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137 Cards in this Set

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A is 3 more (greater)than B
A=B+3
A is less (fewer) than B
A=B-3
A is 2/3 more(greater) than B
A=B+2/3B = B(1+2/3)
A is 2/3 less(fewer) than B
A=B-2/3B = B(1-2/3)
A is percent more (greater) than B
A=B+50/100B =B(1+50/100)
A is percent less (fewer) than B
A=B(1-50/100)
B 가 A 보다 300% 만큼클경우, B가A의 4배이다
B=A(1+300/100)=4A
A is 3 times as many (much) as B = 3 times as many (much) A as B
A=3B
A is no more than B
A is less than or equal to B
A is at most B
A>=B
A is no less than B
A is more than or equal to B
A is at least B
A<=B
정수 X의개수

a=<x<=b
b-a+1
정수 X의개수

a<x<=b
b-a
정수 X의개수

a=<x<b
b-a
정수 X의개수

a<x<b
b-a-1
even + even

even + odd

odd + odd
even

odd

even
even x even

even x odd

odd x odd
even

even

odd
연속인 세 정수
n, n+1, n+3
연속인 세 짝수
2n, 2n+2, 2n+4
연속인 세 홀수
2n+1, 2n+3, 2n+5
prime number
1과 자기자신만을 약수로 갖는수

2는 유일한 짝수
N=A^a * B^b * C^c

약수의개수 (N을 나눌수 있는 수)
약수의개수는
=(a+1)(b+1)(c+1)
4의multiple
끝의두자리숫자가 4위배수
25의배수
끝의두자리숫자가 25의배수
3의배수
각자리 숫자의합이 3의배수
9의배수
각자리 숫자의합이 9의배수
2의배수
일의자리수가 2의배수
5의배수
일의자리수가 0또는 5인수
6의배수
2의배수& 3의배수
12의배수
4의배수 & 3의배수
15의배수
5의배수 & 3의배수
몫과 나머지

N을 8로 나누었을때, 몫이 q이고 나머지가 r 이다
N= 8q +r

N/4 =2q +r-4
주기가 2인수
4, 9
주기가 4인수
2,3,7,8
주기가 1인수
1,5,6
0.333 in fraction
1/3
1/3 in decimal
0.333
0.5 in fraction
1/2
1/2 in decimal
0.5
0.75 in fraction
3/4
3/4 in decimal
0.75
0.167 in fraction
1/6
1/6 in decimal
0.167
sqr root of 2 is
1.4
sqr root of 3 is
1.7
sqr root of 5 is
2.2
work rate
1. 전체 일의 양을 1로 놓고 1사긴당 하는 일을 개산

2. 1시간에 함께 하는일 X t =1

(1/6+1/3)t=1
catch
B가 A를 따라 잡는 시간
B의 거리 =A가 먼저간 걸리 (이전의 거리) + A거리 (이후의 거리)

20t =10X2 + 10t

find t for the 따라 잡는 시간
meet
총거리는 300km
A와 B가 마주보는 지점에서 9시에 출발 11시에 만났다

A는 B보다 6km 더 빠르다

B의 속도는
총거리 =A거리 + B 거리

300 = Va.t + Vb.t
300 = (Vb+6)2 + Vb2

find Vb
avg speed =
total distance/ time

=total distance/갈때 시간 + 얼때시간

=2d/ d/Vavg +d/Vavg =2/1/Vavg+1/Vavg
prime number
number divisible by 1 and itself

2,3,5,7,11,13,17,19,23 and 29
% change
p 95

= difference/original x 100
parcentage=
part/whole X 100
new price if 20% change (inc)
R=S(1+20/100)

