I am practicing for the exam GRE General. I met this question in the book Manhattan 5lb but I cannot get its answer.
Question:
Maribel must divide 60 candies among herself and her 12 cousins, although there is no requirement that the candies be divided equally. If Maribel is to have more candies than everyone else, what is the least number of candies she could have?
Manhattan 5lb's answer:
- One good way to solve this problem is to first evenly divide the candies, and then give Maribel more (by taking candies away from the others) until the conditions of the problem are met. 60 candies divided by 13 people = 4.615….
So, if Maribel had 5 candies, would she have more than everyone else? Well, if the 12 cousins each had only 4 candies, that’s 48 candies total plus Maribel’s 5 = 53 candies. 7 candies are unaccounted for, meaning that some other cousin or cousins will have to have the same as or more than Maribel.
If Maribel had 6 candies, would she have more than everyone else? Well, if the 12 cousins each had only 5 candies, they would have 60. Since there are only 60 candies total, Maribel could have 6 and the other cousins could have 4 or 5 each. The answer is 6.
Keep in mind that when the question asks for a minimum, you can’t just go messing around until you find a case that works — to find the smallest case that works, you need to start small and work up from there.
However, I disagree with this answer. In my opinion, the correct answer should be 9.
If Maribel has 5 candies, there are 7 candies left unaccounted, so one of her cousins may have 7 + 4 = 11 candies.
If Maribel has 6 candies, there are 6 candies left unaccounted, so one of her cousins may have 6 + 4 = 10 candies.
If Maribel has 7 candies, there are 5 candies left unaccounted, so one of her cousins may have 5 + 4 = 9 candies.
If Maribel has 8 candies, there are 4 candies left unaccounted, so one of her cousins may have 4 + 4 = 8 candies.
If Maribel has 9 candies, there are 3 candies left unaccounted, so one of her cousins may have 3 + 4 = 7 candies.
So, where am I misunderstanding?
Thank you very much in advance!
The question specifies that Marabel is to have more candies than everyone else. This means that you can divide the candies any way you like, which also means that you are free not to give another girl the ten candies you mentioned. Simply giving 6 cousins 5 candies and 6 cousins 4 candies, you are sure that Marabel, having 6 candies, has more than all of her cousins. Note that if you want to take into account the worst-case scenario, one of the cousins could potentially have all remaining candies; in this case, Marabel would require 31 candies in order to be sure to have more than all her cousins.