Judy had a number of raisins. after eating one, she gave half the remainder to her sister. after eating another raisin, she gave a third of what was left to her brother. judy now had only six raisins left. how many raisins did she start with?

Suppose she started with x raisins
after eating 1 she had (x-1) rasiins
then she gave 0.5(x - 1) to her sister so she herself had 0.5(x-1) raisins.
after eating another 1 she had 0.5(x - 1)- 1 raisins.= 0.5x - 1.5 raisins
she gave a third  of these to her brother   so she had 2/3 left 
= 2/3 (0.5x - 1.5) raisins

2/3[ 0.5x - 1.5) = 6
2(0.5x - 1.5) = 18
x  - 3 = 18
x = 21

She started with 21 raisins.