Difference between revisions of "2021 Fall AMC 12B Problems/Problem 3"

Latest revision as of 12:51, 30 December 2022

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:

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

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

Video Solution by Interstigation

https://youtu.be/p9_RH4s-kBA?t=291

~Interstigation

Video Solution

https://youtu.be/480KnrVnbOc

~Education, the Study of Everything

Video Solution by WhyMath

https://youtu.be/y4U3wv2euH8

~savannahsolver

Video Solution by TheBeautyofMath

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

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

~IceMatrix

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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

2021 Fall AMC 12B (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 12 Problems and Solutions

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

@@ Line 1: / Line 1: @@
-{{duplicate|[[2021 Fall AMC 10B Problems#Problem 4|2021 Fall AMC 10B #4]] and [[2021 Fall AMC 12B Problems#Problem 3|2021 Fall AMC 12B #3]]}}
+{{duplicate|[[2021 Fall AMC 10B Problems/Problem 4|2021 Fall AMC 10B #4]] and [[2021 Fall AMC 12B Problems/Problem 3|2021 Fall AMC 12B #3]]}}
 ==Problem==
@@ Line 6: / Line 6: @@
 <math>\textbf{(A)}\: 10\qquad\textbf{(B)} \: 30\qquad\textbf{(C)} \: 60\qquad\textbf{(D)} \: 100\qquad\textbf{(E)} \: 120</math>
-==Solution 1==
+==Solution 1 (Two Variables)==
-Let the temperature of Minneapolis be <math>M</math>, and that of St. Louis be <math>L</math>. We have <math>M=L+N</math>.
+At noon on a certain day, let <math>M</math> and <math>L</math> be the temperatures (in degrees) in Minneapolis and St. Louis, respectively. It follows that <math>M=L+N.</math>
-At <math>4{:}00</math>, either
+At <math>4{:}00,</math> we get
+<cmath>\begin{align*}
+|(M-5)-(L+3)| &= 2 \\
+|M-L-8| &= 2 \\
+|N-8| &= 2.
+\end{align*}</cmath>
+We have two cases:
+<ol style="margin-left: 1.5em;">
+  <li>If <math>N-8=2,</math> then <math>N=10.</math></li><p>
+  <li>If <math>N-8=-2,</math> then <math>N=6.</math></li><p>
+</ol>
+Together, the product of all possible values of <math>N</math> is <math>10\cdot6=\boxed{\textbf{(C)} \: 60}.</math>
-<math>(M-5)=2+(L+3)</math>
+~Wilhelm Z ~KingRavi ~MRENTHUSIASM
+== Solution 2 (One Variable) ==
+At noon on a certain day, the difference of temperatures in Minneapolis and St. Louis is <math>N</math> degrees.
+At <math>4{:}00,</math> the difference of temperatures in Minneapolis and St. Louis is <math>N-8</math> degrees.
+It follows that <cmath>|N-8|=2.</cmath>
+We continue with the casework in Solution 1 to get the answer <math>\boxed{\textbf{(C)} \: 60}.</math>
-or
+~Steven Chen (www.professorchenedu.com) ~MRENTHUSIASM
+==Video Solution by Interstigation==
+https://youtu.be/p9_RH4s-kBA?t=291
-<math>(M-5)+2=(L+3)</math>
+~Interstigation
-Plug in <math>M=L+N</math> to solve the two equations respectively to get <math>N=10</math> or <math>N=6</math>. Hence the answer is <math>60 \Rightarrow \boxed{(\textbf{C})  }.</math>
+==Video Solution==
+https://youtu.be/480KnrVnbOc
-~Wilhelm Z ~KingRavi ~MRENTHUSIASM
+~Education, the Study of Everything
+==Video Solution by WhyMath==
+https://youtu.be/y4U3wv2euH8
-== Solution 2 ==
+~savannahsolver
-<cmath>
-\begin{align*}
-| N - 5 - 3 | = 2 .
-\end{align*}
-</cmath>
-Hence, <math>N = 10</math> or 6.
+==Video Solution by TheBeautyofMath==
+For AMC 10: https://youtu.be/lC7naDZ1Eu4?t=670
-Therefore, the answer is <math>\boxed{\textbf{(C) }60}</math>.
+For AMC 12: https://youtu.be/yaE5aAmeesc?t=379
-~Steven Chen (www.professorchenedu.com)
+~IceMatrix
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