2018 AMC 10A Problems/Problem 3: Difference between revisions

Revision as of 09:40, 28 September 2022

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

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 dividing by $60$ again: $9\cdot 8\cdot 7\cdot 2$ 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 in $\fbox{\textbf{(E) }\text{February 12}}$ .

~nosysnow ~Max0815

Solution 2 (Concise)

There are $60 \cdot 60 \cdot 24 = 86400$ seconds in a day, which means that the blood expires in $42$ days. $10! \div 86400 = 42$ days. Since there are $31$ days in January (consult a calendar), then $42-31+1$ (Jan 1 doesn't count) is $12$ days into February, so $\boxed{\textbf{(E) }\text{February 12}}$ .

~MrThinker

Video Solutions

https://youtu.be/vO-ELYmgRI8

https://youtu.be/FbSYTL8tPwo

~savannahsolver

https://youtu.be/bPfLeXu9kx0

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2018 AMC 10A (Problems • Answer Key • Resources)
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 ==Solution 2 (Concise)==
-There are <math>60 \cdot 60 \cdot 24 = 86400</math> seconds in a day, and so <math>10! \div 86400 = 42</math> days. Since there are <math>31</math> days in January (consult a calendar), then <math>42-31+1</math> (Jan 1 doesn't count) is <math>12</math> days into February, so <math>\boxed{\textbf{(E) }\text{February 12}}</math>.
+There are <math>60 \cdot 60 \cdot 24 = 86400</math> seconds in a day, which means that the blood expires in <math>42</math> days. <math>10! \div 86400 = 42</math> days. Since there are <math>31</math> days in January (consult a calendar), then <math>42-31+1</math> (Jan 1 doesn't count) is <math>12</math> days into February, so <math>\boxed{\textbf{(E) }\text{February 12}}</math>.
 ~MrThinker

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