Pre-Algebra

Pre-Algebra
Negative Numbers
Fractions, Proportions
Decimal Numbers and Rounding
Square Roots
Length, Mass, Volume Units
Decimals, and Percentages
Word Problems

Negative numbers, like positive numbers, may be whole numbers or fractions, and all are less than
zero in value. A negative number may be represented algebraically as -N, where N represents the
number of units that -N is less than 0. The difference between -N and N equals 2N (Try to verify this
on paper!)

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DECIMALS AND ROUNDING

NEGATIVE NUMBERS

Quadratic Equation

0.5 rounds to 1
1.66669 rounds to 1.7 (to nearest decimal place)
-0.00000215    rounds to -0.0 (rounding to one decimal place)
6.6666666 rounds to (2 decimal places) 6.67
$98,223.08 to the nearest dollar    $98,223
square root of 2 to 3 decimal places    1.414

My property is twice the AREA of Nick's. If his property is 100' by 50', then what is the area of my property? The area of nick's property is what percent of the area of mine?
    Am = Area (my property) =   2 * Area (Nick's property)
                                                  =   2 * (100')(50') = 10,000 sq. ft.

    An = Area (Nick's property)   = (100)(50) = 5000 sq.ft.

    % = 5000 sqft * 100 = 0.5*100 = 50% or one-half the area of mine.
              10000 sqft

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     The square root of a number is equal to the factor which, when multiplied by itself, equals that number.

For example, (5)(5) = 25; therefore, (25)^0.5 = 5.        The square root of 25 = 5.

   The square root of a variable squared or of a number squared is equal to the absolute value of the variable or number. Thus, the square root is always positive.

This page does not have radical signs; the square root of "x" ?will be represented as          (x)^0.5 or simply x^.5   .

     (x^2)^0.5 = | x |
    (16^2)^.5 = | 16| = 16
    (-7^2)^ .5 = | -7 | = 7

Another example of rounding ( 0.9995 round to 3 decimal places):

0.9995     Last digit is 5 >= 5; therefore, round up to 1.000. In this case, by adding 1 to the last digit desired, 0.999
becomes 1.000 (or 0.999+.001 = 1.000). When the last decimal place desired is a 9, and rounding up is clearly indicated, this decimal place becomes a "0", and 1 is added to the decimal immediately preceding the last decimal place (of the FINAL ANSWER). When, as in this case, more than one 9 appears at the final positions (consecutively), the above logic is applied as though each digit were a final digit.

Round 6.499 to 1 decimal place.

   6.49 (drop last digit). 6.49 = 6.5 (final answer)

Round 0.49999 to 3 decimal places:

   0.4999 (after dropping final digit "9")   =    0.500   (i.e. 0.499 + 0.001)

Decimals: Addition, Subtraction, Multiplication and Division
                           1.20
                                2.66
                                5.00
                                  .14
                                8.00
   First add furthest decimal place to right: 0+6+0+4=10; then next furthest: 2+6+0+1=9; after all decimal places are added up, units: 2+5=7.   Now, 7.00 + .90 + .10 = 8.00 or 8.                                                           subtracting decimals
Subtraction of decimals is identical to ordinary subtraction of integers, except that we must
line up the decimal point properly:
                                                       4781.22
                                                     - 22.88
                                                       4758.34
                    (Use a calculator to check your answer.)

                                                  multiplying decimals

     When multiplying decimal numbers, multiply as if there were no decimal points; then,
adding the total number of decimal places of each number, move the decimal point from right to left, that many spaces.
                                   2.4 * 1.6 = 3.84
                             1 dec + 1 dec = 2 dec

     When dividing, subtract # of dec. places of divisor from the number divided:

                               2.25 = 1.5     (2 DEC - 1 DEC = 1 DEC)
                                 1.5

     1. Round 2.005 to 2 decimal places.

     2. Add 1.02 + 0.96 + 7 = ?

    3.   3.333 - 1.667 = ?

     4. (1.4)(1.4) = ?

     5. 256.00 = ?
           16

    Answers:
    1. 2.01
    2. 8.98
    3. 1.666
    4. 1.96
    5. 16.00

                                                            fractions

     Any fraction can be thought of as a ratio of two numbers and expressed in the general format:
                                              f = n/d , d is not = 0
In the above expression, "n" is defined as the numerator which is divided by "d", the denominator.
    Division by zero is undefined, and, for our purposes, will never happen.

     A proper fraction is one in which the numerator is less than the denominator.

     An improper fraction is one in which the denominator is greater than the numerator.

     A complex fraction is defined as a fraction having a fraction or mixed number in the numerator and/or denominator, or both.

    There are other types, which we need not get into here.
     Let us start with some simple examples:

          1/4 1/2 2/3 9/10 -5/6 are all proper fractions (n and d are both integers). and n<d
          4/3 9/5 11/9 6/5 12/11 are all improper fractions (denominator > numerator).

