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?

A. $\frac{4}{7}$

B. $\frac{3}{5}$

C. $\frac{2}{3}$

D. $\frac{3}{4}$

E. $\frac{7}{8}$

1. 2025 AMC 8 官方解析

Answer (A): In each round, Sarika eats twice as much cheese as Dev, and Dev eats twice as much cheese as Rajiv. Once they have eaten all the cheese, Dev will have eaten about twice as much cheese as Rajiv, and Sarika will have eaten about 4 times as much cheese as Rajiv. If $r$ is Rajiv's portion of cheese, then Sarika's portion is about $4r$ and Dev's portion is about $2r$. Together, the three will eat a portion approximately of size $7r$, so the fraction of the original block Sarika eats is about $\frac{4r}{7r} = \frac{4}{7}$ of the total amount.

OR

The following infinite geometric series represent each person's share of the cheese:

Sarika: $\frac{1}{2} + \frac{1}{16} + \frac{1}{128} + \cdots$

Dev: $\frac{1}{4} + \frac{1}{32} + \frac{1}{256} + \cdots$

Rajiv: $\frac{1}{8} + \frac{1}{64} + \frac{1}{512} + \cdots$

Using the infinite geometric series sum formula $S = \frac{a}{1 - r}$ with $a$ equal to the first term $\frac{1}{2}$ and ratio $r$ equal to $\frac{1}{8}$, the fraction of the block of cheese eaten by Sarika is about
$\frac{\frac{1}{2}}{1 - \frac{1}{8}} = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{1}{2} \cdot \frac{8}{7} = \frac{4}{7}.$