Difference between revisions of "2014 AMC 8 Problems/Problem 8"

 
(27 intermediate revisions by 14 users not shown)
Line 1: Line 1:
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid thei rguest speaker $ \underline{1} \underline{A} \underline{2}. What is the missing digit A of this 3-digit number?
+
==Problem==
 +
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker <math> \textdollar\underline{1} \underline{A} \underline{2}</math>. What is the missing digit <math>A</math> of this <math>3</math>-digit number?
  
 
<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4</math>
 
<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4</math>
 +
 +
==Solution 1==
 +
Since all the eleven members paid the same amount, that means that the total must be divisible by <math>11</math>. We can do some trial-and-error to get <math>A=3</math>, so our answer is <math>\textbf{(D) }3</math>.
 +
~SparklyFlowers
 +
 +
==Solution 2==
 +
We know that a number is divisible by <math>11</math> if the odd numbers added together minus the even numbers added together(or vice versa) is a multiple of <math>11</math>. So, we have <math>1+2-A</math> = a multiple of <math>11</math>. The only multiple that works here is <math>0</math>, as <math>11 \cdot 0 = 0</math>. Thus, <math>A = \textbf{(D) }3</math>
 +
~fn106068
 +
 +
==Video Solution==
 +
https://youtu.be/mHWTWk-xt0o ~savannahsolver
 +
 +
==See Also==
 +
{{AMC8 box|year=2014|num-b=7|num-a=9}}
 +
{{MAA Notice}}

Latest revision as of 09:15, 27 April 2022

Problem

Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\textdollar\underline{1} \underline{A} \underline{2}$. What is the missing digit $A$ of this $3$-digit number?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

Solution 1

Since all the eleven members paid the same amount, that means that the total must be divisible by $11$. We can do some trial-and-error to get $A=3$, so our answer is $\textbf{(D) }3$. ~SparklyFlowers

Solution 2

We know that a number is divisible by $11$ if the odd numbers added together minus the even numbers added together(or vice versa) is a multiple of $11$. So, we have $1+2-A$ = a multiple of $11$. The only multiple that works here is $0$, as $11 \cdot 0 = 0$. Thus, $A = \textbf{(D) }3$ ~fn106068

Video Solution

https://youtu.be/mHWTWk-xt0o ~savannahsolver

See Also

2014 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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 AJHSME/AMC 8 Problems and Solutions

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

Invalid username
Login to AoPS