     Examples of a complex fraction as follows:

          (1/2)/3 (1/5)/(1/2) (3 1/2)/7     Each contains a fraction or mixed number in denominator and/or numerator.

                                             1. addition and subtraction of fractions

                                                a/c   +    b/d   = ( ad + cb)/cd
                                      a/c   -   b/d   = (ad - cb)/cd
     For our purposes here, a,b,c, and d are all constants. However, variable expressions follow the same rules,
        as in this example:
                                   x/y + y/z = (xz + y^2)/yz
     Here we give a few examples of addition/subtraction of constant fractions:

                                           1/2 + 1/4 = (4 + 2)/8 = 6/8 reduces to 3/4 .
                                           1/3 + 1/4 = (4 + 3)/12 = 7/12 .
                                           1/2 - 1/5 = (5-2)/10 = 3/10 .

                                         2. multiplication of fractions

                                         a/c * b/d = ab/cd
       Example:   (1/2)(1/4) = 1/8

                                  3. division of fractions

       Division of two fractions is the equivalent of multiplying by its reciprocal:

                                         (a/c)/(b/d) = (a/c)(d/b) = ad/cb

          Example 1:     2/3 * 1/2 = 2/6 = 2/(2*3) reduces to 1/3. It is preferred to always reduce our answer to simplest terms.

          Ex. 2:     (4/5)/(1/3) = (4/5)*(3/1) = 12/5 .

     These two operations apply in exactly the same way to variable expressions:

          (x/2y) * (x+ y/2) = (x^2 + xy)/4y

                       4.converting fractions to decimals and percentages:

   A fraction (a/b; a=numerator, b=denominator) is a number, first of all.
   That number is calculated by dividing the numerator "a" by the denominator "b".

   Example 1:   4/5 = 0.8 (from long division)

   Example 2:   -4.6/2.3 = -2 (long division and retain the "sign")

   Example 3:   ab/a = b (a/a = 1; 1*b=b)

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In our discussion of decimals and division, some mention needs to be made regarding the correct rounding of decimals.

   When a decimal is "rounded", we mean that the decimal number has more "decimal" places than we require, and we need to eliminate one or more digits starting from the far right, and ending just after the first digit immediately after the last decimal place ; then the last digit is now used only for rounding. If this digit is >= 5, we round "up", meaning add 1 to the last digit desired, and drop the final digit; if it is <5, drop the final digit and leave the last remaining digit alone.

   For example, suppose we are asked to round the number 22.25006 to 2 decimal places:

   Since we need 2 decimal places, we can eliminate the last two digits, leaving 22.250. The final digit is 0 < 5 , so answer is 22.25



