Question

How do I set up the equation for this word problem? A takes 1/2 of the diamonds plus 1 from the table. B takes 2/3 of what remains. C takes 2/3 of what remains plus 1. One diamond is left. How many diamonds had originally been on the table? Answer is 38 but I don't know how to set it up.

Answer 1

The first thief takes (1/2 x + 1) .  What remains ?   x - (1/2x + 1)

So the 2nd thief takes  2/3 of  [  x - (1/2x + 1) ]

   What remains ?    x - 2/3 [ x - (1/2x + 1) ]

So the 3rd thief takes    2/3 of  { x - 2/3 [ x - (1/2x + 1) ] }  and he takes 1 more .

   What remains ?    x - ( 2/3 { x - 2/3 [ x - (1/2x + 1) ] } + 1 )

And that whole ugly thing is equal to ' 1 ', so you can solve it for 'x'..

The whole problem from here on is an exercise in simplifying
an expression with a bunch of 'nested' parentheses in it.  
===============================================
This is a lot harder than just solving the problem with logic and
waving your hands in the air.  Here's how you would do that:

Start from the end and work backwards:

-- One diamond is left.
-- Before the 3rd thief took 1 more, there were 2.
-- That was 1/3 of what was there before the 3rd man took 2/3.
So he found 6 when he arrived.
-- 6 was 1/3 of what was there before the second thief helped himself.
So there were 18 when the 2nd man arrived.
-- 18 was 1 less than what was there before the first thief took 1 extra.
So he took his 1 extra from 19.
-- 19 was the remaining after the first man took 1/2 of all on the table.
So there were 38 on the table when he arrived.

Thank you for your generous 5 points.