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Difference between revisions of "2023 AMC 10B Problems/Problem 5"

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==Problem==
  
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Maddy and Lara see a list of numbers written on a blackboard. Maddy adds <math>3</math> to each number in the list and finds that the sum of her new numbers is <math>45</math>. Lara multiplies each number in the list by <math>3</math> and finds that the sum of her new numbers is also <math>45</math>. How many numbers are written on the blackboard?
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<math>\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10</math>
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==Solution==
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Let there be <math>x</math> numbers in the list of numbers, and let their sum be <math>S</math>. Then we have the following
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<math>S+3x=45</math>
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<math>3S=45</math>
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From the second equation, <math>S=15</math>, so <math>15+3x=45</math>. Solving, we find <math>x=\boxed{\textbf{(E) }10}.</math>

Revision as of 15:03, 15 November 2023

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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