	Variables A variable is a symbol that represents a number. Usually we use letters such as x,y,z,s,etc., for variables. For example, we might say that s stands for the side-length of a square. The perimeter of the square is given by 4 � s. The area of the square is given by s2. When working with variables, it is useful to use a letter that will remind you of what the variable stands for: let n be the number of people in a movie theater; let t be the time it takes to travel somewhere; let d be the distance from my house to the park. Expressions An expression is a mathematical statement using numbers, variables, or both. Example: The following are examples of expressions: 6 x 3 + 7 6 � y + 5 (9-5) � 14 Example: Roland weighs 70 kilograms, and Mark weighs k kilograms. Write an expression for their combined weight. The combined weight in kilograms of these two people is the sum of their weights, which is 70 + k. Example: A car travels down the freeway at 55 kilometers per hour. Write an expression for the distance the car will have traveled after h hours. Distance equals rate times time, so the distance traveled is equal to 55 � h.. Example: Evaluate the expression 4z+ 16 when z = 15. We replace each occurrence of z with the number 15, and simplify using the usual rules: parentheses first, then exponents, multiplication and division, then addition and subtraction. 4z + 16 becomes (4 � 15) + 16 = 76 Equations An equation is a statement that two numbers or expressions are equal. Equations are useful for relating variables and numbers. Many word problems can easily be written down as equations. Example: The following are examples of equations: 6 = 6 21 = 6 + 15 x = 7 7 = x t + 3 = 8 3 (n +16) = 100 w + 4 = 16 - w y - 1 - 6 - 9.3 = 34 3 (d + 4) - 11 = 4 Example: Translate the following word problem into an equation: My age in years y plus 60 is equal to four times my age, minus 12. The first expression stands for "my age in years plus 60", which is y + 60. This is equal to the second expression for "four times my age, minus 12�, which is 4y � 12. Setting these two expressions equal to one another gives us the equation: y + 60 = 4y - 12 Solution of an Equation When an equation has a variable, the solution to the equation is the number that makes the equation true when we replace the variable with its value. Example: We say y = 3 is a solution to the equation 4y + 7 = 19, since replacing y with 3 gives us 4 � 3 + 7 = 19 ==> 12 + 7 = 19 ==> 19 = 19 which is true. Examples: x = 100 is a solution to the equation (x /2) - 40 = 10 z = 16 is a solution to the equation 5 � (z - 6) = 50 Counterexample: y = 10 is NOT a solution to the equation 4y + 7 = 19. When we replace each y with 10, we get 4 � 10 + 7 = 19 ==> 40 + 7 = 19 ==> 47 = 19 not true! Counterexamples: x = 600 is NOT a solution to the equation x � 6 - 40 = 10 z = 6 is NOT a solution to the equation 5 � (z - 6) = 30 Simplifying Equations To find a solution for an equation, we can use the basic rules of simplifying equations. These are as follows: 1) You may evaluate any parentheses, exponents, multiplications, divisions, additions, and subtractions in the usual order of operations. When evaluating expressions, be careful to use the associative and distributive properties properly. 6) You may combine like terms. This means adding or subtracting variables of the same kind. The expression 6x + 4x simplifies to 10x. The expression 13 - 7 + 3 simplifies to 9. 3) You may add any value to both sides of the equation. 4) You may subtract any value from both sides of the equation. This is best done by adding a negative value to each side of the equation. 5) You may multiply both sides of the equation by any number except 0. 6) You may divide both sides of the equation by any number except 0. Hint: Since subtracting any number is the same as adding its negative, it can be helpful to replace subtractions with additions of a negative number. Example: This problem illustrates grouping like terms and dealing with subtraction in an equation. Solve x - 12 + 20 = 37. x + 8 = 37. Now we may subtract 8 from each side of the equation, (we will actually add a -8 to each side). x + 0 = 29 x = 29 Example: This problem illustrates the proper use of the distributive property. Solve 2 � (x + 1 + 4) = 20. 2(x + 5) = 20 The equation now becomes 2x + 10 = 20. Subtracting a 10 (adding a -10) to each side gives us 2x = 10. Since the x is multiplied by 2, we divide both sides by 2 to solve for x: x = 5. We can check this solution in the original equation: 2 � (5 + 1 + 4) = 20 ==> 2 � 10 = 20 ==> 20 = 20 so our solution is correct. Combining like terms One of the most common ways to simplify an expression is to combine like terms. Numeric terms may be combined, and any terms with the same variable part may be combined. Example: Simplify the expression 4+ 2x + 8 - 3x � 5 = (4 + 8 � 5) + (2x � 3x) = 7 + (-x) = 7 - x Examples: For the equation 3x + 4 = 12, we can isolate the variable term on the left by subtracting a 4 from both sides: 3x + 4 - 4 = 12 - 4 ==> 3x = 8 x = 8/3 Simplfying by multiplication When a variable is divided by some number, we can multiply both sides to solve for the variable. Example: Solve for x in the equation x /12 = 5. Multiplying both sides by 12 will cancel the 12 in the denominator: x/12 � 12 = 5 � 12 ==> x = 60. Simplifying by division When solving for a variable, we want to get a solution like x = 3 or z = -6. When a variable is multiplied by some number, we can use division on both sides to solve for the variable. Example: Solve for x in the equation 7x = 133. Since the x on the left side is being multiplied by 7, we can divide both sides by 7 to solve for x: 7x � 7 = 133 � 7 (7x)/7 = 133 � 7 x = 19. Note that dividing by 7 is the same as multiplying both sides by 1/7. Word problems as equations When converting word problems to equations, certain "key" words tell you what kind of operations to use: addition, multiplication, subtraction, and division. The table below shows some common phrases and the operation to use. Word Operation Example As an equation sum addition The sum of my age and 20 equals 52. y + 20 = 52 difference subtraction The difference between my age and my younger brother's age, who is 9 years old, is 3 years. y - 9 = 3 product multiplication The product of my age and 14 is 280. 14y = 280 times multiplication Three times my age is 20. 3y = 20 less than subtraction Seven less than my age equals 21. y - 7 = 21 total addition The total of my pocket change and 10 dollars is $10.67. y + 10 = 10.67 more than addition Eleven more than my age equals 43. 11 + y = 43 Sequences A sequence is a list of items. We can specify any item in the list by its place in the list: first, second, third, fourth, and so on. Many sequences have patterns so we know what items occur in each place in the list. There are 2 kinds of sequences: A finite sequence is a list made up of a finite number of items. An infinite sequence is a list that continues without end. Examples: The following are examples of finite sequences. The sequence 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is the sequence of the first 10 odd numbers. The sequence 0, 1, 4, 9, 16, 25, 36, 49 is the sequence of the squares of the first 8 whole numbers. Examples: The following are examples of infinite sequences: The sequence 2, 4, 6, 8, 10, 12, 14, 16, ... is the sequence of even whole numbers. 1, 1, 1, 1, 1, 1, ... is the sequence where every item in the list is the number 1. 1, 2, 3, 4, 5, 6, 7, ... is the sequence of counting numbers. Each item in the list is its place number in the list. 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ... is the sequence of reciprocals of the whole numbers.