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multiple

The product of a specified number and some integer. For example, 3, 12 and 90 are all multiples of 3. 4 is not a multiple of 3 because there is no integer that can be multiplied by 3 and yield 4

1


An integer is divisible by 2 if...

...if its last digit is divisible by 2

2


An integer is divisible by 3 if...

...if its digits sum to a multiple of 3 6,930 is a multiple of 3 because 6+9+3+0=18 which is a multiple of 3

3


An integer is divisible by 4 if...

...if its last two digits are a multiple of 4 4,716/4 = 1179

4


An integer is divisible by 5 if...

...if its last digit is 0 or 5

5


An integer is divisible by 6 if...

...if it divisible by 2 and 3

6


An integer is divisible by 9 if...

...if its digits sum to a multiple of 9 6,930...6+9+3+0=18, which is a multiple of 9

7


factors (aka divisors)

The factors of an integer are the positive integers by which it is evenly divisible. 36 has 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36

8


5% as a decimal and fraction

0.05 and 1/20

9


12.5% as a decimal and fraction

0.125 and 1/8

10


20% as a decimal and fraction

0.2 and 1/5

11


33 1/3% as a decimal and fraction

.3333 and 1/3

12


10% as a decimal and fraction

.1 and 1/10

13


16 2/3 % as a decimal and fraction

0.16666 and 1/6

14


percent formula

PART/WHOLE x 100 = PERCENT

15


percent increase (or decrease)

amount of increase/original whole X 100

16


to multiple powers with the same base...

add the exponents and keep the base 7^3 x 7^5 = 7^8

17


to divide powers with the same base...

subtract the exponents and keep the base the same 4^5/4^2 = 4^3

18


to multiple powers (or raise a power to a power)

multiply the exponents 7^2(^3) = 7^6

19


a negative number raised to an even power...

...yields a positive result (1)^2 = 1

20


a negative number raised to an odd power...

...yields a negative result (1)^57 = 1

21


raising a fraction between zero and 1 to a power...

...yields a smaller result (1/2)^2 = 1/4

22


what happens when an exponent is negative?

Take the reciprocal of the base and change the sign of the exponent (2)^2 = 1^2/2^2 = 1/4

23


Express 9^1/2 as a radical

= the square root of 9 = 3

24


Express 8^1/3 as a radical

= the cube root of 8 = 2

25


to simplify a radical...

factor out the perfect squares and move them to the front of the radical sign. For example, the square root of 50 = 5 square root 2

26


when can radicals be added and subtracted?

only when the number under the radical is the same! 6radical7 + 2radical7 = 8radical7

27


to multiply radicals...

...multiply the numbers under the signs and then put a single radical sign over them the new number

28


to divide radicals...

...divide the two numbers in question and then put them under a single radical

29


if multiplying or dividing an inequality by a negative number...

REVERSE the inequality sign 3x < 6 = x > 2

30


supplementary angles

two angles are supplementary if their measures sum to 180

31


complementary angles

two angles are complementary if their measures sum to 90

32


adjacent angles

angles that are adjacent (next to each other) are supplementary because they lie along a straight line

33


vertical angles

two angles that are not adjacent to each other are opposite, or vertical, and are equal in measure

34


perimeter of a triangle

the sum of the lengths of all three sides

35


area of a triangle

area of a triangle = 1/2(base)(height)

36


isosceles triangles

an isosceles triangle has two equal sides and the angles opposite these sides are equal as well

37


equilateral triangle

all three sides of an equilateral triangle are equal and the interior angles equal 60

38


right triangles

triangles with one interior angle of 90. The hypotenuse lies opposite the right angle. The other two sides are legs. leg^2 + leg^2 = hypotenuse^2

39


pythagorean triplets (2)

3:4:5 (leg:3 leg:4 hypotenuse:5) and 5:12:13

40


isosceles right triangles

angles = 45, 45, and 90 the ratio of sides is always 1:1:root2

41


306090 right triangles

the ratio of sides is always 1:root3:2 paired as follows...the side opposite the 30 degree angle is 1, etc.

42


define: quadrilateral

a four sided polygon where the four interior angles add up to 36, regardless of the quadrilateral's shape

43


define: parallelogram

a parallelogram has two pairs of equal sides. Opposite angles are equal. Consecutive angles add up to 180

44


define: rectangle

a quadrilateral with four right angles. Opposite sides are equal.

45


perimeter of a rectangle

perimeter = 2(length + width)

46


area of a rectangle

area = length x width

47


area of a square

area = (side)(side)

48


area of a parallelogram

area = base x height, but the height is NOT the length of the side. You must draw a line from the one base to the other to form a right angle  that line = the height.

