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Pre Algebra Contents

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Pre Algebra Operations

1)  Parenthesis

2)  Powers

3)  Divisions / Multiplication > (left to right)

4)  Subtraction / Addition > (left to right)

Algebraic expression involving multiplication and division

   9xm= 9*m= 9(m)= 9m

   7xaxb= 7*a*b= 7(a)(b)= 7ab

   y/3= y/3

	ex.1) Evaluate m(7+w); m=9

                       9(7+3); w=3

                        9(10)

                          90



	ex.2) Evaluate 8+ab/40; a=8

                       8+8*10

                          40

                         8+80

                           40

                          8+2

                           10



	ex.3) Evaluate 3y+9 ; y=9

                        2y

                       27+9

                        18

                        36

                        18

                         2



Patterns and Variables

Translating Phrases to Algebraic Expressions

sum +

difference -

product *

quotient /

Translate to a numerical expression

     ex.1) the difference of 10 and 5 = 10-5

     ex.1) 4 less than 10 = 10-4

     ex.3) the product of 8 and 7 = 8*7

     ex.4) 45 divided by 3 = 45/3

     ex.5) a number increased by 7 = h+7

     ex.6) 5 less than a number = n-5

     ex.7) 35 increased by twice a number = 35+2*d

     ex.8) 1 less than a number divided by 10 = d/10-1

Properties

Commutative:

a+b = b+a

a*b = b*a

     ex.1) 5*7 = 7*5

     ex.2) m+7 = 7+m 



Associative:



     ex.1) (98+47)+53 = 98+(47+53)

                          98+100

                           198

     ex.2) 7*(5x) = (7*5)x

                      35x



Identity:



a+0 = a

a*1 = a



Distributive:



a*(b+c) = a*b+a*c

     ex.1) a*(3+5) = 9*3+9*5

                      27+45

                        72

     ex.2) 6(r+s) = 6r+6*5

                    6r+30

     ex.3) (m+7)4 = 4m+28

     ex.4) 86*48+86*52
    

Simplifying Algebraic Expressions

Simplify - replace with a simpler equivalent expression



     ex.1) Simplify  3(2n)

                    (3*2)n

                      6n

     ex.2) (x+7)+6

            x+(7+6)

             x+13



Term - parts of an expression separated by + or -



     4m+2n-3

      terms



Like terms - terms with the same variable 2n and 3n are like terms, 

4m and 2m are unlike terms



     ex.3) 2x+3x

           (2+3)x

             5x



     ex.4) 25g+15g

           (25+15)g

              40g

     ex.5) 2m+3m+m

             6m

     ex.6) 4e+3f+e

            5e+3f

Integers

Integers {.-3,-2,-1,0,1,2,3,.}

Numbers such as 4 and -4 are opposites.

The opposite of -4 is written -(-4) = 4.

Absolute Value - distance from zero



     ex.1) | 7 | = 7

     ex.2) |-4 | = 4

     ex.3) | 0 | = 0

     ex.4) Find m if | m | = 8.  m = -8,8



Order from least to greatest

     ex.5) -1,-6,4 = -6,-1,4

     ex.6) -2,|-3 |, -1,| 4 |

                3         4

     -2,-1,|-3 |,| 4 |



Adding and Subtraction Integers

Think about money (+ : $ you have)

                                (- : $ you spent)

     ex.1) 3+-2 = 1

     ex.2) -3+2 = -1

     ex.3) -2+-3 = -5

To subtract, we add the opposite of the second number.



     |------|   |------|   |------|

     | 9-7  | = | 5-1  | = | 2-3  | =

     | 9+-7 | = | 5+-1 | = | 2+-3 | =

     |------|   |------|   |------|



     NOTE: -5 = -5

     ex.4) 3-2 = 1

     ex.5) -3-2 = -5

     ex.6) 3- -2 = 5

     ex.7) -3- -2 = -1



Multiplying and Dividing Integers

If the signs are the same the answer is positive.  If the signs are 

different the answer is negative.



     ex.1) 4*-5 = -20

     ex.2) -4*5 = -20

     ex.3) -4*-5 = 20

     ex.4) (-5)3 = -125

     ex.5) 42/-7 = -6

     ex.6) -42/7 = -6

     ex.7) -42/-7 = 6

If n = + then n5 = +

If n = - then n5 = -

If n = + then n8 = +

If n = - then n8 = -

    

Factors and Multiples


  factors        product

    7*4      =      28

  factors of 8 = 1,2,4,8

  factors of 84 = 1,2,3,4,6,7,12,14,21,28,42,84

Divisibility - one number is divisible by another if their quotient 

is a whole number (no remainder)

     ex.1) Is 11 a factor of 143?

     ex.2) Is 9 a factor of 326?



