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  seven out of nine are not shaded      7/9

sqrt (10*5*16 p*p² *p⁴) = sqrt (800 p⁷) = 20p³ sqrt(2p)

(x,y) = (3,-4).

p = 17.

I solved this before; there are two circles that work.

probability none will speak p0 = (3/4)⁶  

probability exactly one will speak p1 = 6*(1/4)*(3/4)⁵ 

probability at least two will speak p2 = 1 - p1 - p0

You can crunch the numbers

The Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability 3/4 and there are no ties, what is the probability that the Cubs will win the World Series? Express your answer as a percent rounded to the nearest whole percent.    

Is this a trick question or just worded poorly? 

Consider the third sentence:  If the Cubs win each game....  

If the Cubs win each game, then the probability they win the Series is 100%.  

Thanks Guest.  It is always good to have an answer to compare mine too.

Andrew chooses a number from 1 to 100, and Mary also chooses a number from 1 to 100. (They may choose the same number.) It turns out that the product of their numbers is divisible by 3. In how many ways could Andrew and Mary have chosen their numbers?

Here is the mathematical way.

One or both of them has to choose a multiple of 3.  There are 100 numbers and 33 multiples of 3 (for each of them)

There are 33*100 ways for Mary to chose a multiple of 3 and Andrew to chose any number

There are 33*100 ways for Andrew to chose a multiple of 3 and Mary to chose any number

But I have double counted.

There are  33*33 ways that they can both chose a multiple of 3.

So altogether there are     33*100  +  33*100  -  33*33  =  5511 ways.    Just as answering guest found.

This could be displayed in a ven diagram.  2 overlapping circles.  

33*33= 1089  is the overlapping bits

and  100*33 - 1089 = 2211 in each of the other two arias.

I will take a crack at this one! Here is my thinking:

Mary and Andrew can choose any number between 1 and 100. The maximum "sample space" of the products of their two numbers should be: 100 x 100 =10,000. So, the question becomes: how many of those 10,000 products are divisible by 3?

I wrote a short computer code to determine that and the answer came out as: 5,511 out of those 10,000 products. So, the conclusion I come to is: there are 5,511 ways that Mary and Andrew could have chosen their numbers.

Note: Melody, CPhill and others should take a good look at my attempt. Is there a simple solution to it, or a simple formula one could use? I don't know!.

Bob had 160 and Jay had 240 at first. Each boy spent an equal amount of money. Then the ratio of the amount of money bob had left to the amount of money Jay had left was 3:5. How much money did each boy spend?

(160 - x) / (240 - x) = 3 / 5

x = 40

Let  A  be the equal amount spent  by each....and we know that

[ 160  - A ]          3

_________  =   ___             cross-multiply

[ 240 - A ]           5

5 [ 160  - A ]   =  3 [ 240 - A]     simplify

800 - 5A =  720 - 3A      add 5A to both sides, subract 720 from each side

80  =  2A     divide  both side  by 2

$40  = A  = amt each spent

Very nice, jugoslav  !!!!

A 288-degree circular sector with a radius of 18 is rolled to form a cone. Find the height of the cone.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Shortcut:

The radius of a sector:      R = 18

The radius of a cone:      r = R * (288 / 360) = 14.4

Cone slant height:          L = 18

Cone height:                  h = sqrt(18² - 14.4²) = 10.8

Let's see how the guest derived his/her answer.....

Look at the following:

The area we seek  is (1/2)  the area of the octagon  less the area of  triangle CDE

Note that angle CDE = 135°

Area of triangle  CDE =  (1/2) (CD) (DE) sin (135)  = (1/2) (12) (12) ( 1/ sqrt(2))  = 72/sqrt (2)  = 36sqrt (2)   [1]

Area of (1/2) of the octagon =  ( 1 + sqrt (2) ) * side^2  =  (1 + sqrt (2) ) (12)^2  =  (1 + sqrt (2) ) (144)  =

144 + 144sqrt (2)     [2]

Area of trapezoid =     [2]  - [1]   =    [ 144 + 108 sqrt (2)] cm^2  ≈  296.735 cm^2

Look at the graph here, Thanks4Helping :  https://www.desmos.com/calculator/pxlandri3i

The over-lapping areas of the  two  inequalities represent the  feasible region....(it will appear as a grey area)

Note that we can immediately rule out any  points  lying  below the x axis  [ points  B, F, J , H  and D ]

E = (5,6)   not in  feasible region

A = (3,3)   not in feasible region

I = (2,4)    this point is on the border of the solid line....it is included in the region

G= (-3, 5)  in the feasible region

C = (-4,1) not in feasible  region

So....the  points are   ( I , G )

Thx, jugoslav  !!!!

Consider "tens digit" as blue die and the "ones digit" to be "yellow" die, then we have:

11 , 21 , 22 , 31 , 32 , 33 , 41 , 42 , 43 , 44 , 51 , 52 , 53 , 54 , 55 , 61 , 62 , 63 , 64 , 65 , 66 , Total =  21 ways

Or: Probability is: 21 / 6^2 =21 / 36 = 7 / 12

Thx, guest....here's another way

If AB  passes through  (0,0)  and has a slope of 2/3, the equation of a line through AB  is

y= (2/3)x

Since  B is on  the line y =5, then  the y coordinate of B  = 5

To  find the  x coordinate we have

5 = (2/3) x    multiply both sides by 3/2

(3/2) 5  = x  =  7.5

B= (7.5 ,5)   and the sum of the coordinates  =  12.5

I'm listing ALL combinations from 1 letter to 4 letters.

{A}, {H}, {M}, {T}, {A, H} | {A, M} | {A, T} | {H, M} | {H, T} | {M, T} , {A, H, M} | {A, H, T} | {A, M, T} | {H, M, T} {A, H, M, T} > Total = 15 such “words”

(c+d*i)*(e+f*i)=ce-df+i*(cf+de)

since c,d,e,f are positive real number, their product always in first or second qudrant

To have their product in seond qudrant, ce-df must ;ess than 0, that is ce>df

pick c=1,e=2,d=3,f=4, problem solved

(1+3i)(2+4i)=-10+10i,which is in the second qudrant of the complex plane.

That is EXACTLY what CPhill has given you!

50 t0 250    is     201 numbers

perfect squares in this range are 64 81 100 121 144 169 196 225      you can do the math from here

perfect cubes are 64 125 216         

Excellent work, Phill !!!

Slope is   rise / run    =  2/3     

    rise 5    would be run  7.5    to equal   2/3

(7.5 , 5 )