R= new price
S= original price
X^0 =
1
X^-n=
1/X^n
X^a x X^b =
X^a+b
(X^a)^b =
X^ab
X^a / X^b =
X^a-b
(XY)^a
X^a Y^a
30^50=
30^25 x 30^25
6^25 =
(3x2)^25 = 3^25 x 2^25
지수의 합차는 낮은지수로 통일
p103
2^x+ 2^x =
4^x
when derivative of point is zero?
meaning that the slope is zero

it is possible that point is a local min or a local max
[] = squr root

[a] x [b]
[ab]
[]= squr root

a[c] + b[c] =
a+b[c]
[] squr root

[25] =
only 5 not -5
x^2 = 25
x is either 5 or -5
[]= squr root

[x] =
x^1/2
x^1/3
{}=cube root

{x}
log 10 (X) = a
10^a =x
log (x) is same is
log 10 (x)
log (10) =
log (10) =1

same as log 10 (10)=1

10^a =10
a= 1
log (ab)
log (a) + log (b)
log (a/b)
log (a) - log (b)
log (1/a) =
-log (a)
log (a^B)
b log (a)
a2 + 2ab + b2
(a+b)^2
a2 + 2ab + b2
(a+b)^2
a2-2ab+b2
(a-b)^2
a2-b2
(a-b)(a+b)
(x-1)(x-3)>0

x is?
0보다 클땐 항상 바깥범위

x>3
x<1
(x-1)(x-3)<0

x is ?
0 보다 작을땐 사이 범위

1<x<3
부등식의 연산

3<x<10
1<y<2

x+y ?
4<x+y<12
부등식의 연산

3<x<10
1<y<2

xy?
3<xy<20
부등식의 연산

3<x<10
1<y<2

x-y?
x-y = x+(-Y)

3<x<10
-2<y<-1

1<x+(-Y)<9
부등식의 연산

3<x<10
1<y<2

x/y ??
x/y = x(1/y)

3<x<10
1/2<y<1/1

3/2<x(1/y)<10
probability

동시에 일어날수 없는 두 사건?

a 또는 b
더한다 +
a 가 b와 함께 일어나는 경우

and
곱한다 X
probability that an event occur + probability that the event does not occur =?
1
4i
24
5i
120
6i
720
7i
5040
varience
v^2=(x-mena)^2/n
standard deviation is
squr root of varience
csc
1/sin
sec
1/cos
cot
1/tan
i
squr root (-1)
i^2
-1
i^3
i^2 x i

-1 x i = -i
i^4
1
11X11
121
12x11
132
12x12
144
12X9
108
12X8
96
vector

a+bi
a= on the x axis

b= is on the y axis
vector equation

what is the r for a point that hs 3 dimenional coordinates of (3,6,8)
r= 3i+6j+8k
slope =
rise / run
if f(x) then

f'(x) = d/dx f'(x) = dy/dx
first derivitive

sign of slope of graph at any given point is told by the sign of the derivative
dy/dx = 0
slope is 0
fucntion is constant
dy/dx > 0
postive slope
function is inc
dy/dx <0
negative slope
function is dec
for f(x)

f''(x) = d^2/dx^2 f(x)
=d/dx f'(x)
second derivitive
tell concave up(local min) or down (local max)
f''(x) >0
concave up (local min)
f''(x)<0
concave down (local max)
f''(x) = 0
constant slope
d/dx [f(x)/g(x)] =

quotient rule
g(x)f'(x)-f(x)g'(x)/g^2(x)
velocity v(t)
first derivativ of position (a(t)
acceleration (how fast speed change)

a(t)
second derivative of position
first derivative of velocity
d/dx sin(x)
cos(x) 1

1 is der of x
d/dx cos (x)
-sin (x) 1

1 is der of x
d/dx tan (x)
sec^2(x) 1

1 is der of x
d/dx csc(x)
-csc(x)cot(x) 1

1 is der of x
d/dx sec (x)
sec(x)tan(x) 1

1 is der of x
d/dx cot(x)
-csc^2(x) 1

1 is der of x
product rule

d/dx[f(x)g(x)]=f(x)g'(x)+f(x)g'(x)
f(x)g'(x)+f(x)g'(x)
if f(x)=e^x

then f'(x)
f'(x) = e^x
if f(x)=ln(x)

then f'(x)
f'(x) = 1/x
[ = intergral

[cos(x)dx
sin (x) + C
[ = intergral

[sin(x)dx
-cos (x) + c
[ = intergral

[sec^2(x)
tan (x) +c
[ = intergral

[csc(x)cot(x)dx
-csc(x) +c
[ = intergral

[sec(x)tan(x)dx
=sec(x)+c
[ = intergral

[csc^2(x) dx
-cot(x)+c