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No prob....!!!

Yeah, I was a little confused. All good, thanks!

OK.....let's take it from here

( [ (3/2) (a) +  5 ]  -  [ (3/2) (b) + 5 ] )  / [ a - b ]  =

[ (3/2)a + 5 - (3/2)b - 5 ]  / [ a - b ]  =

[ (3/2) a - (3/2)(b)] / [ a - b]  =

 (3/2) (a - b) / ( a - b)  =

3/2

I think I forgot to close the parentheses  on the (a) term above....sorry....!!! 

Hey CPhill, thanks for the help!

I still have one question based on your work : )

Here is the work:

[  f-1(a)  - f-1(b) ] / [a - b]  =  (  [ (3/2)(a + 5]  -  (3/2)(b) + 5] / [ a - b]  =

[ (3/2) (a - b) ] / [ a - b ]  =    3/2

I don't understand the bolded out part, as in how you went from [ (3/2)(a + 5]  -  (3/2)(b) + 5] / [ a - b] to the next part..

Could you please further explain your steps, I am a little confused about that

Just found the correct answer. Thank you for the help! 

There are 7! ways to put the beads on the grid, not considering rotations and reflections. Arrangements can be reflected or not reflected and can be rotated by 0, 60, 120, 180, 240, or 300 degrees, so they come in groups of twelve equivalent arrangements. Correcting for the symmetries, we find that there are 7!/12 or 420 distinct arrangements.

Taking what you have literally written, we have

{ |-2(-5)-38|+2} / 3 * [-9+7]^3  =

(1/3)  {   l  10 - 38  l  + 2 }  * ( -9 + 7)^3  =

(1/3)  { l - 28 l + 2 }  * (-2)^3  =

(1/3) ( 30) * (-2)^3  =

10 * ( -8)  =

-80

Thank you! 

We simplify: \(\left|-2\left(-5\right)-38\right|+\frac{2}{3}\left(-9+7\right)^3=\left|2\cdot \:5-38\right|+\frac{2}{3}\left(-9+7\right)^3\), so we get \(28+\frac{2}{3}\left(7-9\right)^3\) or \(\boxed{\frac{68}{3}}.\)

what he said

Excellent, PM....!!!!

Good job, PM....!!!

Thanks, EP....Here's another way

Just add the equations and we get that

2x  = 4 + 4

2x = 8

x = 4

Sub this back into the first equation to find y

4 + y  = ( 5 - 4) + ( 5 - y)

4 + y   =  6 - y

2y = 2

y = 1

Let's write  y = (2/3) ( x - 5)       multiply both sides by 3/2

(3/2)y =  x - 5       add 5 to both sides

(3/2)y + 5 =  x        "swap" x and y

(3/2)x + 5   =  y  = f^-1(x)       this is the inverse

So

a) f^-1 (-2)   =  (3/2)(-2) + 5  =  - 3 + 5  =  2

b)  f^-1 ( a + 1) - f^-1 ( a)  =   [ (3/2)(a + 1)   + 5 ] - [ (3/2)a + 5 ]  = 3/2

c )  [  f^-1(a)  - f^-1(b) ] / [a - b]  =  (  [ (3/2)(a + 5]  -  (3/2)(b) + 5] / [ a - b]  =

[ (3/2) (a - b) ] / [ a - b ]  =    3/2

Simplify the first eq:

x+y = 5-x+5-y

2x +2y =10

Simplify the second eq:

x-y=x-1+y-1

-2y = -2

y =1            Sub this into the first equation   2x+2(1)=10    2x+2=10  x = 4       ordered pair = (4,1)

Sam = s     Jane = j

1/3 s = 1/2 j

s = 3/2 j

s+j = 10   (more than 10 actually)

3/2j + j = 10

5/2j =10

j = 4    so   s = 3/2j = 6   if it EQUALS 10....we want the next whole numbers that are greater than 10

j=6      and s = 9        would be the next whole number amounts that would work    total  $ 15

Lowest possible solution is 3,2

If we just multiply the numbers by the same factor which would be large enough that the sum of both integers is more than 10,

3,2 x 4

we get 12,8 

if a = -40   the equations become the same    2x+5y=-8    or    6x+15y=-24     

         there are infinite solutions to this equation (two variables , one equation)

Good afternoon for me lol 

Let's use X to represent adult dinos and 3X to represent baby dinos. 

Our equation is 7X + 3X(15) = 208. Simplifying, we have 52X = 208. If we divide both sides by 52, we are left with X = 4.

If X = 4, then 3X = 12. 

There are 12 baby dinos and 4 adult dinos.

Hope this helps,

- PM 

Here is the tree for  part a).

0 child              1st child              2nd child

                                                     | -- Boy (27.04%)

                           _ Boy (52%) -------[

Start                   |                          | -- Girl (24.96%)

    X -------------------|                          

                          |                          | -- Boy (24.96%) 

                          |_ Girl (48%) --------[

                                                     |_ Girl (23.04%)

- PM

A = apples

oranges = 9A

A+9A=90

10A=90

A=apples = 9

B = boy, G = girl

a) 
Probability of 2B is (0.52)(0.52) = 0.2704.
Probability of BG is (0.52)(0.48) = 0.2496 
Probability of GB is the same as the probability of BG, so it is 0.2496.
Probability of 2G is (0.48)(0.48) = 0.2304.

b) 
Number of girls born: 

Table for p(x) 
For zero girls and 2 boys, it is 0.2704.
For 1 girl and 1 boy, it is 0.4992.
For 2 girls, it is 0.2304. 

- PM

There are 7 points, so there are 7! = 5040 ways to arrange the points without considering overcounting.

Let's first consider rotations. If we number the 6 beads on the outside as ABCDEF and the center bead as G, the arrangement of ABCDEFG is considered the same as BCDEFGA. Since there are 6 different locations the outer bead A can be, (A cannot be in the center) we divide 5040 by 6 to get 840.

Let's now consider reflections. We can draw six axes for the beads to reflect on, and ABC G(center) DEF is considered the same as DEF G(center) ABC. We divide 840 by 6 to get our answer, \(\boxed{140}\).

Hope this helps, 

- PM 

\(\dfrac{7\cdot(5!)}{6} = \boxed{140}\)

- PM