CPhill

R = number of red

R + 6 =  number of white

2R =  yellow

2( R + 6) =  blue

R +  (R + 6) + 2R  + 2 ( R + 6)  =  48

6R + 18  = 48

6R  =  48 - 18

6R  = 30

R = 30 / 6 =   5

R + 6 =  11

2R =  10

2(R + 6)  =  22

This  is the sum of an arithmetic series

n ( a1  + a1 + (n -1) d)  / 2  >  300

a1  = 6  = the first term

n = the number of days

d  = the common difference =  8       

So we have

n ( 6 + 6  + (n-1) * 8 )   /  2 >  300

n ( 12 + 8n - 8)  >  600

n ( 4 + 8n)  >  600

4n + 8n^2  > 600       divide through by 4

2n^2 + n >  150

2n^2 + n - 150  >  0

Solve this

2n^2  + n  -150  = 0

Using the Quadratic Formula

n   =        -1 + sqrt ( 1^2  - 4 (2) (-150) )                -1 + sqrt ( 1201)

              _________________________ =        _______________ ≈  8.41

                               2 * 2                                                4

He will  run out  sometime  during the 9th day

Let n be the smallest and  n + 4 the largest.....and we  have that,,,,,

n >  (3/4)(n + 4)

n >  (3/4)n + 3

n - (3/4)n > 3

(1/4)n > 3       multiply both sides by 4

n > 12

So  n =  13

By the triangle inequality  either

80  + 25  >  x               or          25  +  x    >  80

105 > x                                          x  >   80  - 25

x < 105                                          x >   55

Then  the remaining side's shortest length   =   56  inches

See here   :   https://web2.0calc.com/questions/last-two-digits_1

$4   is 1/2 of his remaining money

So......  4 / (1/2) =  $8  =   1/2 of his paycheck

So....his whole paycheck  =   8 /  (1/2)  =   $16

Find the slope   [   5 - 11 ] / [ -4 - 1 ]  =  -6 / -5   =  6 / 5

Write the equation of the line in point-slope

y = ( 6 /5) ( x - 1) + 11

y = (6/5)x - 6/5  + 11

y = (6/5)x  - 6/5 + 55/5

y = (6/5)x + 49/5        multiply both sides by  5/49

(5 / 49)y =   (6 /49)x  + 1

- (6 / 49)x  + ( 5/49) y  =  1

And we can write  this as

x / ( - 49/6)  + y / (49 / 5)   =  1

a =  -49 /  6

x^2 - 1  = (x + 1)  ( x - 1)

Multiplying both sides by the right side above

- 9x  =  2 (x - 1) - 6(x + 1)  + 3(x - 1)      simplify

-9x = 2x - 2  -6x - 6 + 3x  - 3

-9x =  -x -  11

-8x =  11

x =   -11 /  8

The "2x"     compresses f(x)  horizontally  by a factor of 2

So.....the interval width  =   [ 9 - - 1 ]  / 2  =   10 / 2  =   5

Vowels   A , E , O , U

Consonants   D M  P  S

Choose  any 1 of the 4 vowels  = C(4,1) =  4

Choose any 2 of the 4 consonants  and  permute them in 2 ways =   P (4,2)  =  12

Total arrangements =   4 * 12   =    48