# 2021 Fall AMC 12B Problems/Problem 3

The following problem is from both the 2021 Fall AMC 10B #4 and 2021 Fall AMC 12B #3, so both problems redirect to this page.

## 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 (Two Variables)

At noon on a certain day, let $M$ and $L$ be the temperatures (in degrees) in Minneapolis and St. Louis, respectively. It follows that $M=L+N.$

At $4{:}00,$ we get \begin{align*} |(M-5)-(L+3)| &= 2 \\ |M-L-8| &= 2 \\ |N-8| &= 2. \end{align*} We have two cases:

1. If $N-8=2,$ then $N=10.$
2. If $N-8=-2,$ then $N=6.$

Together, the product of all possible values of $N$ is $10\cdot6=\boxed{\textbf{(C)} \: 60}.$

~Wilhelm Z ~KingRavi ~MRENTHUSIASM

## Solution 2 (One Variable)

At noon on a certain day, the difference of temperatures in Minneapolis and St. Louis is $N$ degrees.

At $4{:}00,$ the difference of temperatures in Minneapolis and St. Louis is $N-8$ degrees.

It follows that $$|N-8|=2.$$ We continue with the casework in Solution 1 to get the answer $\boxed{\textbf{(C)} \: 60}.$

~Steven Chen (www.professorchenedu.com) ~MRENTHUSIASM

~Interstigation

## Video Solution

~Education, the Study of Everything

~savannahsolver

## Video Solution by TheBeautyofMath

For AMC 10: https://youtu.be/lC7naDZ1Eu4?t=670

For AMC 12: https://youtu.be/yaE5aAmeesc?t=379

~IceMatrix