 6-6. Z+ 5) {-3, 7, 1.1} Z 3) Find the value of n 1) -8 + n = -8 2) n + 5 = 0 3) n + -3 = -8 Study hard: Multiplication of Integers 15
Possibility of multiplication in Since the sum of two integers is an integer and we know that the multiplication operation is a repeated addition operation. Thus, the multiplication of two integer. i.e. Multiplication operation is always possible in 1 The product of two integers with the same sign is positive. i.e. = and = + + + - - + 2 The product of two integers with different signs is negative. i.e and = + - - - + - o Multiplying any integer by zero equals zero. The properties of multiplication in z 1) Closure property The product of any 2 integers is an integer. 2) Commutative property: If a and b are two integers, then: a b = b a 16
3) Associative property: If a, b and c are three integers, then: a b c = (a b) c = a ( b c) 4) The existence of the additive identity (neutral) element integer: For any integer a, we have: 1 a = a 1 = a i.e. the number ''1'' is the multiplicative identity ( neutral) element. 5) Multiplication is distributed over addition and subtraction in Z: If a, b and c are three integers, then: a (b + c) = a b + a c and (b + c ) a = b a + c a a (b _ c) = a b _ a c and ( b _ c ) a = b a _ c a Some exercises about multiplication 1) Multiply 1) (- 131) (- 3) 2) 9 (7) 3) - (- 6) (- 2) 4) - 10-3 17
2) Find the value of x : 1) - 5 x = 45 2) x -9 = - 45 3) 8 x = - 48 3) If x =2, y = - 4 find the n. Value of the following : a) x y b) 2 x + 3 y c) x y _ 2 y 4 )Use the properties of multiplication to find : 1) (- 4) 57 (- 25) 2) 8 2 125 (-50) 3) 77 8 (- 125) 4) 5 (- 45) 2 5) Solve by 2 different ways [8 + (-5)] 6 6) Use the distributive property to find: 1) 120 19 + 120 (-9) 2) 11 35 + 11 64 + 11 3) 5-3 _ 5 7 18
Division of integers Possibility of Division Division is not always possible in z or z is not closed under division. But: division is not commutative and not associative Study hard: 1 The quotient of integers with the same sign is positive i.e. = and = + + + + For example : 8 2 = 4, - 10 (-2) = 5 2 i.e. The quotient of two integers with different signs is negative. and + _ + _ For example : 40 ( - 5) = - 8, - 40 5 = - 8 Note V. I: Divisibility by zero has no meaning i.e. 8 0 has no meaning but 0 8 = 0 19
Some Exercises: 1) Divide: a) 8 2 b) -64 8 c) 49 (-7) d) ( - 36 ) (- 4 ) e) 0 100 f)77 (-11) g) - 100 25 h) 18 2 i) 36 12 j) 18 6 k) -45-5 l) - 42 6 20
Repeated multiplication 5 5 5 5 means 5 is multiplied by itself 4 items, we can write it as 5 4 (i.e. 5 5 5 5 = 5 4 ) and it is read as Base 5 4 Power 5 to the power 4 or 5 to the fourth power 5 is called the base and 4 is called the power or index or exponent. Generally If a is an integer and n Z +, then a a a x. to n times = a n Where a is called the base and n is called the power or index or exponent. Remarks 1) Any number to the first power is that number itself. For example: 9 1 = 9, ( 3) 1 = 3 and x 1 = x where x Z 2) Any number to the zero power, except zero, is 1 For example: 5 0 = 1, ( 7) 0 = 1 and a 0 = 1 where a Z {0} 21
3) If the base is one and n Z, then 1 n = 1 For example: 1 5 = 1, 1 12 = 1 (a) n if n is even 4) If a Z and n Z +, then ( a) n = (a) n if n is odd i.e. A negative integer raised to the power of even integer gives a positive integer. A negative integer raised to the power of odd integer gives a negative integer. For example: ( 4) 2 = 4 2, ( 4) 3 = (4) 3 5) If a Z {0}, n, m Z +, then: a m a n = a m + n i.e. the product of numbers with a common base, raised to powers, equals the common base raised to the sum of the powers. 6) If a is an integer and a 0, n, m Z +, m > n, then : a m i.e. when dividing numbers with the = same a m-n base, raised to powers, we a n subtract the power of the divisor from that of the dividend. 22
