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?
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