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2021 Fall AMC 12B Problems/Problem 3

Revision as of 02:02, 24 November 2021 by Wilhelmz (talk | contribs) (Created page with "==Problem== At noon on a certain day, Minneapolis is <math>N</math> degrees warmer than St. Louis. At <math>4{:}00</math> the temperature in Minneapolis has fallen by <math>5<...")
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Problem

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$

$(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

Solution 1

Let the temperature of Minneapolis be $M$, and that of St. Louis be $L$. We have $M=L+N$.

At $4{:}00$, either

$(M-5)=2+(L+3)$

or

$(M-5)+2=(L+3)$

Plug in $M=L+N$ to solve the two equations respectively to get $N=10$ or $N=6$. Hence the answer is $\boxed{(\textbf{C}) 60}.$

~Wilhelm Z

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