Math Review

"adding and subtracting decimals

line up the decimals on their decimal points to ensure place of value
Adding fractions

all fraction in must have the same denominator. Given common denominators, simply add or subtract the numerators. To convert to same 1st determine the lowest common denominator which is the smallest # that divides evenly into all denominators in the problem.
The average of a group of numbers is the sum of the numbers, divided by the number of numbers added
Ex. The average of 5, 9, and 10 is calculated as follows
5+9+10= 24 then 24/3= 8
Basic percentage problems have three components
1. The percent
2. The base
3. The percentage
ex. 25% of 200 is 50
25%= the percent
200= the base, and
50= the percentage
Common cubes
2`3=8 8`3=512
3`3=27 9`3=729
4`3=64 10`3=1,000
5`3=125 10`-2=.01
6`3=216
7`3=343
Common squares

1`2=1 6`2=36 10`-1=.1
2`2=4 7`2=49
3`2=9 8`2=64
4`2=16 9`2=81
5`2=25 10`2=100
complex fraction

a fraction, where the numerator and/or denominator are fractions
Convert decimal to a percentage

multiply by 100
Ex. 0.426 x 100 = 42.6 %
Convert percentage to decimal
divide by 100
Ex. 45% / 100 = 0.45
Ex. 12.5 / 100=0.125
Counting system is based on place of value unit of ten

ones, tens, hundreds, thousands,
ten thousand, hundred thousands,
million
Decimals

use place values to the right of the decimal point.
decimals by the power of 10

Move the decimal point to the left in the dividend by the same number of places as there are zeros in the power of 10. Ex. 45.67 / 100 move the decimal point two places to the left
45.67 /100= 0.4567
decimals converted to fractions

with denominators of 10,100,1,000 so on
Decimal to fraction

To convert a decimal into a fraction, you must know the place value of the decimal. The place value tells you the denominator of the fraction.
Dividing Decimals

If the divisor is not a whole number, move decimal point to right to make it a whole number and move decimal point in dividend the same number of places.
Divide as usual. ...
Put decimal point directly above decimal point in the dividend
Dividing Fractions

Invert the second fraction, and then multiply the two fractions.
easiest way to subtract integers is to turn subtraction into addition

subtracting a positive is the same as adding a negative
ex. +15-+5=+15+-5=10
Subtracting a negative is the same as adding a positive
ex. +15--5=+15++5=20
equivalent fractions

equal fractions
An exponent ( or power) tells you how many times to multiply the base by itself.

when a exponent is 2 is called a square
if is 3 is called a cube
Find the base when you know the percent and the percentage, set up an equation and divide both sides by the percentage.
ex. five ( the percentage0 is 10% ( the percent) of what number ( the base)?
is following equation, the base is x.
0.1 x (x) =5
0.1x / 0.1 = 5/0.1 X= 50
Divide both sides by 0.1 to solve for x
Find the percentage when you know the base and the percent
ex. what percentage of 50 is 5? Make the percentage of 50(x) the percentage of 50
Then divide both sides by 50
(x%) x 50 = 5 x= 0.1
Then turn 0.1 into a percentage by multiplying by
100
0.1 x 100= 10%
Fractions

may not have a denominator of 0, because it is not possible to divide by 0.
fractions and decimals

to convert a fraction into a decimal, divide the denominator into the numerator
ex. 2/5 = 2 divided by 5 = 0.4
ex. 2_3/4 = 2+ (3 divided by 4) = 2.75
fraction to percentage

1. change the fraction to percentage
2. change the decimal to a percentage
Ex. 2/5 = 0.4 = 40%
Ex. 9/2 = 4.5 = 450%
greatest common factor

the largest number that divides evenly into two or more numbers
Ex:  the greatest common factor of 36 and 54 is 18
( their common factors are 1,2,3,6,9,18)
improper fraction

a fraction in which the numerator is greater than or equal to the denominator
improper to mixed numbers

Divide the numerator by the denominator.
The quotient (without the remainder) becomes the whole number part of the mixed number. ...
The new denominator is the same as the denominator of the improper fraction.
9/5= 1_4/5
integer is any positive whole number, negative whole number, or zero.