Divisibility rules

2 - even

3 - add digits, test 3

4 - test last 2 digits

5 - end in 0 or 5



Multiples - counting by a number

     ex3.) List the multiples of 5

                    0,5,10,15.

     ex.4) List the first 3 nonzero multiples of 11\

                    11,22,33.



Adding and Subtracting Fractions


     ex.1) 2/5 + 1/3 = 6/15 + 5/15 = 11/15

     ex.2) 3/10 + 6/15 = 9/30 + 12/30 = 21/30 = 7/10

     ex.3) -2/5 - 3/6 = -12/30 + -15/30 = -27/30 = -9/10

     ex.4) 5/6 - -3/4 = 10/12 + 9/12  = 19/12



Reducing Fractional Expressions



     ex.1) 3mb/7mk = 3b/7/k

     ex.2) 4ef/12f = 1e/3

     ex.3) 15ac2/25a2c   15acc/25aac   15c/25a   3c/5a



Adding and Subtracting Mixed Numbers



     ex.1) The west wall measures 60 9/16" and 217 7/8". Find the 

total length?

                        





     ex.2) 3M stock rose from 50 _ to 51 5/8. What was the increase?







     ex.3) GM stock fell from 52 1/8 to 50 3/4






Multiplying Rational Numbers



Multiply / Divide rules

+ + = +

+ - = -

- + = -

- - = +



     ex.1) 2/5 * 5/12 = 10/60

     ex.2) -6/4 * -3/5 = 9/10

     ex.3) 1/2 of 3/4 = 3/8

     ex.4) 2 2/3 (1 1/5) = 16/5

     ex.5) 3 1/12 (-2) = 37/6

     ex.6) (5/8)2 = 5/8 * 5/8 = 25/64



Reciprocals and Dividing Rational Numbers

  - two numbers are reciprocals if their product is 1

     ex.1) 2/3 in the reciprocal of 3/2 because 3/2 * 2/3 = 1

     ex.2) what is the reciprocal of -2000m/3, _______

       how many halves are in 3?

       3/ 1/2 = 6

       2 1/2 / 1/4 = 10

     ex.3) 2/3 / 1/6 = -4

     ex.4) -5/24 / 5 = -32/15

     ex.5) 8/ (-3 _) = -1/24

     ex.6) -3 1/8 / -2 1/12 = 3/2 (1 «)



More About Exponents

Review: Exponents are used to show how many times the some factor is 

repeated.

                              exponent

                        43 = 4 x 4 x 4 = 64

               base



     ex.1) 82 = 8*8 = 64

     ex.2) 24 = 2*2*2*2 = 16

     ex.3) (-.03)2 = -.03 x -.03 = .0009

Rule: To multiply two powers with the same base, we add the 

exponents.

     ex.1) 32 * 34 = 36

     ex.2) 103 * 102 = 105

     ex.3) 24 * 2 = 25



Try the following: 45/42 = 43



This suggests a rule for simplifying expressions in this form.

Rule: To divide two powers with the same base, we subtract the 

exponents.



     ex.1) 25/23 = 22                       ex.4) x6/x2 = x4

     ex.2) (-3)7/ (-3)4                     ex.5) m5/m

     ex.3) 104/10                           ex.6) 34/34



     you can also use the rule given above to simplify the expression 

52/54



      52/54 = 1/52                                52/54 = 5-2



     This shows that 1/52 is the same as 5-2



     ex.1) 3-2 = 1/32 = 1/9      ex.3) (-2)-4 = 1/-2-4 = 1/16



     ex.2) 5-3 = 1/53 = 1/125    ex.4) 4-1 = 1/-4



Scientific Notation

Review     

103 = 1000                10-3 = 1/1000 = .001



105 = 100000              10-5 = 1/1000000 = .00001



106 = 1000000             10-6 = 1/10000000 = .000001



Scientific notation is used to simplify work with very small or very 

large numbers.