Exercise 1)Find the value of each of the following : 1) 2² 2³ = 2) 5 3 5 2 = 3) (-1) 50 = 4) (-1) 51 = 5) 17 10 17 9 = 6) (-6) 7 (-6) 5 = 7) a 4 a 3 = 8) (-x) 5 (-x) 3 = 9) (-3)² -(3)² = 10) (-1) 5 (-1) 7 = 11) (-1)³ + (-1) 4 = 12) (-1) 100 + (-1) 101 = 13) 2³ + 2² = Remember that: a 0 = 1, where a 0 14) 7 3 7 = 2) If a = 2, b = -3, find the numerical value of : (a) 3a² (b) 2a + 3b 23
3) Find 1 5 4 5 3 5 2 = 5 5 5 2 2 3 4 5 3 3 3 3 3 = 3 2 3 6 (-3) 4 3 5 = 3 3 (-3) 3 7 2 3 4 7 5 = 3 3 7 5 (-5) 5 3 4 = ( -3) 3 5 3 6 (-2) 2 3 12 2 3 = (-3) 5 2 4 (-3) 6 4) PUT (<, >,or = ) (a) 4 2.. 8 (b) (-6) 2 (-12) (c) (9) 2. (-3) 4 (d) 1 75 7 5... 7 5 24
1) Choose the correct answer from between brackets : 1) 3 15 3 5 = ( 3 20, 3 10, 1 10, 0 ) 2) 2 5 2 5 =. ( 1, 2 10, 4 5, 4 10 ) 3) (3 7 ) 3 5 = ( 3 10, 3 7, 6 5, 3 2 ) 4) 5 13 5 12 =. (5, 1, 5 0 ) 5) 3x 0 =. (0, 3x, no meaning ) 6) 3y 0 =. ( 0, 3y, no meaning ) 7) a 6 a 2 =.. ( a³, a 4, a 8 ) 8) 1³ = (1, 3, 30 ) 2) Find the result of each of the following : 7 5 1 6 = 5 2 4 3 3 2 3 3 = 8 7 15 15 3 14 15 = a a 4 a a 5 4 7 2 = 7 ( 2) ( 2) ( 2) 5 9 4 = 6 ( 7) 7 ( 7) 6 3 4 25
Numerical patterns 1) Complete in the same pattern : a) -2, 0, 2, 4,.,.,.. b) -15, -10, -5,,.., c) 6, 18, 54,...,... d) 2, 8, 32,...,... e) 1, 2, 4, 7, 11, 16,...,...,... f) 2, 6, 18, 54, --------, --------- g) 4, 7,, 13, 16,,. h) 0.5, 1,., 2, 2.5,..,. i) 3, 9, 27, 81,..,.., j) k) 1, 4, 9, 16,.,.. l) 16, 12, 8, 4,.,. m) 9, 5, 1,.,.. n) 27, 9, 3, 1,.,. o) 10000, 1000, 100,,.. p) 1, 2, 4, 8, 16,,., 26
Revision 1) Complete using,,, 1) N. Z 2) Z +.. Z 3) {0} N 4) {0} Z 5) 5..Z 6) 0 Z + 7) {-1, 2} {3, 4}.. Z + 8) 40 20 Z 9) {5, 2 5 }. Z 10) - 9 + 3 2. Z 2) Complete : 1) Z + - Z - = 2) Z - Z + = 3) Z - Z - =.. 4) Z - N =. 5) Z + Z - = 6) Z + Z - {0} = 7) Z + Z - =.. 8) Any positive integer is.. Than any negative integer. 9) is neither positive nor negative 27
3) Using the properties of adding and multiplication find the result of 1) 6 x (-2 + (- 7) ) 2) 112 17 + 112-17 3) 2015 + 180 + (-1015) 4) 24 + 19 + (-24) 4) Find the value of X 1) X 2 = 0 2) 1 X = - 6 3) -5 {-1, 0, -3, X} 4) X {2, 5,-3} {-5, -3, -8} 5) { 2, x } U {-4, 0, 4 } = {0, -2, 2, -4, 4 } 5) If x =2, y = -4, z = -1 find the value of 1) 2x + y z 2) 2 x y + z 3) 2 ( x + 4 ) 5 ( z 2 ) 28
6) Complete 1) The sets of integers between -3,2 is.. 2) The greatest negative integer is and the smallest positive integer is. 3) x = 7 then x=..or.. 4) 5 X = - 45, X =. 5) 2 [X + 8] = 2-3 + 2 8 then X=.. 6) The integer between -2, 0 is. 7) The additive inverse of -3 is. 8) 3 + -3 = 9) The adding natural (Identity) element is.. 10) The multiplying neutral (identity) element is. 11) The non positive integer set is.. 12) The non negative integer set is.. 13) -9 = X then X =. 14) (-3) 7 =7 (-3) (..property) 15) (-2 6) 8 =8 (-2 6) (..property) 16) 6 + (3+9) = (6+3) +9 (.. property) 29
7 ) Simplify: a) b) 5 4 + 2 8 5 2 2 4 ( 2 ) 6 ( 2 ) 2 2 4 = c) ( -2 ) 4 + ( - 3 ) 3 = d) ( - 1 ) 100 + ( - 1 ) 101 e) 3 4 ( 3 ) 5 3 7 = 8) If a = 3 2, b = 2 3,find (a b ) 5 9) Complete in the same pattern:- a) 3, -6, 12, -24,, b) 1, 2 3 3, 1, 4 3 30
10) Choose the correct answer from those between the brackets: a) Z N =. [Z +, {0}, Z -, 0] b) Z N = [Z,N,Z -,Z + ] c) Z - Z - =.. [ Z -,Z +,N,{0} ] d) Z Z + = [Z - {0},Z,N, 0 ] e) Z- N =.. [ Z, Z -, Z + ] f) An integer included between -2, 3 is [-2, -1, 3, -3] g) The number of integer numbers between -2 and 3 =. [ 3,4, 5,6 ] h) 3 + -3 =.. [ zero, 3, -6, 6 ] i) If a { 2, -5, -3 } { 5, -2, -3 } then a =.. [ {2}, {-3}, {-5} ] j) Z = Z -.. [ {0} Z + or Z + ] k) Z = Z [ N or C ] l) N.. [ Z +, Z +, Z ] m) Which of these numbers is the greatest? -1, -6, -10, -100 [-1, -6, -10, -100 ] n) Which of these numbers is the smallest? -1, -6, -10, -100 [-1, -6, -10, -100 ] 31
