Math Review Sheet #1

 

Signed numbers

 

Addition/subtraction:

same sign ่ add                     

Example:          -3 + -4 = -7

different sign ่ subtract          

Example:          -3 +  4 =  1

sign of result = sign of number with largest absolute value 

Example:          -4 + 3 = -1

The absolute value of a number n is either n or –n, whichever is non-negative. The notation is .

Example: If >3 then either n>3 or n<-3.

 

Multiplication / Divison :

Same sign ่ result positive                 

Example:          (-3) (-4) = 12 ; -3/-4 = .75

Different sign ่ result negative

Example:          (-3) 4    = -12 ; -3/4 = -.75

 

Fractions

 

Multiplication:

To multiply two fractions, multiply the numerators with each other, and multiply the denominators with each other.

            Example:

 

Division:  To divide by a fraction, multiply by its reciprocal.

Example:  (whether x is a fraction or not)

To add/subtract two fraction, you must first bring them to a common denominator, then add/subtract the numerators.

Example:

To find a common denominator, you need to find a common multiple. The easiest way is to multiply the two denominators (5 and 7 in the above example). Sometimes, multiplication gives you a common denominator that is larger than necessary. Example: Since 12 is a common multiple of 4 and 6, you can subtract as follows:

 

Median and mean

 

The median of a set of numbers is the number in the middle after the numbers have been sorted (either ascending or decending). If two numbers are in the middle, then the median is their average.

Example: The median of {x,x+2,x-3,x+4} is x+1, regardless of the value of x, because the sorted order is {x-3,x,x+2,x+4} and the average of x and x+2 is x+1.

The mean (or average) of a set of numbers is their sum divided by their count.

Example: The mean of {x,x+2,x-3,x+4} is x+0.75 because their sum is 4x+3, and then we have to divide by 4.

 

Variables and constants

 

A variable is a quantity that can have different values, depending on the situation.  A constant is a quantity with a fixed value.

Examples of variables: today’s temperature, my age, the US population. Examples of constants: number of pounds per kilo (2.2), number of US senators (100)

Point of view: Many quantities can be constant or variable, depending on point of view.  Example: Historically, the number of senators  was not always 100 (there were 96 senators when the US had 48 states).

 

Expressions

An expression is a sequence of arithmetic operations involving quantities that are constants (numbers) or variables. An expression may have pairs of parentheses.

Example: -5(3-x) + 2x – 1

To evaluate an expression means to replace a variable by a given value. Often, you need to put parentheses around the replacement so as to respect order of operation.

Example: To evaluate -5x for x = -1 you calculate -5(-1) =5.

To create a table of values for an expression, evaluate it for many different values of the variable. Example:

 x

3(x-2)+5

-2

3(-2-2)+5 = -12+5 = -7

-1

3(-1-2)+5 = -9+5   = -4

 0

3(0-2)+5  = -6+5   = -1

 1

3(1-2)+5  = -3+5   = 2

 2

3(2-2)+5  =  0+5   = 5

 

To calculate a table of values by Graphing Calculator:

Press the button Y=

Under \Y1= enter the expression

Press (yellow) TABLE

 

Legends

Example: see translating word problems

 

Translating word problems to equations

  1. read carefully
  2. underline key information
  3. decide on a variable and write a legend (usually: what is asked)
  4. think: What is equal to what? What do I know?
  5. translate what I know into an equation

 

Example: “If a video store charges a $10 membership plus $ 4 per video, How many videos did someone rent who paid a total of $90 ?”

 

Key info: $10 fee, $4 per video, $90 total.

Legend: Let x be the number of videos rented.

Total cost is equal to membership plus rental fees. Rental is $4 times x .

$90      =      $10   +  $4 x

To solve: first subtract 10 from both sides, then divide both sides by 4. (If you divide first, it works, but you get fractions).

 

Translating Percent problems.

Example: 80 is 3% of what?. Translate piece-by-piece:

80 remains 80, “is” translates to =, 3% translates to 0.03 (move the decimal point twice), “of” translates to multiplication, “what” translates to x. Thus the resulting equation is

80 = 0.03 x

The solution is found by dividing both sides by 0.03, thus

 

Solving Linear equations

Use the tools from  the “toolbox” until you get the variable alone on one side of the equal sign.

Toolbox:

Add the same quantity to both sides

Subtract the same quantity from both sides

Multiply both sides by the same quantity

Divide both sides by the same quantity

Distribute (multiply out) parenthesis

Combine Like terms

Move all terms with variables to one side

Cross Multiply

See example in section on proportions.

 

 

Proportions

A ratio is a relationship of two numbers by division.

Examples: three to five  is a ratio,   -7 / 12  is another ratio. 

A proportion is the statement that two (or more) ratios have the same value.

 

Examples of proportions:

If you have a fixed hourly wage, then the ratio of money earned to hours worked is always the same, no matter how many or few hours you work.

For all objects, the ratio of length in inches to length in centimeters is always the same:

     

Two ratios form a proportion if and only if the two cross products have the same value.

Example:

and vice versa.

Thus, from a proportion involving variables you obtain an equation which can then be solved for x.

 

Example: solve and check the equation

 

To eliminate the denominators, cross multiply:

  7(2x + 1)        =          4(4x + 3)        

Solve this equation using the tools from the toolbox:

Distribute:                     14x  + 7          =          16x  +  12

Move terms with x:       -14x                             -14x         

Result:                          7          =            2x    +  12

Subtract:                                   -12                               - 12

Result:                          -5         =            2x                 

Divide:                                      - 2.5     =              x                 

Check:                                     ึ

 

Percent equations are proportions.

Example: 3% of what number is 80? 

3% is the ratio . Let  be the number.

Then 3 out of 100 must be the same proportion as 80 out of n:

. Cross product:  so .

 

Scale drawings

The scale of a drawing (or a plan) is the ratio

Length on plan  :  length in nature 

The ratio is the same for all objects on the plan.

The three ways to indicate a scale are

  1. as a sentence. Example: “One inch on the plan represents 5 feet in nature.”
  2. As a numeric ratio. Example:  1 :  60
  3. Graphically

 

 

 

 

 

0’

5’

10’

15’

(draw an axis with marks at fixed distances, eg. 1 inch apart, and label them with the distances they represent in nature)

 

To determine the scale of a plan:

find the ratio of plan to nature ( using same units !!!)

 

Example: 1” (one inch) on plan corresponds to 5’ (five feet) in nature.

Convert 5’ to inches: 5’ = 60”. Therefore the scale is 1 : 60

 

To find the real length of an object shown on plan:

Multiply the plan length by the scale factor, then convert to appropriate units.

Example. A drawn object of 2 พ ” on a scale of 1 : 80 has a real length of (2+3/4)*80 inches = (2.75)*80 = 220” .

Now convert to feet and inches: 220๗12 = 18.33333 = 18’4”.

 

To determine how big a element of the drawing should be:

Divide the real length by the scale factor. Must be consistent with units !!!

Example: How long should you draw a 50’ (50 foot) house on a plan of scale 1 : 80 ?

50’ = 50*12” = 600”.  600๗80 = 7.5” or 7 ฝ ”