CPhill

Triangle HQN sinilar to triangle FQG           Triangle MPF similar to triangle GPH

HQ/HN = FQ/FG                                                MF/PF = GH/PH                                       

HQ/2 = FQ/4                                                       2/PF = 4/PH

HQ/FQ = 2/4 = 1/2                                              PF/PH = 2/4 = 1/2

HQ = ( 1/3)FH

PF = (1/3)FH

FH - HQ - PF  = PQ

FH - (1/3)FH - (1/3)FH = (1/3)FH = PQ

PQ / FH =  (1/3)FH / FH  = 1/3  

\(\cfrac{1}{1 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{2}}}} \)

Work from bottom  to  top

4 + 1/2 = 9/2

1/ (9/2)  = 2/9

4 + 2/9  = 38/9

1/(38/9) = 9/38

1 + 9/38  = 47/38

1/(47/38) = 38 / 47

17b =   (1b + 7)

(1b + 7)^2  =  b^2 + 14b + 49

211b  = 2b^2 + 1b + 1

b^2 + 14b + 49  = 2b^2 + 1b + 1

b^2 - 13b - 48 = 0

(b - 16) ( b + 3)  = 0

b - 16  = 0

b = 16

\(4+ { \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4}}}} \)

Work from the bottom-up

4 + 1/4 =  17/4

1/(17/4) = 4/17

4 + 4/17  = 72/17

1/(72/17)  = 17/72

4 + 17/72  =  305 / 72

2.57 repeating

[257 - 2 ] / [990 ] =  255 / 990 = 17 / 66

11012 / 100000   =   2753 / 25000

12 people can  count 1/4 box in  one hour

So 1 person  can count  1/ 4 * 1/12  =    1/48 box in  1 hr

So

30 people can count   30 * (1/48)  = 30/48 =  5/8  of a box in  1  hour

So

30  / (5/8) =   48 hrs

1/x - 1/(xy) - 1 / (xyz)  =  1/3

Let x  =1

1 - 1/3 =  1/(y) + 1/(yz)

2/3  = 1/y + 1/(yz)

2/3 = [ z + 1] / [ yz]

2yz = 3z + 3

2yz - 3z  = 3

z (2y - 3)  =  3

  If y = 2  then z  = 3

x,y,z = ( 1,2,3)

Check

1/1 - 1/2 - 1/6  =   1/3

1/977  =  .001........

2^4 * 3 =   48