Unit 2 The equation and inequality of the first degree 4+6=10 is called anumerical sentence The numerical sentence is called closed mathematical sentence because we can determin whether they are right or wronge. 5x y=20 is called symbolic sentence. The symbolic are called open mathematical sentence because we cannot determine whether they are right or wrong due to the existence of symbole y. The mathematical sentence which contain the symbol = is called equations. The equation: is a mathematical scentence includes equality relation between two sides. The inequality is a mathematical sentence includes equality relation between two sides. Degree of the equation: The degree of the equation is equal to the highest power of the unknown (symbol) in this equation. 1. Determine whether each of the following represntes an equation or not and say why? a) Y + 6 b) 10 3 = 7 c) Z + 12 = 25 2. Determine wich of the following represents an equation or an inquality giving reason. a) a 8 4 b) b + 25 c) 2c 9 3. Determine the degree: a) X + 8 = 17 b) X 4 3 c) 3x 2 + 2 = 14 d) a + 2b = 5 e) 3y 4-4 f) 3 3 3x 2 = 0 4. Given the substitution set is X = { -1, 0, 1, 2, 3 } a) Find the solution set of the equation 3x 2 =7 b) Find the solution set of the inequality x + 1 3 32
5. Find the solution set for each of the following equations and inequalities: a) 1 + x = 21 if the substitution set is {2, 4, 6, 8, 10} b) 3x + 5 = 20 if the substitution set is {-2, 0, 3, 5} c) 5x 6 =19 if the substitution set is {0, 3, 5} d) 2(x 4) = x + 2 if the substitution set is {4, 6, 8, 10} e) X + 4 6 if the substitution set is {-1, 0, 1, 2} f) 4x - 3-7 if the substitution set is {-2, -1, 0, 1, 3} g) x + 2 5 if the substitution set is {-3, -1, 0, 1, 3} h) 4x + 3 9 if the substitution set is {-2, -1, 0, 1, 2} 33
Solving first degree equation in one unknown If a, b and c are three integers, and a = b then:- a + c = b + c a c = b c If a, b and c three inteders and a = b then, a c = b c a c = b c, c 0 (any number divided zero has no meaning) 1) Solve the following equations :- a) X + 4 = 10 in N b) X 12 = 8 in Z c) Y 2 = 11 in N d) Z + 11= 9 in Z e) b + 9 = 7 in N f) a + 5 = 5 in N 2) Find the solution set of this equation in N :- a) 5x + 1 = 26 b) 6x + 7 = 25 c) 4y = 5y + 21 d) M + 8 = 23 e) 7n = 56 f) 3x + 4 = x + 20 g) (4L + 3) + 4 = 31 34
Applications on solving first degree equation in z 1- If 7 is addedto anumber, then the the result is 25. What is the number? 2- If 10 is added to anumber, we get the result is 15, find the number. 3- If we subtracte 8 from a number the result was 5. Find the number. 4- If we subtracted 6 frme duble a number, the result was 2. Find the number. 5- A number was addded to 17, the result become 3. What is the number? 6- Three consecutive natural numbers their sum is 36.find these numbers 7- Three consecutive natural numbers, their sum is 45. Find these numbers? 8- Two consecutive numbers their sum is 13.find these two numbers 9- Two even consecutive numbers, thier sum is 18. Find these two numbers? 10- two odd consecutive numbers, thier sum is 20. Find these two numbers? 11- Three even consecutive numbers, there sum is 18. Find these numbers 12- If mohamed s age exceed george s age by 2 years and next year the sum of their ages will become 50 years. What is the age each of them now? 13- A rectangle its length is three times its width and the perimeter is 46 cm. Find the area. 14. A rectangle its width is half its length and the perimeter is 42 cm. Find its dimensions. 15. A metal wire of length 40 cm is divided into two parts, one of them is of length nine times the leng of the other. Calculate the length of each part. 16. If 8 is added to twice a number, the result is 12. Find this number. 17. Two integers numbers, one of them is twice the other and their sum is 39. Find the two numbers. 18. Two consecutive odd numbers, their sum is 8. Find these two numbers. 19. The sum of the three dimensions of a cuboid is 70 cm. If the length is four times the width and its height is five times its width. Calculate the volume of this cuboid. 20. If the number of pupils is 360 pupils and the number of boys is five times the number of girls. Calculate the number of each boys and girls. 35
