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# Pre Algebra

Pre algebra is mathematics course studied in middle school consisting of basic algebra operations such as factorization, simplification, solving equations. Get Pre Algebra help online with TutorVista. Solve problems, work on basic concepts and get help with your homework too. TutorVista's intensive Pre Algebra tutoring prepares you for high school algebra. Study with highly qualified tutors with many years of experience in tutoring students who will work with you to ensure that you understand the subject and score well in it.

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## What is Pre Algebra?

Pre algebra is the branch of the mathematics used to prepare the student for the study of algebra. Topics covered in pre algebra are Factors and Primes, GCF and LCM, Fractions, Square Roots, Basic Equations.

### Pre Algebra Review

Our intensive Pre Algebra tutoring covers topics like:

• Relations and functions
• Factorization
• Linear equations
• Simple roots and powers
• Variables

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## Pre Algebra Tutoring Program

Given below are some of the benefits of taking up an online Pre Algebra tutoring program:

• Expert tutors
• 24/7 live tutor available
• Sharing whiteboard facility
• Usage of VoIP
• Free demo session

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## Pre Algebra Curriculum

Curriculum of pre algebra is given below:

Whole Numbers
1. Introducing Whole Numbers
2. Naming Numbers
3. Rounding Whole Numbers
4. Operations on Whole Numbers
5. Word Problems with Whole Numbers.
Integers
1. Comparing Integer Values
2. Arithmetic Operations on Integers
3. Introducing Exponents
Fractions
1. Reducing Fractions
2. Converting Between Improper Fractions and Mixed Numbers
3. Multiplying Fractions and Mixed Numbers
4. Dividing Fractions and Mixed Numbers
5. Finding Common Denominators
6. Arithmetic Operations on Fractions and Mixed Numbers
7. Comparing Fractions
Decimals
1. Naming Decimals
2. Rounding Decimals
3. Operations on Decimels
4. Converting Between Fractions and Decimals
Powers, Exponents, and Roots
1. Powers and Exponents
2. Square Roots and Cube Roots
3. Scientific Notation
Percent
1. Converting Between Percents, Decimals, and Fractions
2. Solving Percent Problems
Expressions and Equations
1. Evaluating Algebraic Expressions
2. Combining Like Terms
3. Solving Equations
Ratio and Proportion
1. Introducing Ratios and Proportions
2. Solving Proportions
3. Solving Problems Using Ratios and Proportions
Polynomials
1. Adding and Subtracting Polynomials
2. Multiplying and Dividing Polynomials
Geometry and Measurement
1. Introducing Geometric Figures
2. Finding Perimeter and Circumferences
3. Finding Areas and Volumes
4. Working with Right Triangle
5. Measuring Lengths, Weight, Capacities and Time.
Graphing
1. Plotting Points and Lines on the Coordinate Plane
2. Graphing Linear Equations
3. Finding the Slopes of Lines
4. Solving Systems of Linear Equations

## Pre Algebra Terms

Various terms are used in pre algebra like coefficient, variable, factor, expression, quotient, power, exponent, constant, integers and absolute values etc. Let us discuss some terms:
• Coefficient is a numerical part of a variable term.
• Variable represents one or more numbers.
• Factor takes place when two numbers are multiplied.
• Expression is mathematical sentence which contain more than one operation.
• Quotient is an answer to a division problem.
• Power is made up of an exponent.
• Exponent is a number or a variable that represents the number of times a base is used as a factor in a repeated multiplication.

## Formulas

List of pre algebra formulae is given below:Polynomial Factoring Formulas:

(x + y)$^2$ = $x^2 + y^2 + 2xy$

(x - y)$^2$ = $x^2 + y^2 - 2xy$

$x^2 - y^2 = (x + y)(x - y)$

$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

The value of x for the quadratic equation, $ax^2 + bx + c = 0$, $x \neq 0$ is

x = $\frac{-b \pm \sqrt{b^2} -4ac}{2a}$.
Law of Exponent

$(x^a)^b$ = $x^{ab}$

$x^a \times x^b$ = $x^{a + b}$

$(xy)^a = x^ay^b$

$x^{-a}$ = $\frac{1}{x^a}$
Some Other Formulas:

Area of a Square = Side $\times$ Side

Perimeter of Square = 4Side

Area of a rectangle = Length $\times$ Breadth

Perimeter of a rectangle = 2(Length+ Breadth)

Area of Triangle = $\frac{1}{2}$ Base $\times$ Height

Area of a circle = 2$\pi r^2$, r = radius of the circle

Circumference of the Circle = 2$\pi$ r, r = radius of the circle

Sum of the angles in a polygon = (n - 2)180, n = sides of the polygon

## Pre Algebra Equations

Lets discuss some pre algebra equations with the help of examples.

Single step Addition And Subtraction Equations

Let us solve x - 4 = 0

x - 4 = 0

$\Rightarrow$ x = 4.

