Amattress underwent the following changes in price before being removed from the store.

20%
markdown
35%
markup
24%
markdown
13%
markup
31%
markdown

if the final price of the mattress was $227.83, what was the original price of the mattress, to the nearest dollar? round all dollar values to the nearest cent.

Since we are given with the final price already, we calculated for the initial price backwards.
  (1)                          227.83 = (1 - 0.31)x₁
                                   x₁ = 330.19

  (2)                          330.19 = (1 + 0.13)x₂
                                     x₂ = 292.20

   (3)                           292.29 = (1 - 0.24)x₃
                                      x₃ = 384.48
 
   (4)                           384.48 = (1 + 0.35)x₄
                                       x₄ = 284.80
   
   (5)                           284.80 = (1 - 0.20)x₅
                                        x₅ = 355.99 or 356

Therefore, the original price of the mattress is ~ $356.