Ratios

Let b =initial number of blue tiles and w = initial number of white tiles.

Initially

b/w = 5/9  or b = 5w/9   (1)

After

(b+9)/(w-9) = 13/15  or 15(b+9) = 13(w-9)    (2)

Using (1) in (2)

15(5w/9 + 9) = 13(w - 9)

15*5*w/9 + 15*9 = 13*w - 13*9

25w/3 + 135 = 13w - 117

Multiply through by 3

25w + 405 = 39w - 351

39w - 25w = 405 + 351

14w = 756

w = 756/14 = 54

Put this back into (1)

b = 5*54/9 = 30

Hence total number of tiles: b + w = 30 + 54 = 84

. 

The number of blue and white tiles  = total tiles

B +  W   =   T     

And we're told that:    9B = 5W  ...so...   B   = (5/9)W   .....which implies that....

(5/9)W  + W = T         

And if 9 white tiles are replaced with blue tiles, we have.....

[(5/9)W + 9 ]  : [ W - 9 ]  = 13 : 15    ....or.....expressing another way.....

[(5/9)W + 9 ] *15  = 13*[W - 9]  ..... solve for W

(75/9)W + 135 =  13W -  117

75W + 1215  = 117W  - 1053

2268  = 42W

So, W = 54 = the number of original white tiles ....  and B = (5/9)W =  30 = number of original blue tiles....so...... there are  84 tiles total

Check......  before   =   30/54   = 5/9    ...    after =  39/45  =  13/15    √

         Before                                    After

      5(b) : 9(w)                            13(b) : 15(w)

       5+9=14                                13+15=28        (The total number of tiles must be evenly divisible by 28)  

1/  28/14=2    2*5=10,  2*9=18   if we took 9 tiles out 18 and add them up to 10, the ratio will be 19:9  WRONG!

2/  28*2=56,    56/14=4,      4*5=20,    4*9=36  in this case, the ratio is 29:27  WRONG!

3/  28*3=84,     84/14=6,     6*5=30,    6*9=54   if we subtruct 9 from 54 and add it to 30, we get the desired

ratio:  39 : 45  or   13 : 15   RIGHT!

There are 84 tiles on the floor!