Multi Step Equationscdd

Let us solve 4x - 12 = 0

4x - 12 = 0

$\Rightarrow$ 4x = 12 (by adding 12 both the sides)

$\Rightarrow$ x = $\frac{12}{4}$ = 3 (divide each side by 4)

x = 3.

Let us solve x$^2$ + x - 6 = 0

x$^2$ + x - 6 = 0

$\Rightarrow$ x$^2$ + 3x - 2x - 6 = 0 (by factoring)

$\Rightarrow$ x(x + 3) - 2(x + 3) = 0

$\Rightarrow$ (x - 2)(x + 3) = 0

$\Rightarrow$ either x - 2 = 0 or x + 3 = 0

$\Rightarrow$ x = 2, -3.

## Examples

Given below are some of the problems of pre algebra:

### Solved Examples

Question 1: Find the remainder when $x^3 - 2x^2 + 4x + 5$ is divided by $x - 3$.
Solution:
Let $P(x) = x^3 - 2x^2 + 4x + 5$

Step 1:

Put x - 3 = 0

$\Rightarrow$ x = 3

Step 2:

$P(3) = 3^3 - 2 \times 3^2 + 4 \times 3 + 5$

= 27 - 2 $\times$ 9 + 12 + 5

= 27 - 18 + 12 + 5

= 26

When P(x) is divided by (x - 3), then remainder is 26.

Question 2: Find a fraction such that if 2 is added to the numerator and 1 to the denominator, it reduces to $\frac{1}{2}$ and becomes $\frac{3}{5}$, if added 3 to the denominator.
Solution:
Let the fraction be $\frac{x}{y}$

Step 1:

Case 1:

When 2 is added to the numerator and 1 to the denominator it reduces to $\frac{1}{2}$

$\Rightarrow$ $\frac{x + 2}{y + 1} = \frac{1}{2}$

2x + 4 = y + 1

2x - y + 3 = 0 .........................(i)

Case 2:

Fraction becomes $\frac{3}{5}$, if we add 3 to the denominator.

$\frac{x}{y + 3} = \frac{3}{5}$

5x = 3y + 9

5x - 3y - 9 = 0 .......................(ii)

Step 2:
Solve equation (i) and (ii).

Multiply equation (i) by 3 and subtract from (ii).

5x - 3y - 9 = 0
-(6x - 3y + 9) = 0
---------------------------
-x + 0 - 18 = 0
---------------------------

$\Rightarrow$ x = - 18

Plug the value of x in equation (i)

$\Rightarrow$ 2 $\times$ (-18) - y + 3 = 0

-36 - y + 3 = 0

y = - 33

The required fraction is $\frac{18}{33}$.

## Word Problems

Given below are some of the word problems on Pre Algebra.

### Solved Examples

Question 1: If three times the square of a number is decreased by 10 and equals 15 more than 10 times the number, what is the number?
Solution:
Let the number be 'x'.

Step 1:

The problem states that:
3(square of a number) - 10 = 10 $\times$ number + 15

$\Rightarrow$ 3x$^2$ - 10 = 10x + 15

$\Rightarrow$ 3x$^2$ - 10x = 15 + 10

$\Rightarrow$ 3x$^2$ - 10x = 25

3x$^2$ - 10x - 25 = 0

Which is quadratic equation

Step 2:

Solve for x, 3x$^2$ - 10x - 25 = 0

3x$^2$ - 10x - 25 = 0

$\Rightarrow$ 3x$^2$ - 15x + 5x - 25 = 0

$\Rightarrow$ 3x(x - 5) + 5(x - 5) = 0

$\Rightarrow$ (3x + 5)(x - 5) = 0

Step 3:

Either 3x + 5 = 0 or x - 5 = 0

$\Rightarrow$ 3x = - 5 or x = 5

$\Rightarrow$ x = $\frac{-5}{3}$ or x = 5

Hence, the number is $\frac{-5}{3}$ or 5.

Question 2: Find the largest of three consecutive odd integers, if the sum of the first and the third integers is 7 more than the second integer.
Solution:
Let the three consecutive odd integer be x, x + 2 and x + 4.
That is,
First integer = x
Second integer = x + 2
Third integer = x + 4

Step 1:

The problem states that:

Sum of the first and third integers = Second integer + 7

$\Rightarrow$ x + (x + 4) = (x + 2) + 7

$\Rightarrow$ x + x + 4 = x + 2 + 7

Step 2:

Combine the like terms:

$\Rightarrow$ 2x + 4 = x + 9

Subtract x from both side

$\Rightarrow$ 2x - x + 4 = x + 9 - x

$\Rightarrow$ x + 4 = 9

Subtract 4 from both sides

$\Rightarrow$ x + 4 - 4 = 9 - 4

$\Rightarrow$ x = 5, first integer.

Step 3:

Largest or third integer = x + 4 = 5 + 4 = 9

Hence, the largest integer is 9

## Practice Problems

Question 1: Solve the quadratic equation 2x$^2$ - 7x - 15 = 0.