Difference between revisions of "2023 AMC 10B Problems/Problem 5"
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<math>\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }8\qquad\textbf{(D) }6\qquad\textbf{(E) }9</math> | <math>\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }8\qquad\textbf{(D) }6\qquad\textbf{(E) }9</math> | ||
Revision as of 19:23, 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?
On my copy of the AMC 10B, the order of the answers is different:
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
~Mintylemon66
Solution 2
Let where
represents the
th number written on the board. Lara's multiplied each number by
, so her sum will be
. This is the same as
. We are given this quantity is equal to
, so the original numbers add to
. Maddy adds
to each of the
terms which yields,
. This is the same as the sum of the original series plus
. Setting this equal to
,
~vsinghminhas