Difference between revisions of "2025 AMC 8 Problems/Problem 20"
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| + | ==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? | 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? | ||
| − | <math>\ | + | <math>\textbf{(A)}\ \frac{4}{7} \qquad \textbf{(B)}\ \frac{3}{5} \qquad \textbf{(C)}\ \frac{2}{3} \qquad \textbf{(D)}\ \frac{3}{4} \qquad \textbf{(E)}\ \frac{7}{8}</math> |
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| + | ==Video Solution== | ||
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| + | Key Idea: Let <math>x</math> be the fraction eaten by Sarika. Then Dev eats <math>\frac{x}{2}</math> and Rajiv eats <math>\frac{x}{4}</math>. Hence <math>x + \frac{x}{2} + \frac{x}{4} = 1</math> so solving for <math>x</math> we get <math>x = \frac{4}{7}</math>, <math>\boxed{\textbf{(A)}}</math>. | ||
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| + | Video Link: https://www.youtube.com/watch?v=VUR5VYabbrc | ||
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| + | ==Solution 1== | ||
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| + | WLOG, let the amount of total cheese be <math>1</math>. Then Sarika eat <math>\dfrac{1}{2}</math>, Dev eats <math>\dfrac{1}{4}</math>, Rajiv eats <math>\dfrac{1}{8}</math>, Sarika eats <math>\dfrac{1}{16}</math> and so on. After a couple for attempts, we see that Sarika eats cheese in an infinite geometric sequence with first term <math>\dfrac{1}{2}</math> and common ratio of <math>\dfrac{1}{8}</math>. Therefore, we use the infinite geometric sequence formula and get <cmath>\dfrac{\dfrac{1}{2}}{1-\dfrac{1}{8}}=\dfrac{\dfrac{1}{2}}{\dfrac{7}{8}}=\dfrac{4}{7}</cmath> To find how much Sarika eats, we just divide this by our original total and get <math>\dfrac{\dfrac{4}{7}}{1}=1</math>. | ||
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| + | Therefore, Sarika eats <math>\frac{4}{7}</math> <math>\boxed{\textbf{(A)}}</math> of the cheese. | ||
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| + | ~athreyay | ||
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| + | ==Solution 2 (If you forgot the infinite geometric series formula)== | ||
| + | At first, Sarika eats <math>\frac{1}{2}</math> of the cheese, and then after <math>2</math> more people eat, there is <math>\frac{1}{8}</math> of the cheese remaining, so Sarika eats <math>\frac{1}{16}</math> of the cheese. Continuing this pattern until a reasonable amount, we get new fractions of <math>\frac{1}{128}</math> and <math>\frac{1}{1024}</math>. Adding these fractions together yields <math>\frac{1}{2}+\frac{1}{16}+\frac{1}{128}+\frac{1}{1024} = \frac{512+64+8+1}{1024} = \frac{585}{1024}</math>. Approximating this answer yields about <math>0.5713</math> which is about <math>\frac{4}{7}</math> <math>\boxed{\textbf{(A)}}</math> of the cheese. | ||
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| + | ~Soupboy0 | ||
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| + | ==Video Solution 1 by SpreadTheMathLove== | ||
| + | https://www.youtube.com/watch?v=jTTcscvcQmI | ||
| + | |||
| + | ==Video Solution by Thinking Feet== | ||
| + | https://youtu.be/PKMpTS6b988 | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2025|num-b=19|num-a=21}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 19:11, 30 January 2025
Contents
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?
Video Solution
Key Idea: Let 
be the fraction eaten by Sarika. Then Dev eats 
and Rajiv eats 
. Hence 
so solving for we get 
, 
.
Video Link: https://www.youtube.com/watch?v=VUR5VYabbrc
Solution 1
WLOG, let the amount of total cheese be 
. Then Sarika eat 
, Dev eats 
, Rajiv eats 
, Sarika eats 
and so on. After a couple for attempts, we see that Sarika eats cheese in an infinite geometric sequence with first term and common ratio of . Therefore, we use the infinite geometric sequence formula and get ![\[\dfrac{\dfrac{1}{2}}{1-\dfrac{1}{8}}=\dfrac{\dfrac{1}{2}}{\dfrac{7}{8}}=\dfrac{4}{7}\]](http://latex.artofproblemsolving.com/0/2/5/02549b7cb383d8ca2ba8555f8013f5a9649395fc.png)
To find how much Sarika eats, we just divide this by our original total and get 
.
Therefore, Sarika eats 
of the cheese.
~athreyay
Solution 2 (If you forgot the infinite geometric series formula)
At first, Sarika eats 
of the cheese, and then after 
more people eat, there is 
of the cheese remaining, so Sarika eats 
of the cheese. Continuing this pattern until a reasonable amount, we get new fractions of 
and 
. Adding these fractions together yields 
. Approximating this answer yields about 
which is about of the cheese.
~Soupboy0
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution by Thinking Feet
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