Integers may not contain fractions or decimals
integers

...-3,-2,-1,0,1,2,3.....
least common denominator

the smallest denominator that two or more fractions have in common
least common multiple

the smallest number that two or more numbers divide into evenly
Ex. The multiples of 5 are 10,15,20,25,30.....
The multiples of 15 are 30,45,60....
30 is the least common multiple
lowest terms

in a fraction, where the numerator and denominator are relatively prime
mixed numbers

the sum of a whole number and a fraction, or just another way of writing an improper fraction
mixed number to improper fraction

Multiply the whole number by the denominator. Add the answer from Step 1 to the numerator. Write answer from Step 2 over the denominator.
1_4/5 = 1 x 5 +4= 9/5
Multiply decimals by the powers of 10

Move the decimal point to the right the same number of places as there are zeros in the power of 10
Ex. 45.67 x 100 the decimal point moves two places to the right = 4,567
Multiplying decimals

multiply then count the number of decimal places in each number , and add them
Multiplying fractions

multiply the numerators, and multiply the denominators
Multiplying integers
and dividing integers

positive (x or /) positive=positive
negative (x or /) negative = positive
Negative (x or /) positive = negative
Multiplying mixed numbers

convert them to improper fractions and then multiply
remember you may need to reduce the final answer
The opposite of a prime number is a composite number, which

is a whole number ( other than 0) with more than two factors. Note
0 and 1 are neither prime nor composite numbers.
( Percentage Discount and Tax Increase )

Find the amount of discount or increase when the percentage is known

1. Change the percentage to a decimal ( or fraction)
2. Multiply by the original cost
3.add or subtract accordingly
ex. a $250 stereo is discounted 18%. find the new sale price.
convert 18% to a decimal by dividing by 100, and multiply the decimal ( 0.18) by $ 250.
18% / 100 = 0.18
0.18 x $250. = $ 45
Then subtract $ 45 (the 18% decrease) from $250 ( the original cost) to find the new cost. $250- $45 =$205
You can also subtract 18% from 100% to find that the discounted price is 82% of the original price and then calculate 82% of 250 ( 0.82 x $250)
( Percentage Increase and Decrease)
Make fractions out of the info provided
1. Write the amount of increase or decrease as the numerator
2. write the original amount as the denominator
3. change the fraction to a percentage
ex. My little brother grew from 60 to 66 inches in the last year. What percentage increase is this?
The amount of increase is 6 in. Make this the numerator, and make his original height ( 60in) the denominator. Then change the fraction to a percentage.
6/60 = 1/10 = 0.1 = 10% Increase by 10%
Percentages

A fraction with a denominator of 100. Ex. 25% is really 25/100 or 3% is 3/100
Ex:  Percentages greater than a whole 150/100
or less than 1 whole 10/100
percentage to a fraction
1. change the percentage to a fraction
2. Reduce the fraction
ex. 35% = 35/100 = 7/20
Prime numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, and 29
proper fraction

a fraction in which the numerator is less than the denominator
Proportion is an equation of two equal ration. All proportion equation have a special property

Cross products are equal
Cross- multiplication
Rate is a ration made up of two different units of measurement or amounts

ex. I can drive my car 2500 miles on 10 gallons of gas. This relationship can by expressed as a ration of miles to gallons
250 to 10, 250
10 or =250 miles/ 10 gallons = 25/1 = 25 miles / gal
Ratio is a comparison of two numbers, usually by division

ex. in a class of 15 people, there are 7 boys and 8 girls. The ratio of boys to girls is 7 to 8, or 7
8, or 7/8
rational numbers

numbers that can be expressed as fractions
real numbers

all numbers on the number line
reciprocal

the resulting fraction when you switch the numerator and denominator.
Note 0/denominator has no reciprocal
reducing fractions

To reduce a fraction divide its numerator and denominator by the largest number that divides evenly into both.
relatively prime
term used to describe two or more numbers whose greatest common factor is 1.
ex. 6 and 35 are relatively prime. Factors of 6 are 1,2,3,6
Factors of 35 are 1,5,7,35. The greatest common factor is 1
Subtracting Fractions

all fraction in must have the same denominator. Given common denominators, simply add or subtract the numerators. To convert to same 1st determine the lowest common denominator which is the smallest # that divides evenly into all denominators in the problem.
The symbol is used is called the radical and the number underneath the symbol is called the radicand

This is
to find the percentage of a given number, multiply the percentage by the number
ex. what is 10% of 50?
'of' = multiply
' is '= means (=)
x= 0.1 x 50=5
unit fraction

a fraction, where the numerator = 1
When the signs of integers are the same adding two or more works just as you might remember

positive+ positive= positive
negative+negative= negative
when working with negative exponents, express it as the reciprocal of the base to the exponent changed to a positive number.

Ex: 10 exponent -3 = 1/10 exponent +3 = 1/1,000 = .001"