2025 AMC 8
Complete problem set with solutions and individual problem pages
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?
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Solution 1
Let the total amount of cheese be . We will track the amount of cheese Sarika eats throughout the process.
First Round: Sarika eats half of the total cheese, so she eats:
Second Round: Dev eats half of what remains after Sarika's turn, which is:
Third Round: Rajiv eats half of the remaining cheese after Dev's turn, which is:
At the end of the first round, the total cheese eaten is:
We observe that Sarika's consumption follows a geometric sequence. In the first round, she eats , and in subsequent rounds, she eats half of what remains from the previous round. This gives the following series for Sarika;s total consumption:
This is a geometric series with first term and common ratio . The sum of this infinite geometric series is given by the formula:
where is the first term and is the common ratio. Substituting and :
Thus, Sarika eats of the original block of cheese. The correct answer is:
Solution 2
Sarika eats of the original cheese, and then because the others eat and , she eats next, and then , and then so on. Since the values later are going to be too small to make a huge difference, we can use these values. She ate . We can replace the with a for now, so , which simplifies to around . Since there is a little bit more of the cheese to be accounted for, the amount that she eats will be around .
