2018 AMC 10A Problems/Problem 3

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, and so $10! \div 86400 = 42$ 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

Education, the Study of Everything

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
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2018 AMC 10A Problems/Problem 3

Contents

Problem

Solution

Solution 2 (Concise)

Video Solutions

See Also