Question 331081
Let {{{c}}} = number of whole cookies on plate to begin with
I gave Tim {{{c/2 + 1/2}}}, so I have left
{{{c - c/2 - 1/2 = c/2 - 1/2}}}
I gave Beth {{{(1/2)*(c/2 - 1/2) + 1/2 = c/4 - 1/4 + 1/2}}}, so I have left
{{{c/2 - 1/2 - c/4 + 1/4 - 1/2 = c/4 - 3/4}}}
I gave Michelle {{{(1/2)*(c/4 - 3/4) + 1/2 = c/8 - 3/8 + 1/2}}}
There are no cookies left on the plate
That means I gave away all the cookies, so
Tim, Beth, and Michelle have them all, so
{{{c/2 + 1/2 + c/4 + 1/4 + c/8 + 1/8}}} = all the cookies
{{{7c/8 + 7/8}}} = all the cookies
and, since 
{{{c}}} = number of whole cookies on plate to begin with,
{{{c = 7c/8 + 7/8}}}
{{{c/8 = 7/8}}}
{{{c = 7}}}
There were 7 cookies on the plate to begin with
check answer:
I gave Tim {{{7/2 + 1/2 = 4}}}, so I have {{{3}}} left
I gave beth {{{3/2 + 1/2 = 2}}}, so I have {{{1}}} left
I gave Michelle {{{1/2 + 1/2 = 1}}}, so I have {{{0}}} left
OK