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rational #'s

any # that can be written as a fraction, terminating (ending) decimal (0.25), or repeating decimal (0.212121....)

irrational #'s

#'s that cannot be written as fractions (square roots, cube roots)

calculation of percents

step 1. change the written statement into a mathematical equation. Keep in mind:

- the word of translates into multiply

- the word is translates into equals

- the decimal form of the percent should be used in the equation

step 2. solve for the unknown quantity

step 3. rewrite the statement & make sure that the answer is reasonable (example: 15% of 500 is what #?= 0.15 x 500= 75 )

- the word of translates into multiply

- the word is translates into equals

- the decimal form of the percent should be used in the equation

step 2. solve for the unknown quantity

step 3. rewrite the statement & make sure that the answer is reasonable (example: 15% of 500 is what #?= 0.15 x 500= 75 )

solving for percent decrease

percent decrease = original value minus new value

divided by original value

then multiply by 100

divided by original value

then multiply by 100

solving for percent increase

percent increase = new value minus original value

divided by original value

then multiply by 100

divided by original value

then multiply by 100

converting from fractions to decimals

divide the numerator by the denominator

converting from decimals to fractions

the decimal # expressed becomes the numerator of the fraction & the # of decimal places to the right of the decimal determines the value of the denominator

converting from fractions to percents

1st, convert the fraction to a decimal - convert the decimal to a percent by multiplying by 100 & adding the % symbol

5/8= 0.625 = 62.5%

5/8= 0.625 = 62.5%

converting from percents to fractions

step 1. remove the % sign

step 2. write the number from step 1 in the numerator of the fraction & write 100 in the denominator

step 3. simplify the fraction

step 2. write the number from step 1 in the numerator of the fraction & write 100 in the denominator

step 3. simplify the fraction

converting from percents to decimals

remove the % symbol & move the decimal point left 2 places

(example: 0.045= 4.5 %)

(example: 0.045= 4.5 %)

determining which fraction is greatest

find a common denominator for the fractions- the fraction with the greater numerator is the greater fraction

estimation

the approximate value= the 1st digit in the # will not be zero but all the other digits will be zeros

to balance a checking or savings

1st group the deposits & add them together -then group the checks & add them together- next add the deposits to the previous balance & subtract the checks from the result- then subtract the service charge & add interest

proportion

this states that 2 ratios are equal- when setting up this, the numerators must be in the same units & the denominators of both ratios must be in the same units- use this formula to solve for this:

units of an item = units of an item

divided by units of a different # Divided by units of a different #

units of an item = units of an item

divided by units of a different # Divided by units of a different #

ratio

this is used to express a relationship between 2 quantities

rate of change problems

use proportions to determine the difference in completion times for a given task

example: 10 pages / 1 hour = 288 pages / A hours

use the method of cross products to solve:

10 x A= 288 x 1 - 10A / 10 = 288 / 10 A= 28.8 hrs

example: 10 pages / 1 hour = 288 pages / A hours

use the method of cross products to solve:

10 x A= 288 x 1 - 10A / 10 = 288 / 10 A= 28.8 hrs

roman numerals

M, D, C, L, X, V & I

1000

roman numeral M=

500

roman numeral D=

100

roman numeral C=

50

roman numeral L=

10

roman numeral X=

5

roman numeral V=

1

roman numeral I=

subtraction from the larger values

the use of I, X, & C to the left of larger values indicates what

2.54 cm

1 inch is how many cm?

2.2 pounds

1 kilogram is how many pounds?

3 feet

1 yard is how many feet?

4 quarts

1 gallon is how many quarts?

calipers

this is used to measure very small lengths, usually 6 inches or fewer with greater precision than a ruler

rulers

this is used for measurements no longer than 12 inches

yard sticks

this is used for measurements no longer than 1 yard, 3 feet, or 36 inches

scales

these are used to measure weight

beakers

these are used for measurements in teaspoons

graduated cylinders

these are used for measurements in tablespoons

measuring cups

these are used for measurements in pints

pipettes

instrument used for measuring small volumes of fluid

independent variable

the variable that is put into the set of data, or the input, (statistics) a variable whose values are independent of changes in the values of other variables

dependent variable

is the output based on the input, (statistics) a variable in a logical or mathematical expression whose value depends in the independent variable

line graphs

these graphs show changes over a period of time or compares the relationship between 2 quantities (compare time of day to temperature)

pie (circle) graphs

a circular graph divided into sectors representing the frequency of an event- percentage of a whole, where the whole circle or pie equals 100%- shows how much of the whole each part represents

bar graphs

this graph is used to compare the frequencies of an event

(the number of inches of rain that fell in a certain city in the each month)

