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.