When 0.5 cm was planed off each of the six faces of a wooden cube, its volume decreased by 169cm^3. Find its new volume.

Let's call the original volume V_o ......... and let's call the new volume V_n 

And V_o  = s³  = V_n + 169    where s is the original side length

Note that, if .5 cm is shaved off each face, then each side is now just (s-1)cm long......so we have...

V_n = (s - 1)³ =  s³ - 3s² + 3s - 1       and substituting V_n + 169 for s³   we have

V_n = [ V_n + 169] - 3s² + 3s - 1      and subtracting V_n  from both sides and rearranging, we have

3s² - 3s - 168  = 0     divide both sides by 3

s² - s - 56 = 0   factor this

(s - 8)(s +7) = 0   so, s = 8   or  s = -7     (reject the negative answer)

 So, s = 8  and that's the original side length

So the new side length = (s - 1) = (8 - 1 )  = 7 .....  and the new volume is 7³ = 343cm³

Note that the original volume = 8³ = 512cm³

And 512cm³ - 343cm³ = 169cm³   .....and that's the difference between the old volume and the new volume !!!