Andy, Bert, and Charles have 20 pieces of chocolate. Some of the chocolates have nuts, some are plain, and some are filled with caramel (they are split in a ratio of 1:2:2, respectively). Neither Andy nor Charles can eat nuts, and Charles will always take caramels over plain chocolates if he can (the others are indifferent to plain or caramels). If Andy, Bert, and Charles split the chocolates among themselves in a ratio of 3:1:1, how many pieces of chocolate with caramel does Andy receive?
Here is my attempt at this:
1/5 x 20 =4 pieces that have nuts in them
2/5 x 20 =8 pieces that are plain.
2/5 x 20 =8 pieces that have caramel in them.
3/5 x 20 =12 pieces that Andy will receive.
1/5 x 20 =4 pieces that Bert will receive.
1/5 x 20 =4 pieces that Charles will receive.
Since Andy cannot eat any nuts, his 12 pieces must be plain or caramel.
And since Charles also cannot eat nuts, his 4 pieces must be plain or caramel. Since he prefers caramel his 4 pieces must be caramel.
Since Bert is indifferent to plain or caramel, his 4 pieces must be nuts.
The remaining 4 caramels + 8 plain ones must have gone to Andy.