CPhill

Here's my best effort :

Note that, getting common denominators, we have that

9/11 = 54/66 = 108 / 132         

5/6  =  55/66 =  110 / 132

The fraction between  these two  with integer numerator / denominator  =   109 / 132 

Interested to see if someone else has a better answer  !!!!!

5x^2  -kx  + 8  = 4x^2          .....subtract 4x^2  from both sides

x^2  - kx + 8  =  0

We will have no solutios  when the discriminant ....   (-k)^2  - 4 (1) (8)  < 0

So....simplifying  and rearranging  we have

k^2   <   32

The integer values  that  make this  true   are  [   -5 ...... 5 ]

So.....5  is the largest  integer that produces no real solutions

We can do this one by exclusion

Possible 4-letter "words"  =  (6)^4

Possible "words"  containing only the consonants listed  = (4)^4

So.....possible "words" with at least one vowel  =  ( 6)^4 - (4)^4  =  1296 - 256  =   1040

Multiples of 6   =  102 , 108 ........498

So.....the number of multiples divisble by 6 =    [498 - 102 ]  /  6  + 1  =   66 + 1  =  67

Multiples of  8  =   104, 112 .......496

So  ...      [ 496 - 104 ] / 8  +   1 =  49 + 1   = 50

We  need  to add  all those divisible  by 6,8...i.e.,   the LCM of 6,8  =  24

So....Multiples of 24   =   120, 144 ......480

So....  [ 480  - 120 ]  / 24   +  1   =  15 + 1  = 16

So

Divisible by 6   +  Divisible by 8  +  Divisible by 24 =

     67               +          50           +        16    =                  133

x + y + 3z  = 10

-4x + 3y + 6z  = 8

kx + z =  3

Strategy :  Eliminate  y

Multiply the first equation by -3  and add to the second.....you should get

-7x  -3z  =  -30         (1)

Multiply the  third equation by  -3  to match its z term with  (1)  =  - 3kx  - 3z  = -9      (2)

So using(1) and (2) we have

-7x -3z  =  -30

-3kx - 3z  =  -9

We  want to now solve this for k

-3kx  = -7x            ....divide through by -x

3k  = 7

k = 7/3 gives  us  no solutions

No prob  !!!

Thanks , EP.....here's another way to see this

Note  that the first two terms factored  =   3m ( n - 2)

Then using (n - 2)  as  one of the linear  factors, the last two  terms  can be written as  15 ( n - 2)

So  the factorization into  2 linear terms  =   (n - 2)  ( 3m + 15)  = 3mn - 6m  + 15n  - 30

Just as EP found  !!!

The slant height of the cone = the radius of the circle

The  circumference  of  the of the  whole circle =   2pi * 25  =  50 pi

Since  90 degrees is taken out - (1/4) of the circumference- then the circumference of the  cone = (50)pi  * (3/4) =   37.5 pi

The radius of the cone  = 1/2 of this =     ( 37.5 pi) / (2pi)  =  18.75 

The height of the cone =   sqrt ( slant height ^2 - radius^2 )  =  sqrt ( 25^2 - 18.75^2)   ≈ 16.54

The volume is   (1/3) * pi * ( 18.75)^2 * ( 16.54) ≈  6089.3  ( cubic units)

We can put the  1 chocolate chip anywhere   = 12 ways

Since the plain are indistinguishable.......they don't matter

So.....12 different  ways

Note :  We have 26 possible letters   and  0-9  =  10 possible numbers

The number of posssible plates  = 

(26)^3  (10)^3   +   (26)^4 ( 10)^2  =  63,273,600