(the number of inches of rain that fell in a certain city in the each month)

tables

these are helpful for organizing raw data

constant

a quantity that does not change

adding integers

positive + positive = positive

negative + negative = negative

when signs are different = subtract the smaller # from larger & give the sign of the larger #

negative + negative = negative

when signs are different = subtract the smaller # from larger & give the sign of the larger #

subtracting integers

- subtracting a positive is the same as adding a negative

- subtracting a negative is the same as adding a positive

- subtracting a negative is the same as adding a positive

multiplying integers

positive x positive = positive

negative x negative = positive

positive x negative = negative

negative x negative = positive

positive x negative = negative

percentage discount & tax increase

to find the amount of discount or increase when the % is known

1. change the percentage to a decimal (or fraction)

2. multiply by the original cost

3. add or subtract accordingly

1. change the percentage to a decimal (or fraction)

2. multiply by the original cost

3. add or subtract accordingly

percentage increase & decrease

often the problem asks you to determine the percentage of increase or decrease. This type of problem is easily solved by making a fraction out of the information provided

1. write the amount of increase or decrease as the numerator

2. write the original amount as the denominator

3. change fraction to percent

1. write the amount of increase or decrease as the numerator

2. write the original amount as the denominator

3. change fraction to percent

ratio

a comparison of 2 #'s, usually by division

(ex: in a class of 15 people, there are 7 boys & 8 girls. The ratio pf boys to girls is 7 to 8 or 7:8 or 7/8)

(ex: in a class of 15 people, there are 7 boys & 8 girls. The ratio pf boys to girls is 7 to 8 or 7:8 or 7/8)

rate

a ratio made up of 2 different units of measurement or amounts

(ex: I can drive my car 250 mi on 10 gal of gas. This relationship can be expressed as a ratio of miles to gallons: 250 to 10, 250:10, or 250 mi/ 10 gal = 25/ 1 or 25 miles per gal)

(ex: I can drive my car 250 mi on 10 gal of gas. This relationship can be expressed as a ratio of miles to gallons: 250 to 10, 250:10, or 250 mi/ 10 gal = 25/ 1 or 25 miles per gal)

proportion

an equation of 2 equal ratios. All proportions equations have have a special property: cross products are equal. When you multiply the numerator on the left side of the equation by the denominator on the other side & then multiply the left-side denominator by the right-side numerator, the are products equal to each other

(ex: if a car gets 25 miles to the gallon, then how many gallons do I need to drive 125 miles?

set up a proportion to solve this problem. Make sure you align your units correctly on both sides of the equation: in this case, miles across from miles & gallons across from gallons

25 mi / 1 gal = 125 mi / x gal

now cross multiply: 25x = 125

now divide both sides by 25 to solve for x. x = 5

this is your answer- you need 5 gal to go 125 mi

(ex: if a car gets 25 miles to the gallon, then how many gallons do I need to drive 125 miles?

set up a proportion to solve this problem. Make sure you align your units correctly on both sides of the equation: in this case, miles across from miles & gallons across from gallons

25 mi / 1 gal = 125 mi / x gal

now cross multiply: 25x = 125

now divide both sides by 25 to solve for x. x = 5

this is your answer- you need 5 gal to go 125 mi

evaluate

means to find the value of something

addition

"increased by", "more than" & "total"

subtraction

"decreased by", "less than," or "less"

"is"

means "="

the formula for area

A = 1/2 bh A = area, b = base, h = height

exponent rules

1. when multiplying similar bases, add the exponents

2. when dividing similar bases, subtract the exponents

3. when raising a power to another power, multiply the exponents

(ex:(s^2)^3 = s^ 6

4. when the exponent is negative, move its base to the denominator & make the exponent positive

5. any base (except zero) to the 0 power equals 1

2. when dividing similar bases, subtract the exponents

3. when raising a power to another power, multiply the exponents

(ex:(s^2)^3 = s^ 6

4. when the exponent is negative, move its base to the denominator & make the exponent positive

5. any base (except zero) to the 0 power equals 1

area of a rectangle

A= lw A = area, l = length, w = width

area of a square

A = s^2 A = area, s = side

area of a triangle

A = 1/2 bh A = area, b = base, h = height

perimeter of a rectangle

P = 2l + 2w

perimeter of square

P = 4s

perimeter of a triangle

P = s + s + s