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2023 AMC 10B Problems/Problem 5

Revision as of 16:03, 15 November 2023 by Mintylemon66 (talk | contribs)

Problem

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard?

$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Solution

Let there be $x$ numbers in the list of numbers, and let their sum be $S$. Then we have the following

$S+3x=45$

$3S=45$

From the second equation, $S=15$, so $15+3x=45$. Solving, we find $x=\boxed{\textbf{(E) }10}.$

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