2023 AMC 10B Problems/Problem 5
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
~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
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=SUnhwbA5_So
See also
2023 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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