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

Latest revision as of 13:37, 3 July 2023

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 Yasin's blood expires in $10! \div 86400 = 42$ 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 Solution (HOW TO THINK CREATIVELY)

https://youtu.be/bPfLeXu9kx0

Education, the Study of Everything

Video Solutions

https://youtu.be/vO-ELYmgRI8

https://youtu.be/FbSYTL8tPwo

~savannahsolver

@@ Line 3: / Line 3: @@
 A unit of blood expires after <math>10!=10\cdot 9 \cdot 8 \cdots 1</math> seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
-<math>\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{Febuary 11}\qquad\textbf{(E) }\text{Febuary 12}</math>
+<math>\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}</math>
-== Solution 1 ==
+== Solution ==
-There are <math>10!</math> seconds that the blood has before expiring and there are <math>60 \cdot 60 \cdot 24</math> seconds in a day. Dividing <math>\frac{10!}{60 \cdot 60 \cdot 24}</math> gives <math>42</math> days. <math>42</math> days after January 1 is <math>\fbox{\textbf{(E) }\text{Febuary 12}}</math>.
+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.
+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.
+Convert minutes to hours by dividing by <math>60</math> again: <math>9\cdot 8\cdot 7\cdot 2</math> hours.
+Convert hours to days by dividing by <math>24</math>: <math>3\cdot 7\cdot 2 = 42</math> days.
-== Solution 2 ==
+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 in <math>\fbox{\textbf{(E) }\text{February 12}}</math>.
-<math>10! = 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1</math>.
+~nosysnow ~Max0815
-Convert <math>10!</math> seconds to minutes: <math>9\cdot 8\cdot 7\cdot 5\cdot 4\cdot 3\cdot 2</math> minutes.
-Convert minutes to hours: <math>9\cdot 8\cdot 7\cdot 2</math> hours.
-Convert hours to days: <math>3\cdot 7\cdot 2 = 42</math> days.
-<math>42</math> days after January 1 is <math>\fbox{\textbf{(E) }\text{Febuary 12}}</math>
+==Solution 2 (Concise)==
-   ~Nosysnow | wonsysoN~
+There are <math>60 \cdot 60 \cdot 24 = 86400</math> seconds in a day, which means that Yasin's blood expires in <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
+==Video Solution (HOW TO THINK CREATIVELY)==
+https://youtu.be/bPfLeXu9kx0
+Education, the Study of Everything
+==Video Solutions==
+https://youtu.be/vO-ELYmgRI8
+https://youtu.be/FbSYTL8tPwo
+~savannahsolver
 == See Also ==

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