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The answer is 6/23

Ellen can choose 1*2*3= 6 tiles that are different from steve.

There is a total of 23 different tiles because Steve already took one and 24-1=23. 

so it is 6/23

I think the two equations should be like this....

If Alice receives n dollars from Bob, then she will have 3 times as much money as Bob.

A + n  =  3 * (B - n)

If she gives n dollars to Bob, then she will have 2 times as much money as Bob.

A - n  =  2 * (B + n)

So we have..

A + n  =  3B - 3n    →     A  =  3B  - 4n     →     3A  =  9B   - 12n

A - n  =  2B + 2n    →     A  =  3B + 3n     →     4A  =  12B + 12n        Add these two to get..

7A  =  17B     →     7A / B  =  17     →     A / B  =  17 / 7
 

this one was hard

Consider                                                                ab4  =  48  

We will divide both sides by ab.  

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab4         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab4 = (ab) * (b3)  

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b3           2  

                                                                             ——   =   ——  

                                                                                1            3  

                                                                                 b   =   cube root of (2 / 3)  

Seven integers    n  + n+1   n+2 ....... n+6    =   7n + 21

    equals   6 * (n+6) = 6n+36

7n+21  = 6n+36

n = 15    the smallest

thanks to ElectricPavlov

 d−4=5(g−4)................(1)

d==5(g - 4) + 4

d−7=8(g−7)................(2)

5(g - 4) + 4 - 7==8(g - 7)

5g - 20 + 4 - 7==8g - 56

56 - 20 + 4 - 7==8g - 5g

33 == 3g

g==33/3==11 - sub this into (2) above

d - 7==8(11 - 7)

d - 7 ==32

d ==32 + 7 ==39 years - his dad's current age.

Midpoint BC  =  [ (-9 + 1)/2 , (6 - 4) / 2 ] = [ -4, 1]

Midpoint AB  = [ (5 -9)/2 , (4 + 6) / 2 ] = [ -2, 5]   

Midpoint AC = [ (5 + 1) / 2 , (4 -4) / 2 ] = [ 3 , 0 ]   

Slope  AD  =  [ 4 - 1 ] / [ 5 - -4] =  1/3

Slope CE  = [ -4-5] / [ 1 - -2] =  -3

Slope BF =  [ 6-0] / [ -9-3]  = -1/2

Equation of line  through AD

y = (1/3)(x + 4) + 1                   (1)

Equation of line  through CE

y =  -3 ( x + 2) + 5              (2)

Equation of line  through BF

y = (-1/2)(x -3)       (3)

Set (1), (2)  equal  to find their x intersection

(1/3)(x + 4) + 1 =  -3(x + 2) + 5

x + 4 +  3  =  -9(x + 2) + 15

x+ 7 = -9x -18 + 15

x + 7 = -9x -3

10x = -10

x = -1

y = -3(-1+2) + 5 =  2

Intersection pt  =  ( -1 , 2)

Test in (3)

2  = (-1/2) (-1 -3) 

2 = 2         true  

Number in Science and Math but not both =  

Members in Math only  = a - b

Members in Science only = 2a - b

Members in Math or Science  but not both  =      

(a - b)  + (2a - b)   = 

3a  - 2b

Slope between AB  = Slope between AC

AB slope =  [ 4 - 0 ] / [ 3 - 0 ]  =  4/3

AC slope  =  [ y - 0 ] / [ 7 - 0 ] = 4/3

y / [ 7 ] = 4/3

y   =  7 (4/3)

y  =  28/3

This is impossible

 (x + 20)^2  + (y - 33)^2  = 34   is a  circle centered at ( -20, 33) with a radius of sqrt (34)

Every point on the line  y =  4  is   more than   sqrt (34)  units away from the center of this  circle

See the graph below

Since the numbers of workers doesnt change, we can times the time by 480/100, so 50 * 4.8 = 240 minutes.

Here for better visualization:

40 members. 100 sq wall.    50 min

40 members.  480 sq wall.  ?

we just times it by 4.8 to get the time. 

I hope it helps!!!

The point (3,4)  is  on the original graph

f(x)  + 4  shifts this up by 4 units

So   ( 3, 8)  must be on the graph

Midpoint BC  =  [ (-9 + 1)/2 , (6 - 4) / 2 ] = [ -4, 1]

Midpoint AB  = [ (5 -9)/2 , (4 + 6) / 2 ] = [ -2, 5]   

Midpoint AC = [ (5 + 1) / 2 , (4 -4) / 2 ] = [ 3 , 0 ]   

Slope  AD  =  [ 4 - 1 ] / [ 5 - -4] =  1/3

Slope CE  = [ -4-5] / [ 1 - -2] =  -3

Slope BF =  [ 6-0] / [ -9-3]  = -1/2

Equation of line  through AD

y = (1/3)(x + 4) + 1                   (1)

Equation of line  through CE

y =  -3 ( x + 2) + 5              (2)

Equation of line  through BF

y = (-1/2)(x -3)       (3)

Set (1), (2)  equal  to find their x intersection

(1/3)(x + 4) + 1 =  -3(x + 2) + 5

x + 4 +  3  =  -9(x + 2) + 15

x+ 7 = -9x -18 + 15

x + 7 = -9x -3

10x = -10

x = -1

y = -3(-1+2) + 5 =  2

Intersection pt  =  ( -1 , 2)

Test in (3)

2  = (-1/2) (-1 -3) 

2 = 2         true  

Thank you!!

