2025 AMC 8 Problems/Problem 20
Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?
Contents
[hide]Video Solution
Key Idea: Let be the fraction eaten by Sarika. Then Dev eats
and Rajiv eats
. Hence
so solving for
we get
,
.
Video Link: https://www.youtube.com/watch?v=VUR5VYabbrc
Solution 1
WLOG, let the amount of total cheese be . Then Sarika eat
, Dev eats
, Rajiv eats
, Sarika eats
and so on. After a couple for attempts, we see that Sarika eats cheese in an infinite geometric sequence with first term
and common ratio of
. Therefore, we use the infinite geometric sequence formula and get
To find how much Sarika eats, we just divide this by our original total and get
.
Therefore, Sarika eats
of the cheese.
~athreyay
Solution 2 (If you forgot the infinite geometric series formula)
At first, Sarika eats of the cheese, and then after
more people eat, there is
of the cheese remaining, so Sarika eats
of the cheese. Continuing this pattern until a reasonable amount, we get new fractions of
and
. Adding these fractions together yields
. Approximating this answer yields about
which is about
of the cheese.
~Soupboy0