2025 AMC 8 Problems/Problem 20
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2 (Estimation)
- 4 Solution 3 Speed
- 5 Solution 4 (Basically Video Solution)
- 6 Video Solution by Pi Academy
- 7 Video Solution 2 by SpreadTheMathLove
- 8 Video Solution (A Clever Explanation You’ll Get Instantly)
- 9 Video Solution by Thinking Feet
- 10 A Concise Video Solution in 56 Seconds by Dr. Xue's Math School
- 11 See Also
Problem
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?
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:
![\[\frac{1}{2}.\]](http://latex.artofproblemsolving.com/c/8/7/c872e3b1287d7dada916f35aba2e4d63b9087374.png)
Second Round: Dev eats half of what remains after Sarika's turn, which is:
![\[\frac{1}{4}.\]](http://latex.artofproblemsolving.com/7/7/b/77bd8903e7a6fcea82d6ac29097edee710f73659.png)
Third Round: Rajiv eats half of the remaining cheese after Dev’s turn, which is:
![\[\frac{1}{8}.\]](http://latex.artofproblemsolving.com/3/b/d/3bd37d4989807ecd8fd332a36de7a6e447e3afa1.png)
At the end of the first round, the total cheese eaten is:
![\[\frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{7}{8}.\]](http://latex.artofproblemsolving.com/3/7/0/3703d55cb30c8707e14171e46d5779d2fb7124c8.png)
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:
![\[\frac{1}{2} + \frac{1}{16} + \frac{1}{128} + \cdots\]](http://latex.artofproblemsolving.com/3/3/6/336a2421947627836579f96c4b540f54e198afb4.png)
This is a geometric series with first term and common ratio 
. The sum 
of this infinite geometric series is given by the formula:
![\[S = \frac{a}{1 - r},\]](http://latex.artofproblemsolving.com/3/5/1/3510685efb7b9739314cec33c22f45a3f91a1007.png)
where 
is the first term and 
is the common ratio. Substituting 
and 
:
![\[S = \frac{\frac{1}{2}}{1 - \frac{1}{8}} = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{1}{2} \times \frac{8}{7} = \frac{4}{7}.\]](http://latex.artofproblemsolving.com/e/3/7/e3742c1d45b63d9fbe0843d0660d2eb1cb180b36.png)
Thus, Sarika eats 
of the original block of cheese. The correct answer is:
![\[\boxed{\textbf{(A) } \frac{4}{7}}.\]](http://latex.artofproblemsolving.com/a/9/c/a9c994e0d960b9161ac6b48c938da587fadc3636.png)
~ aoum
Solution 2 (Estimation)
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 
.
~Sigmacuber
Solution 3 Speed
Her first 
bites will be and 
. Everything else won't matter, so the answer will be closest to 
, .
~Moonwatcher22
Solution 4 (Basically Video Solution)
Let Sarika eat a total of 
units of cheese. We can then see that Dev will eat a total of 
units and Rajiv will eat 
units. Our desired answer is 
, or .
~Irfans123
Video Solution by Pi Academy
https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK
Video Solution
Key Idea: Let be the fraction eaten by Sarika. Then Dev eats and Rajiv eats . Hence 
so solving for we get ![\[x = \boxed{\textbf{(A) }\frac{4}{7}}\]](http://latex.artofproblemsolving.com/5/4/6/546abf0a9fc90f30673b9835f6b3401950f00a3e.png)
.
Video Link: https://www.youtube.com/watch?v=VUR5VYabbrc
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/VP7g-s8akMY?si=E-EnGyXALPVaXvPy&t=2698 ~hsnacademy
Video Solution by Thinking Feet
A Concise Video Solution in 56 Seconds by Dr. Xue's Math School
Work smarter, work more efficiently, not just harder.
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
