2023 AMC 10B Problems/Problem 5: Difference between revisions
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==Problem== | |||
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? | |||
<math>\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10</math> | |||
==Solution== | |||
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 | |||
<math>S+3x=45</math> | |||
<math>3S=45</math> | |||
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 
to each number in the list and finds that the sum of her new numbers is 
. Lara multiplies each number in the list by and finds that the sum of her new numbers is also . How many numbers are written on the blackboard?
Solution
Let there be 
numbers in the list of numbers, and let their sum be 
. Then we have the following
From the second equation, 
, so 
. Solving, we find 
