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 $3:1:1$, respectively, in a way that makes each person as happy as possible, how many pieces of chocolate with caramel does Andy receive?

step-by-step explanation:

in the absence of parentheses, i will have to make some assumptions:   i interpret your (3x+5/y^6)(4y^3/5)​ to mean:

3x + 5     4y^(3/5)       (3x + 5)(4)

* =

    y^6             1               y^(6-3/5)

if this is not what you expected, repost your question using parentheses to indicate whether you meant

4y^3

    or   4y^(3/5)

    5

what the question so we can you out

for ex: the reciprocal of 3/4 is 4/3. it is the opposite.

reciprocal definition:

a mathematical expression or function so related to another that their product is one; the quantity obtained by dividing the number one by a given quantity.