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Difference between revisions of "2018 AMC 10A Problems/Problem 3"

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== Solution 2 ==
 
== Solution 2 ==
  
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<math>10! = 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1</math>.
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The problem says there are <math>10! = 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1</math> seconds.
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Convert <math>10!</math> seconds to minutes: <math>9\cdot 8\cdot 7\cdot 5\cdot 4\cdot 3\cdot 2</math> minutes.
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Convert <math>10!</math> seconds to minutes by dividing by <math>60</math>: <math>9\cdot 8\cdot 7\cdot 5\cdot 4\cdot 3\cdot 2</math> minutes.
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Convert minutes to hours: <math>9\cdot 8\cdot 7\cdot 2</math> hours.
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Convert minutes to hours by again, dividing by <math>60</math>: <math>9\cdot 8\cdot 7\cdot 2</math> hours.
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Convert hours to days: <math>3\cdot 7\cdot 2 = 42</math> days.
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Convert hours to days by dividing by <math>24</math>: <math>3\cdot 7\cdot 2 = 42</math> days.
  
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<math>42</math> days after January 1 is <math>\fbox{\textbf{(E) }\text{February 12}}</math>
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Now we need to count <math>42</math> days after January 1. Since we start on Jan. 1, then we can't count that as a day itself. When we reach Jan. 31(The end of the month), we only have counted 30 days. <math>42 - 30 = 12</math>. Count <math>12</math> more days, resulting <math>\fbox{\textbf{(E) }\text{February 12}}</math>
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Keep in mind that you start on January 1st so you can't count it as a day. By the time you get to noon on January 31, only 30 days have past. Them you can continue counting like you normally would to February 12.
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~nosysnow and Max0815
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  ~Nosysnow
 
  
 
== See Also ==
 
== See Also ==

Revision as of 12:19, 17 February 2018

Problem

A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?

$\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$

Solution 1

There are $10!$ seconds that the blood has before expiring and there are $60 \cdot 60 \cdot 24$ seconds in a day. Dividing $\frac{10!}{60 \cdot 60 \cdot 24}$ gives $42$ days. $42$ days after January 1 is $\fbox{\textbf{(E) }\text{February 12}}$.

Solution 2

The problem says there are $10! = 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ seconds. Convert $10!$ seconds to minutes by dividing by $60$: $9\cdot 8\cdot 7\cdot 5\cdot 4\cdot 3\cdot 2$ minutes. Convert minutes to hours by again, dividing by $60$: $9\cdot 8\cdot 7\cdot 2$ hours. Convert hours to days by dividing by $24$: $3\cdot 7\cdot 2 = 42$ days.

Now we need to count $42$ days after January 1. Since we start on Jan. 1, then we can't count that as a day itself. When we reach Jan. 31(The end of the month), we only have counted 30 days. $42 - 30 = 12$. Count $12$ more days, resulting $\fbox{\textbf{(E) }\text{February 12}}$ 

~nosysnow and Max0815

See Also

2018 AMC 10A (Problems • Answer Key • Resources)
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Problem 2 		Followed by
Problem 4
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All AMC 10 Problems and Solutions

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