Created by the a.m.r.a.h.yCreated by the a.m.r.a.h.y

groupgroup

first topic: number problemsfirst topic: number problems

Of all the word problems, the number Of all the word problems, the number problems are the easiest to translate problems are the easiest to translate into equations since the relationships into equations since the relationships among the numbers are directly among the numbers are directly stated in the problem.stated in the problem.

ILLUSTRATIVE EXAMPLEILLUSTRATIVE EXAMPLE One number is two more than thrice another. Their sum is One number is two more than thrice another. Their sum is

30. Find these numbers.30. Find these numbers. Solution:Solution: READ: READ: Reading the problem thoroughly, we know two Reading the problem thoroughly, we know two

things about the numbers. things about the numbers. a) their sizes : one of them is two more than thrice the a) their sizes : one of them is two more than thrice the

other andother and b) their sum: the sum is 30.b) their sum: the sum is 30. REPRESENT:REPRESENT: If we represent the numbers using the first If we represent the numbers using the first

sentence we have:sentence we have: let x= first numberlet x= first number then: 3x+2= other numberthen: 3x+2= other number

RELATE : The relationship between the number x and 3x+2 and RELATE : The relationship between the number x and 3x+2 and the other number,30, gives as an equation.the other number,30, gives as an equation.

EQUATE: EQUATE: X+3X+2= 30X+3X+2= 30 SOLVE: In solving the equation we have:SOLVE: In solving the equation we have:

x+3x+2=30x+3x+2=30

4x+2=304x+2=30

4x=284x=28

x=7x=7

Therefore, the first number is 7 and the other number is Therefore, the first number is 7 and the other number is 3x+2=3(7)+2=233x+2=3(7)+2=23

Answer :7 &23Answer :7 &23

Prove: Prove:

a)a) Their sum is 30: 7+23=30Their sum is 30: 7+23=30

b)b) 23 is two more than thrice 7: 3(7)+2=21+2=2323 is two more than thrice 7: 3(7)+2=21+2=23

ExercisesExercises

1.) The sum of two numbers is 29 and 1.) The sum of two numbers is 29 and their difference is 6. find the numbers. their difference is 6. find the numbers.

2.) the smaller of two numbers is thrice 2.) the smaller of two numbers is thrice the larger. The larger number is eight the larger. The larger number is eight more than the smaller one. Find the more than the smaller one. Find the numbers.numbers.

Solution:Solution:1.) x-(29-x)=5 1.) x-(29-x)=5

x-29+x=5x-29+x=5

2x-29=52x-29=5

2x=342x=34

x=17x=17

Other number is Other number is

29-x=29-17=1229-x=29-17=12

2.) x=3x+82.) x=3x+8

3x+8=x3x+8=x

2x=-82x=-8

x=-4x=-4

3x=-123x=-12

The answers are: -4 & -12The answers are: -4 & -12

SECOND TOPIC: ODD, EVEN, AND CONSECUTIVE SECOND TOPIC: ODD, EVEN, AND CONSECUTIVE INTEGERSINTEGERS

The word consecutive means following in order The word consecutive means following in order without interruption. Even numbers are whole numbers without interruption. Even numbers are whole numbers divisible by 2 while whole numbers which are not divisible by 2 while whole numbers which are not divisible by 2 are odd numbers. Consecutive even divisible by 2 are odd numbers. Consecutive even integers are even numbers in uninterrupted order which integers are even numbers in uninterrupted order which is the same as when you count by two’s such as is the same as when you count by two’s such as 2,4,6,8,&10. Consecutive odd integers are odd numbers 2,4,6,8,&10. Consecutive odd integers are odd numbers in uninterrupted order, such as 7,9,11,13,& 15. in uninterrupted order, such as 7,9,11,13,& 15.

Illustrative example:Illustrative example:

1.) The sum of three consecutive integers is 90. 1.) The sum of three consecutive integers is 90. Find the integers.Find the integers.

solution: solution:

let x= first numberlet x= first number

then x+1= next consecutive integer. then x+1= next consecutive integer.

and x+2= third consecutive integer.and x+2= third consecutive integer.

their sum is 90their sum is 90

manipulating the equation we have:manipulating the equation we have:

x+(x+1)+(x+2)= 90x+(x+1)+(x+2)= 90

x+x+1+x+2=90x+x+1+x+2=90

3x+3=903x+3=90

3x= 873x= 87

x=29 1x=29 1stst integer integer

x+x1= 2x+x1= 2ndnd integer integer

x+2= 3x+2= 3rdrd integer integer

answer: the consecutive integers are answer: the consecutive integers are 29,30&3129,30&31

Prove: 29+30+31=90Prove: 29+30+31=90

exercises:exercises:

1.1. Find two consecutive integer such Find two consecutive integer such that if we triple the first and double that if we triple the first and double the second, the sum is 77.the second, the sum is 77.

2. the sides of a triangle are 2. the sides of a triangle are consecutive integers. If the perimeter consecutive integers. If the perimeter of this triangle is 192 cm, find the of this triangle is 192 cm, find the length of each sides.length of each sides.

solution:solution:

1.1. 3(x)+2(x+1)=773(x)+2(x+1)=77

3x+2x+2=773x+2x+2=77

5x+2= 755x+2= 75

x=15 1x=15 1stst integer integer

x+1= 16 2x+1= 16 2ndnd integer integer

Answer: the consecutive integers are 15 & 16Answer: the consecutive integers are 15 & 16

Proof: triple 15: 45Proof: triple 15: 45

double 16:32double 16:32

total:77 2. x+x+1+x+2=192total:77 2. x+x+1+x+2=192

3x+3= 1923x+3= 192

3x= 1893x= 189

x=63 cmx=63 cm

x+1= 64 cmx+1= 64 cm

x+2= 65 cmx+2= 65 cm

answer: the lengths of the sides of a triangle are 63,64,65 cm.answer: the lengths of the sides of a triangle are 63,64,65 cm.

The delineation of a sphere on a flat plane is similar to painting.

For just as the painters seek to imitate objects

exactly...geometricians and astronomers delineate on a flat plane solid objectssuch as octahedrons and cubes and all spherical bodies,

like the stars, the heavens, and the earth.

If all the arts aspire to the condition of music,

all the sciences aspire to the condition of mathematics.

Let no one ignorant of mathematics enter here.

As far as the laws of mathematics refer to reality, they are not certain,

and as far as they are certain, they do not refer to reality.

No, it is a very interesting number, it is the smallest

number expressible as a sum of two cubes in two different ways.

There is no "royal road" to geometry.