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?

$\hspace*{5mm}\text{(A) } \frac{4}{7} \quad \text{(B) } \frac{3}{5} \quad \text{(C) } \frac{2}{3} \quad \text{(D) } \frac{3}{4} \quad \text{(E) } \frac{7}{8}$

Video Solution

Key Idea: Let $x$ be the fraction eaten by Sarika. Then Dev eats $\frac{x}{2}$ and Rajiv eats $\frac{x}{4}$ . Hence $x + \frac{x}{2} + \frac{x}{4} = 1$ so solving for $x$ we get $x = \frac{4}{7}$ , $\boxed{\textbf{(A)}}$ .

Video Link: https://www.youtube.com/watch?v=VUR5VYabbrc

Solution 1

WLOG, let the amount of total cheese be $1$ . Then Sarika eat $\dfrac{1}{2}$ , Dev eats $\dfrac{1}{4}$ , Rajiv eats $\dfrac{1}{8}$ , Sarika eats $\dfrac{1}{16}$ and so on. After a couple for attempts, we see that Sarika eats cheese in an infinite geometric sequence with first term $\dfrac{1}{2}$ and common ratio of $\dfrac{1}{8}$ . Therefore, we use the infinite geometric sequence formula and get $\dfrac{\dfrac{1}{2}}{1-\dfrac{1}{8}}=\dfrac{\dfrac{1}{2}}{\dfrac{7}{8}}=\dfrac{4}{7}$ To find how much Sarika eats, we just divide this by our original total and get $\dfrac{\dfrac{4}{7}}{1}=1$ .

Therefore, Sarika eats $\frac{4}{7}$ $\boxed{\textbf{(A)}}$ of the cheese.

~athreyay

Solution 2 (If you forgot the infinite geometric series formula)

At first, Sarika eats $\frac{1}{2}$ of the cheese, and then after $2$ more people eat, there is $\frac{1}{8}$ of the cheese remaining, so Sarika eats $\frac{1}{16}$ of the cheese. Continuing this pattern until a reasonable amount, we get new fractions of $\frac{1}{128}$ and $\frac{1}{1024}$ . Adding these fractions together yields $\frac{1}{2}+\frac{1}{16}+\frac{1}{128}+\frac{1}{1024} = \frac{512+64+8+1}{1024} = \frac{585}{1024}$ . Approximating this answer yields about $0.5713$ which is about $\frac{4}{7}$ $\boxed{\textbf{(A)}}$ of the cheese.