*** Very important notices: 7 is added to a number 7 + x We subtract 8 from a number X - 8 Subtracted 6 from double a number 2x - 6 A number subtracted from 9 9 - x The sum of three consecutive natural num X + ( X + 1 ) + ( x + 2) = 3x + 3 Three even or odd consecutive numbe X + ( x + 2 ) + ( x + 4 ) = 3x + 6 L=3x, W=x, per.= (L +W)x2 A rectangle its length is three times its w 3x+x)2=( 4x)2=8x A rectangle its length is half its lengt W = x, L = 2x If 8 is added to twice a number 8 + 2x 1- Find the solution of the following inequality and represent the solutions on the number line : a) x + 4 < 7 where x N b) 2 x + 9 < 1 where x N and Z c) -1 < 2 x + 3 < 13 where x Z d) x 3 < 1 where x N e) 4 x + 3 < 6 x + 11 where x N and Z f) 2 x 7 < -1 where x N and Z 2- Solve the following in N and Z : a) 5 x < 15 b) X 3 4 c) -2 x 3 d) 1 x > 4 e) -5 2 x 3 1 f) 4 < x + 4 < 7 g) 1 < 5 x 3 h) -3 4 x 7 5 36
The circle Circumference of the circle: C π = 3.14 or 22 7 C i) 2 π r d π Circumference of circle (c ) = 2 π r or c = d π r = c 2 π d = c π Area of the circle (A) = π r 2 r = A π The circular sector: It is a part of the surface of the circle bounded by An arc and two radii passing through ends of the arc r Find the area and the circumference of the circle in each of the following cases (π = 3.14 ) (1) radius = 8 cm (2) radius = 1.4 cm (3) diameter = 21 m (4) diameter = 1.4 cm 37
In the opposite figure: A circle is drawn in a square its length is 14 cm Calculate the area of the shaded part ( π = 22 7 ) ----------------------------------------------------------------------------------------------------------------- (10) In the opposite figure: ABCD is a rectangle its length is 12 cm And its width 7 cm Calculate the area of the shaded part. 12 cm 38
(1) Find the area of a circle with a radius of length 21 cm. ( π=22/7 ) (2) Find the area of a circle with radius 4 cm. ( π=3.14) (3)A circle its diameter is 14 cm. calculate its area. ( π=22/7) (4) Find the length of the radius of a circle if its area is 314 cm². (5) The area of a circle is 154 cm². Calculate its circumference. ( π=22/7) (6)A circle M its half of area is 100.48 cm². Calculate its circumference. ( π = 3.14) (7)In the opposite figure a circle M is divided into four equal circular sectors. If the area of one sector is 364.5 cm² then calculate the area and the circumference of the circle M. 364.5cm² (8) A table its surface in the form of a circle its diameter is 1.5 m its surface is wanted to be covered by a sheet of glass equal to its surface calculate the cost price if the square meter of the glass costs L.E 60. 39
Lateral surface area of the cube: Area of one face x 4 Edge length x itself x 4 Area of one face = L.S.A 4 Total surface area of a cube = Area of one face x 6 Edge length x itself x 6 T.S.A Area of one face = 6 Lateral surface area of cuboid = Perimeter of base x height Total surface area of cuboid = LATERAL & TOTAL SURFACE AREA L.S.A & T.S.A L.S.A + ( 2 x area of one base) L.S.A + area of one base ( with lid) ( without a lid) 40
Lateral & total surface area 1) A cuboid with dimensions 9.5 cm, 6cm, 8 cm. Find the lateral area and the total area. 2) A cube of edge 8 cm. Find the lateral area and the total area. 3) Acuboid with length = 8.5 cm, height =12 cm and lateral area 168 cm 2. Find the width and the total area. 4) A cube with lateral area 100 cm 2. Find the side length and the total area. 5) If the lateral area of a cube is 81 cm 2. Find the total area. 6) If the total area of the cube is 48 cm 2. Find the lateral area. 7) A cube of edge length= 6 cm. calculate the ratio between the lateral area and its total area. 8) The dimensions of a cuboid is 6 cm, 5 cm and 8 cm.and a cube of edge 7 cm. calculate the difference between their lateral areas. 9) Box without a lid its length= 14 cm, width 5 cm and height = 17 cm. Calculate the lateral area and the total area. 10 ) A box truck in form of cuboid, the inner dimensions are 6m, 3m and 1.5 m, it is wanted to paint inner box, and the cost price of one square meter= 12 L.E. Calculate the cost of paint. 11 ) A room its length = 6m, width =5m and height =3m. its wanted to paint its lateral walls and ceiling. And the cost price of one square meter = 6 L.E. Calculate the cost of painting, knowing that the room has two windows and door their area are 8 m 2. 41