                                             3.45      X       103



     A decimal between 1 and 10                a power of 10



Write in standard form.



     ex.1) 5.8 x 103 = 5800       ex.2) 6.556 x 102 = 655.6

     ex.3) 1.8 x 10-4 = .00018    ex.4) 4 x 10-2 = .04



Write in scientific notation.



     ex.1) 4567 = 4.567 x 103       ex.2) 1,234,000 = 1.234 x 106

     ex.3) 234,000 = 2.34 x 105     ex.4) 50,000,000 = 5.0 x 107

     ex.5) 0.000345 = 3.45 x 10-4   ex.6) 0.0206 = 2.06 x 10-2

     ex.7) 0.000008 = 8.0 x 10-6    ex.8) 0.2004 = 2.004 x -10



Why are these not in scientific notation?



     ex.1)(10 is less than 12.5) 12.5 x 104     ex.2) 2 x 4-5 (the 

base must be 10)



Give two reasons for using scientific notation.

1.)  easier for large numbers

2.)  nicer for comparing



Solving Two Step Equations



                           SOLVE

review:     m-17 = 24       3m = 5





            m+5 = 11         m/4 = 9





Rule:  Solving equations with combined operations

A: Identify the order the operations were applied to the variable.

B: Undo the operations in reverse order.



Functions



Definition: A function is a special relationship between two 

variables.

Domain- number put into the function (input).

Range- answer (output)



Function notation: f(x)

     ex.) f(x) = 3 x +4

     then f(5) = 3(5)+4

               = 15+4

               = 19



More Simplifying to Solve Equations

Like terms- have the same variable and exponent

     ex 3m+2m = 5m

Unlike terms- cannot be simplified

     ex 3m+2k

        3m2+2m



ex.1) 9k+5 (k+7) = -49









ex.2) 2m+3(m-7) = 44









ex.3) 3(5n)+14+6n = 21









ex.4) -2(7c)-12+5c = 51


Solving Equations with Variables on Both Sides

1: Get the variable on one side

2: Solve as always

     ex.1) -4(2m-5) = -4m+10

              -8m+20 = -4m+10

              +8m         +8m

                 20 = 4m+10

                -10          -10

                 10 = 4m

                 4     4

                   5/2 = m



     ex.2) 2(m+2) = 2m+4

     ex.3) 2(m-2) = 2(m+1)

Ratio, Rate, Proportion


Ratio- a comparison of one number to another.

     ex.) 3/5 or 3.5 or 3 to 5

Rate- a ratio that involves two different units.

     ex.) 216miles/4hours = 54m/h

Proportion- An equation starting that two ratios are equal.



We can use cross product to solve proportions.

     ex.1) 2/3 = 24/n     ex.2) 9/12 = 3/n     ex.3) 9/10 = n/22


     ex.4) The lions in the zoo eat 40kg.

          Of food every 7 days. How many

          kg. do they eat in 30 days?


                    Kg./days     40/7 = k/30

                      1200 = 7k

                        7    7

                        171 = k

Percent, Decimals, and Fractions

     Percent means hundredths or "out of 100"

     Change to decimal.

1.)  43% = 43/100 = .43

2.)  90% = .9

3.)  2% = 

4.)  34.7% = .347

5.)  10 «% =



     Change to percent.

