2013 AMC 8 Problems/Problem 17: Difference between revisions

Revision as of 15:08, 23 December 2024

Problem

The sum of six consecutive positive integers is 2013. What is the largest of these six integers?

$\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350$

Solution 1

The arithmetic mean of these numbers is $\frac{\frac{2013}{3}}{2}=\frac{671}{2}=335.5$ . Therefore the numbers are $333$ , $334$ , $335$ , $336$ , $337$ , $338$ , so the answer is $\boxed{\textbf{(B)}\ 338}$

Solution 2

Let the $4^{\text{th}}$ number be $x$ . Then our desired number is $x+2$ .

Our integers are $x-3,x-2,x-1,x,x+1,x+2$ , so we have that $6x-3 = 2013 \implies x = \frac{2016}{6} = 336 \implies x+2 = \boxed{\textbf{(B)}\ 338}$ .

Solution 3

Let the first term be $x$ . Our integers are $x,x+1,x+2,x+3,x+4,x+5$ . We have, $6x+15=2013\implies x=333\implies x+5=\boxed{\textbf{(B)}\ 338}$

Solution 4

Since there are $6$ numbers, we divide $2013$ by $6$ to find the mean of the numbers. $\frac{2013}{6} = 335 \frac{1}{2}$ . Then, $335 \frac{1}{2} + \frac{1}{2} = 336$ (the fourth number). Fifth: $337$ ; Sixth: $\boxed {338}$ .

Solution 5

Let the $6th$ number be $x$ . Then our list is: $x-6+x-5+x-4+x-3+x-x-1=2013$ . Simplifying this gets you $6x-21=2013\implies 6x=2034$ , which means that $x = \boxed{\textbf{(B)}338}$ .

Video Solution by Pi Academy

https://youtu.be/KDEq2bcqWtM?si=M5fwa9pAdg1cQu0o

2013 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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 ==Solution 5==
-Let the <math>6th</math> number be <math>x</math>. Then our list is: <math>x-6+x-5+x-4+x-3+x-x-1=2013</math>. Simplifying this gets you <math>6x-21=2013\implies 6x=2034</math>, which means that <math>x = \boxed{\textbf{(B)}338}</math>
+Let the <math>6th</math> number be <math>x</math>. Then our list is: <math>x-6+x-5+x-4+x-3+x-x-1=2013</math>. Simplifying this gets you <math>6x-21=2013\implies 6x=2034</math>, which means that <math>x = \boxed{\textbf{(B)}338}</math>.
 ==Video Solution by Pi Academy==

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