Total area =  DC * AD

Area of triangles  ADU + BCT  = 2* (1/2)( DC/3) (AD)  =  DC * AD  /  3

Shaded area  =  DC*AD - DC * AD / 3  =  (2/3)DC *AD  = (2/3)(180) =  120

f(x - 3) represents a translation right by 3.

Therefore, the whole graph is shifted right by 3, meaning the vertex shifts right by 3.

Shifting right by 3 means that 3 is added to the x coordinate.

Therefore the new vertex is (1 + 3, -2) = (4, -2).

Alternatively, if you don't know translations: Use vertex form:

\(f(x)=a(x-1)^2-2\) represents a parabola of vertex (1, -2), where a is a constant

\(f(x-3) = a((x-3)-1)^2-2=a(x-4)^2-2\), so looking at this form, you know the vertex is (4, -2).

Call the total number of games to be played  = T

We have played  .60T

We have  .40T  left to play

We have won  .40 (.60)T =  .24T

We need to win .50T

So

[  Games  we need to win - Games we have won ] / [ Games to be played ]   =  %we need to win

[ .50T  - .24T ] / .40T  =

.26 / .40   = 

13 / 20  = 

65%  of  our  remaining games

\(3mn-30m+6n+c\)

\(3m(n - 10) + 6n + c\)

\(3m(n-10)+6(n-10)+60+c\)

\((3m+6)(n-10)+60+c\)

\(c+60 = 0\)

\(c=-60\)

The constant is -60.

x^2 - 22x + c

c =  (22/2)^2  = (11)^2   =121

x^2 - 22x + 121  =   ( x - 11)^2

(1/2)BC = 4

Call the altitude  AD =   sqrt [ 5^2 - 4^2]  = 3

tan ( ABC)  = (3/4)

tan (ABC / 2)  =    1 -cos (ABC) / sin (ABC)  =  [ 1  - 4/5 ] / [(3/5 ] = 1/3

Equation  of angle bisector  of  ABC  =   y =(1/3)x

Intersection of   this line with  angle bisector BAC {line  x = 4 }

y= (1/3) (4)  = 4/3

Height of triangle  AMN  =  3 - 4/3  =  5/3

Area  of  [ABC]  =  (1/2) AD * BC  =   (1/2) (3) (8)  = 12

Triangle AMN  similar to triangle ABC

Scale factor  of  AMN/ ABC  =  (5/3) / 3 =  5/9

[ AMN ]  =  12 *  [ (5/9 ] ^2  = 100 / 27   ≈  3.7

A                     X       S        R          B

XB =  2808 / 2  =  1404

XR = (3/5)XB (3/5)(1404)  = 842.4

XS = (2/17)XR = (2/17)842.4 = 8424/  95

RS =  XR - XS  =  842.4 - 8424/95   = 71604/95 ≈ 753.73

(x + 2) / x  *  ( x + 3) / (x + 1)  *  ( x + 4)  / (x + 2)   = 10

(x + 3) / x * ( x + 4)  / ( x + 1)  =  10

(x + 3) ( x + 4)  = 10 (x) (x + 1)

x^2  + 7x + 12  = 10x^2 + 10x

9x^2 + 3x - 12  = 0

3x^2  + x  - 4  =  0

(3x + 4) ( x - 1)  = 0

(x - 1)  = 0

x = 1

First fraction  = (1 + 2) / 1  =  3 / 1

r =  .036 / .4   =  36 / 400  =  9 / 100 =  .09

Sum  =   first term / [ 1 - r ]  =   .4  / [ 1 - .09 ] =   .4 / .91  =  40 / 91

Slope  of PQ =   [ 4 - -5 ] / [ 5 - -1] =  9 / 6  =  3 / 2

y =(3/2) (x - 5)  + 4        multiply through by  2/3

(2/3)y = x - 5 + 8/3

(2/3)y  = x  -  7/3           rearrange

x - (2/3)y  =  7/3           multiply by 3/7

(3/7)x - (3/7)(2/3)y = 1

(3/7)x - (2/7)y  =1

x / ( 7/3)   y  / ( 7/2)  = 1

x^2  + (-4)^2  = 20

x^2 + 16  =  20

x^2  = 4

x  = 2                x =  -2

A = (2, -4)      B =  (-2 , -2)

AB =   2 - (-2)  =   4