49


volume of a rectangular solid

volume rectangle = length x width x height

50


volume of a cube

volume of a cube = (edge)^3

51


define: diameter of a circle

diameter: a line segment that connects two points on the circumference of a circle and passes through the center

52


define: radius of a circle

radius: 1/2 of a circle's diameter, it's a line segment that connects the center with a point on the circle

53


define: central angle of a circle

central angle: an angle formed by two radii

54


define: circumference

circumference: the distance around a circle = (2)(pi)(r)

55


define: arc length

arcs are the portion of a circle cut off by a particular central angle. The degree measure of an arc is equal to the central angle that cuts it off arc length = n/360 x (2)(pi)(r)

56


area of a circle

area of a circle = (pi)(r^2)

57


area of a sector

area of a sector: a slice of pie, n/360(pi)(r^2)

58


define: cylinder and give the formula for volume

cylinder: a solid whose horizontal cross section is a circle. The volume of a cylinder = (pi)(r^2)(h)

59


distance formula, used to define the distance between two points:

distance formula = square root of (x1x2)^2+(y1y2)^2

60


define: slope

change in y/change in x

61


what is the equation for a straight line?

y = mx + b where, m = slope and b = yintercept

62


which is bigger: 8/3 or the square root of 7?

to figure this out, square both sides. They compare 64/9 to 7 (or 63/9)...64/9 (or 8/3) is a little bigger

63


fractions get smaller as...

...their denominator gets bigger

64


when a positive fraction is less than 1 squared the result is...

...less than the original fraction

65


the smaller the negative number...

...the larger its square

66


which are even:
a. odd +/ odd b. even +/ even c. odd +/ even d. odd x odd e. even x even f. even x odd 
a, b, e, and f are even c and d are odd

67


how do you find the least common multiple?

to find the least common multiple, check out the multiples of the larger number until you find one that's also a multiple of the smaller. LCM of 12 and 15 is 60

68


how do you find the greatest common factor

the find the greatest common factor, break down both numbers into their prime factorization and take all the prime factors they have in common. the GCM of 36 and 48 is 12

69


how do you find the prime factorization of an integer?

to find the prime factorization of an integer, just keep breaking it up into factors until all factors are prime. 72  3 x 3 x 2 x 2 x 2

70


cross multiply: x/y = a/b

bx = ya

71


how do you add or subtract fractions?

to add or subtract fractions, first find a common denominator, then add or subtract the numerators

72


how do you multiply fractions?

to multiply fractions, multiply the numerators and multiply the denominators

73


how do you divide fractions?

to divide fractions, invert the second one and multiply

74


how do you compare fractions?

one way to compare fractions is to reexpress them with a common denominator. Another way to compare fractions is to convert them both to decimals

75


how do you compare decimal fractions?

the simplest way to compare decimal fractions is to add zeros after the last digit to the right of the decimal point in each decimal fraction until all the decimal fractions you're comparing have the same number of digits

76


how do you convert decimals into fractions?

to convert a decimal to a fraction, set the decimal over 1 and multiply the numerator and denominator by 10 raised to the number of digits to the right of the decimal point. Then simplify the fraction.

77


how do you express a % as a fraction in lowest terms?

a percent is a fraction with an implied denominator of 100. set % over 100 and simplify

78


express 1/10, 1/5, 1/4, 1/3, 2/3, and 1/8 as percentages

1/10 = .10 = 10% 1/5 = .2 = 20% 1/4 = .25 = 25% 1/3 = .333 = 33 1/3% 2/3 = .666 = 66 2/3% 1/8 = .125 = 12 1/2%

79


how do you find an average?

to find the average of a set of numbers, add them up and divide by the number of numbers

80


how do you use the average to find the sum? example: the average of 10 numbers is 50. What is the sum of the 10 numbers?

sum = average x number of terms if the average of 10 numbers is 50, then they add up to 10 x 50, or 500

81


how do you use the average to find the missing number? example: the average of 4 numbers is 7. If 3 of the numbers are 3, 5, and 8, what is the fourth number?

to find a missing number when you're given the average, use the sum. if the average of 4 numbers is 7, then the sum of those 4 numbers is 4 x 7 or 28. Three of the numbers (3, 5, & 8) add up to 16 of that 28, which leaves 12 for the fourth number

82


how do you determine the average of consecutive numbers? example: what is the average of all the integers from 13 through 77 inclusive

to find the average of evenly spaced numbers, just average the smallest and largest. the average of all the intefers from 13 through 77 is 13 + 77 / 2 = 90/2 = 45

83


how do you find the median of a group of numbers?

the median is the middle value in a group of numbers. If there's an even number of values, the median is halfway between the two middle values. median of 12, 15, 23, 40, and 40 is 23 b/c its the middle value. median of 12, 15, 23, and 40 is 19 b/c its halfway b/w the two middle values of 15 and 23

84


how do you determine the mode of a group of numbers?

the mode is the number that appears the most often. If two numbers appear equally often, they are both the modes. the mode of 12, 15, 23, 40, and 40 is 40 b/c it appears more often.

85


how do you use a ratio to find a number? example: in a group of 18 people, the ratio of men to women is 1:2. How many women are there?

if the parts add up to the whole, a parttopart ratio can be turned into 2 parttowhole ratios by putting each number in the original ratio over the sum of the numbers. if the ratio of men to women is 1 to 2, then the mentopeople ratio is 1/ 1+ 2 = 1/3. And the womentopeeps ratio is 2/1+2 = 2/3. 2/3 of the 18 are women, or 12.

86


what is the distance, rate, time equation in terms of distance?

distance = rate x time.

87


what is the average rate equation?

Average A per B = total A / total B Average speed = total distance/total time

88