1.)  .27 = 27%

2.)  .06

3.)  .7

4.)  .065

5.)  .7/8 = 87.5%



Finding a Percent of a Number

   Estimating Percents



     ex.1) 63%$ of 61

              « of 60 = 30

     ex.2) 27% of 79

             ¬ of 80 = 20



Using Proportions

     ex.3) 90% of 25

           90/100 = n/25  2250 = 100n

                          100    100

         22.5 = n



Using Decimals

     ex.4) 6.5% of $80

          .065 x 80 = 5.20



     ex.5) 15% of $25

          .15 x 25 = 3.75



Using a calculator

     ex.6) 22% of 150

              33





Finding the Percent One

     Number is of Another

     "what percent" … n/100

     of _ …


     ex.1) 17 is what percent of 25?

               n   =  17

              100     25



              25n   =   1700

              25         25

                  n = 68%



     ex.2) What percent is 8 out of 13?



               n   =   8

              100     13






     ex.3) What percent of 72 is 12?



              n   =   12

             100      72











     ex.4) What % of 72 is 90?



               n   =   90

              100      72




Percent of Increase of Decrease


     change

     original = _ %



     ex.1) My weight on January 1st = 140

              My weight on February 1st = 144



     change

     original = 4/140 = .0285 = 3%



     ex.2) Jenny runs the mile in 9 minutes

              in June. In July she can run it in 8

              minutes.



     change

     original = 1/9 = .111 = 11%



    

Calculating Simple Interest



   When you borrow money from a bank, credit union, or loan company 

you pay for the use of it.  The amount you pay for the use of money 

is called interest.



   The amount of interest you pay depends upon the principle (amount 

borrowed), the rate (percent of interest) charged, and the length of 

time the money is kept (time).



		Interest = Principle * Rate * Time



			I = P * R * T

ex.1) Calculate the interest         ex.2) Principle = $80,000

on a $800 loan at 8% interest              Rate = 8% per year

per year if you paid it back in            Time = 30 years

2 years.









ex.3) You use your credit card       ex.4) Principle = $5000

to buy $500 worth of clothes.              Rate = 1.5% per month

You have to pay 15% interest               Time = 2 years

per year. How much interest

would you pay after 1 year?

What did the clothes really cost?









ex.5) You put $1000 in the bank      ex.6) Principle = $1000

and leave it in for 3 months at            Rate = 8% per year

2.5% interest per year. How much           Time = 3 months

interest do you make? How much

money do you have now?









ex.7) Principle = $1,000,000

         Rate = 8% per year

         Time = 3 months




The Basic Counting Principle


     To find the total number of choices for an event, multiply the 

number of choices for each individual part.



     ex.1) How many outcomes are possible if you first toss a coin, 

then roll a die?

                         2 * 6 = 12 outcomes







     ex.2) How many different pizza combinations can you make with 2 

crusts, 2 sauces, and 4 toppings? (only a one topping pizza)

                         2 * 2 * 4 = 16







     ex.3) How many license plates can be made in Minnesota?

               10 * 10 * 10 * 26 * 26 * 26 = 17576000




Permutations and Combinations 14-2



     An arrangement of a group of objects in a certain order is 

called a permutation.



     ex.1) Andy, Bob and Chris are to be seated at 3 desks arranged 

in a row. How many ways can the 3 students arrange themselves?

                    3 * 2 * 1 = 6



     ex.2) Steve, Rachel, Teddy and Kristin are on our 4x100 relay. 

How many running orders are there for this relay?

                    4 * 3 * 2 * 1 = 24



     ex.3) 20 people are running for student council. How many 

different permutations are there?

                    20 * 19 = 380 (president & vice president)



A selection of objects without regard to order is called a 

combination.



     ex.4) choose 2 of these 4: Art, Band, computer, Drafting.





     ex.5) Almond Joy, Butterfinger, Crunch, Dollar Bar; choose 3 of 4.




Probability


Probability = ways for an outcome to occur

                      total outcome

Flipping a coin:

     ex.1) P(getting tails) = «



     ex.2) P(getting heads) = «



Rolling a Die:



     ex.3) P(rolling a 5) = 1/6

     ex.4) P(not rolling a 5) =

     ex.5) P(rolling a 1 or 5) = 2/6 = 1/3

     ex.6) P(rolling a 7) =

     ex.7) P(rolling a number <7) = 6/6 = 1



Note: If any outcome is certain to happen then

     the probability of that outcome is 1.

      If any outcome is impossible then the

     probability of that outcome is 0.



     ex.8) Find P(winning the daily 3)

              1/1000

        1/10 * 1/10 * 1/10 = 1/1000


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