Which jar contains a higher percentage of the color it started with?

Question

This is a problem from Brilliant.org

Two jars each contain 1000 candies. One jar contains all green candies and the other contains all red candies. Take 200 green candies and move them to the red jar. Then take 200 candies (some red and some green) from the red jar and move them back to the green jar.

Which jar contains a higher percentage of the color it started with?

Answer given by Brilliant.org is

Both percentages are the same

But my reasoning is:

We took 200 all green from Green jar. But only some green and some red from Red jar. So, percentage of green candies in Green jar is less compared to percentage of red candies in Red jar.

So my answer is Red jar.

I'm not sure whether I didn't get the logic or Brilliant.org question is not clear.

Answer 1

Since an equal number of candies are moved in each direction, the number of red candies moved back (in the second step) must equal the number of green candies not moved back, so each jar ends up with an equal number of candies of the other colour. Since both jars have the same number of total candies, they must have an equal number of candies of their own colour.

Answer 2

Let's call the jar of green candies $G$ and the jar of red candies $R$.

After we move $200$ green candies from $G$ to $R$ we have $800$ green candies in $G$ and $1200$ total candies in $R$.

Let $g$ be the number of green candies in the $200$ we transfer from $R$ back to $G$.

Then the number of green candies in $G$ is $800+g$ and the number of green candies is $200-g$ in $R$. That means the number of red candies in $R$ is $1000-(200-g)=800+g$ and since both $R$ and $G$ have $1000$ candies they both have the same fraction of their original color.

Answer 3

Answering my own question, first I didn't believe but after thinking about it again and again I finally got some understanding.