12) Youssef used a rectangular piece of card board, its length=12cm and width=80cm to form a cubic box of edge=30cm. Calculate the remaining paper Area. 13) The inner dimensions of a swimming pool are 25 m, 16 m and 3.5 m. it is wanted to cover with square shape tiles, its side length = 20cm. If the cost price of one square meter = 32 L.E. Calculate the cost of covering the wall and the ground. 14) A cuboid shaped box truck with dimensions 6m, 3.5 m and 2.8m. It is wanted to cover its sides with iron sheet, the cost price of the square meter is L.E 13. Calculate the required cost for iron sheet. 42
L.S.A & T.S.A Complete the following: 1) The number of edges of a cube =.. 2) The L.S.A of a cube whose total area =600 cm² is. 3) The lateral area of a cube is 36 cm² then its total area =. 4) If the total of edges of a cube is 120 cm² then its L.A =.cm² 5) A cube without lid of base perimeter 36 cm then its total area = cm² 6) The T. area of a cube is 150 cm² then its edge length =..cm Choose the correct answer: 1) If the perimeter of one face of a cube is 12 cm then its total area =..cm² ( 35 _ 54 _ 55 _56) 2) If the edge length of a cube is 6 cm then the ratio between its L.A: T.A is. (2:3_3:2_1:4_4:1) 3) The lateral area of one face of a cube =. Its total area. (1/2 _ 1\4 _1\6 _1\8 ) 4) length of a if the edges cube = the side length of an equilateral triangle with perimeter 18 cm then its L.A = cm² (36 _ 144 _ 216 _ 180) 5) A cube with side length 6 cm then its T.A =.. cm² (36 _72 _144 _216) 43
The Probability The random experiment :- It is an experiment in which we can determine all its possible outcomes before carrying it out, but we can t predict in certainty which of these outcomes will occur when the experiment is carried out. Sample space ( outcomes Space ) : It is a set of all possible outcomes for a random experiment and it is usually denoted by the symbol (s) and the number of all elements of the sample space is denoted by n (s). Remarks :- (1) The impossible event [ᴓ] : is the event which cannot occur. i.e. The probability of the impossible event = zero... P (ᴓ) = 0 (2) The certain event (sure event ) (S) : is the event whose outcomes are all the possible outcomes. i.e. The Probability of the certain event =1... P (s) =0 (3) the Possible event : Some of outcomes of the experiment. i.e. The Probability of the Possible event = proper fraction. So,the Probability of any events is not less than zero and it is not more than 1 i.e. For any event A, we found that 0 P (A) 1 Probability of the event If each element of the sample space S has the same chance of being the selected element. The probability of occurrence of being the selected element. The probability of occurrence of an event A S is denoted by P (A) and is Found by using the relation : P (A) = The number of elements of A The number of elements of S = n (A) n (S) 44
Complete the following: (a) The probability of the impossible event=. And the probability of the certain event=. (b) For every event A, we find that. P (A). (c) If a fair coin is tossed once, then the probability of appearance of a head =. (d) 10 cards are numbered from 1 to 10 A card is drawn randomly, then the probability that the card carries an odd number =. (e) A box contains 48 oranges, 4 of them are bad.if we draw an orange at random, then the probability that the drawn orange is bad =., and the probability that it is not bad =. (f) If the probability of the occurrence of an event is 5 8, then the probability of the non occurrence of this event is. (g) In the experiment of rolling a die once, if A is the event of getting a number less than 4, then P(A)=. [a] 5 6 [b] 2 3 [c] 1 2 [d] 1 6 (h) If the probability of the success of a student is 70 %, then the probability of the failure=. [a] 0.7 [b] 0.07 [c] 0.3 [d] 0.03 (i) A regular coin is tossed 1000 times, then the most expected number to get a head equals. [a] 496 [b] 503 [c] 600 [d] 999 45
