2025 AMC 8 Problem 20

Question

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?

\dfrac{4}{7}

\dfrac{3}{5}

\dfrac{2}{3}

\dfrac{3}{4}

\dfrac{7}{8}

Answer: A

Concepts:geometric sequence summation

Difficulty rating: 1580

Small hint:

Sarika eats every third term of the halving sequence

Big hint:

Her shares form a geometric series with ratio

\frac{1}{8}

Video solution:

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Written solution:

Sarika gets

\frac{1}{2}

of the cheese in the first step.

Then Dev gets

\frac{1}{2^2}

of the cheese.

Then Rajiv gets

\frac{1}{2^3}

of the cheese.

Sarika then gets another

\frac{1}{2^4}

of the cheese.

This pattern continues. Ultimately, Sarika gets

\frac{1}{2} + \frac{1}{2^4} + \frac{1}{2^7} + \dots

which is the sum of an infinite geometric series with first term

a = \frac{1}{2}

and common ratio

r = \frac{1}{2^3}.

That sum is

\frac{a}{1-r} = \frac{\frac{1}{2}}{1 - \frac{1}{8}} = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{4}{7}

which is choice A.

2025 AMC 8 Problem 20

Answer: A

Difficulty rating: 1580

Small hint:

Big hint:

Video solution:

Written solution:

Problem 20 in other years