Answer:
To determine the number of cookies in the batch at the beginning, let's break down the problem step by step.
1. Aaron takes 1/6th of the cookies, leaving 5/6th of the original batch.
2. Matthew then takes 1/5th of the remaining cookies, leaving 4/5th of the 5/6th.
3. Paige takes 1/4th of what remains after Matthew, leaving 3/4th of the 4/5th.
4. Jodh takes 1/3rd of what remains after Paige, leaving 2/3rd of the 3/4th.
5. Dylan grabs half of the remaining cookies after Jodh, leaving 1/2 of the 2/3rd.
6. Finally, Gavin swoops in and grabs the last remaining 6 cookies.
Now, let's calculate the number of cookies left at each step:
1. 5/6th of the original batch after Aaron = (5/6) * original batch
2. 4/5th of the 5/6th after Matthew = (4/5) * (5/6) * original batch
3. 3/4th of the 4/5th after Paige = (3/4) * (4/5) * (5/6) * original batch
4. 2/3rd of the 3/4th after Jodh = (2/3) * (3/4) * (4/5) * (5/6) * original batch
5. 1/2 of the 2/3rd after Dylan = (1/2) * (2/3) * (3/4) * (4/5) * (5/6) * original batch
Since Gavin takes the last remaining 6 cookies, we can set the equation:
(1/2) * (2/3) * (3/4) * (4/5) * (5/6) * original batch = 6
Now, we can solve for the original batch:
(1/2) * (2/3) * (3/4) * (4/5) * (5/6) * original batch = 6
Simplifying the equation, we have:
(1/6) * original batch = 6
Multiplying both sides by 6, we find:
original batch = 36
Therefore, the batch initially had 36 cookies.