(1) If a fair die is thrown once and we observe the number on the upper face, find the probabilities of each of the following events : [1] A is the event of appearance of a number greater than 4 [2] B is the event of appearance of an even number. [3] C is the event of appearance of the number 5 [4] D is the event of appearance of the number 7 [5] E is the event of the appearance of a number less than 7 (1) A box contains 10 cards numbered by the even numbers from (2 to 20), one of the cards is selected at random. Calculate the probability of : [A] The event A: appearance of a multiple of the number 4 [b] The event B: appearance of an even number. [c] The event C: appearance of a number that is divisible by 3 (2) A bag contains 25 balls (4 balls are yellow, 7 balls are red and the remainder is black). If a ball is drawn randomly, find the probability that the drawn ball is : (a) Black. (b) Yellow or black. (c) Not yellow. (d) Green. (e) Neither black nor yellow. 46
(3) In the experiment of forming a 2-digit number from the set of digits {5,6}. What is the probability: (a) The event A: the units digit is an odd number. (b) The event B: the sum of the two digits is 11 (c) The event c: the two digits are equal. (4) In the meeting for discussing the problems of the workers in the factory, 100 workers were attending from men and women. If the probability of a man standing to show the problems of the workers is 3 5, calculate the number of the men and women in this meeting. 47
Unit (4) Representing the Statistical data by using the circular sectors Notice that :- (1) Each Circular sector has an angle whose vertex is the centre of the circle which is called a " Central angle ". (2) The Sum of measures of angles accumulating around the centre of the circle is equal to 360 Central angle (3) 25 % 50 % 75 % 25 % can be represented by a sector of area which is a quarter of the area of the circle. 50 % can be represented by a sector of area which is a half of the area of the circle. 75 % can be represented by a sector of area which is three quarters ( 3 ) of the area of the 4 circle. 48
(1) Complete each of the following :- (a) The measure of the angle of a circular sector which represents 1 of the area of the 3 circle equals. (b) The measure angle for the sector of quarter circle is. degree. (c) The measure of the angle of the circular which area represents 1 the area of the circle 8 (d) The sum of the measures of the angles around the point is.. degree. (e) the angle of the sector is called. Because its vertex at the Centre of the circle. (f) A pie chart whose angle measure is 60 g, than its area represents of the circle area ( 1 3 or 1 4 or 1 5 or 1 6 ) 1- The following table shows the percentage of the students participated in the school activities. Activity cultural sport social art Percentage of 25% 30% 20% 25% students Represent these data by a pie chart. ''Giza 2011'' 49
2- The following table shows the percentage of egg production in four farms, a merchant collected these eggs to distribute it on the grocery stores represent these data by using the circular sectors. The farm First Second Third Fourth The percentage of 50% 25% 15% 10% the production «Alex 2011» 3- The table shows the percentage of the four factories: Factory First Second Third Fourth Percentage 35% 15% 25%... (a) Complete the table. (b) Represent these data by a pie chart. 50