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?
Answer Choices:
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E.
Solution:
Let be the length of the list and be the sum of the numbers in the list. Because Lara multiplied each number in the list by 3 , the sum of the numbers in the list also increased by a factor of 3. It follows that . Additionally, because Maddy added 3 to each number in the list, the overall sum became . Setting this equal to 45 and solving yields . For instance, one possible list is .
The problems on this page are the property of the MAA's American Mathematics Competitions