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Rita has a bowl containing three types of candies: cherry, mango, and lime. If there are 82 candies in total, out of which 29 are cherry, how many mango candies are in the bowl?
(1) The minimum number of candies Rita must pick at random from the bowl to ensure getting at least one candy of each flavor is 69.
(2) There are fewer lime candies than mango candies.
(1) The minimum number of candies Rita must pick at random from the bowl to ensure getting at least one candy of each flavor is 69.
(2) There are fewer lime candies than mango candies.
13 Jan 2026, 10:23
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Official Solution:
Rita has a bowl containing three types of candies: cherry, mango, and lime. If there are 82 candies in total, out of which 29 are cherry, how many mango candies are in the bowl?
(1) The minimum number of candies Rita must pick at random from the bowl to ensure getting at least one candy of each flavor is 69.
Total is 82 with 29 cherry, so mango + lime = 53.
For the minimum number of picks needed to guarantee at least one candy of each flavor, we consider the worst-case scenario: Rita could keep picking all candies from only the two largest flavor groups first, before finally getting a candy of the third flavor on the next pick. So the minimum number needed to guarantee all three flavors equals the total of the two largest groups plus 1. Since that minimum is 69, the two largest groups must total 68.
Since mango + lime = 53 and cherry = 29, mango and lime cannot both be greater than 29. So cherry must be one of the two largest groups. So either mango + cherry = 68, giving mango = 39 and lime = 14, or lime + cherry = 68, giving lime = 39 and mango = 14. Not sufficient.
For example, if there are 39 mango, 29 cherry, and 14 lime candies, then to guarantee getting at least one candy of each flavor Rita must pick 69 candies, because in the worst case she could pick 39 mango and 29 cherry first. That is 68 candies and she still would not have all three flavors. The next pick must be lime, so with the 69th pick she gets all three flavors.
(2) There are fewer lime candies than mango candies.
We know mango + lime = 53 and lime is less than mango. That still allows many possible values for mango. Not sufficient.
(1)+(2) From (1) mango = 39 and lime = 14, or lime = 39 and mango = 14. From (2), lime < mango. Therefore, mango must be 39. So there are 39 mango candies. Sufficient.
Answer: C
Rita has a bowl containing three types of candies: cherry, mango, and lime. If there are 82 candies in total, out of which 29 are cherry, how many mango candies are in the bowl?
(1) The minimum number of candies Rita must pick at random from the bowl to ensure getting at least one candy of each flavor is 69.
Total is 82 with 29 cherry, so mango + lime = 53.
For the minimum number of picks needed to guarantee at least one candy of each flavor, we consider the worst-case scenario: Rita could keep picking all candies from only the two largest flavor groups first, before finally getting a candy of the third flavor on the next pick. So the minimum number needed to guarantee all three flavors equals the total of the two largest groups plus 1. Since that minimum is 69, the two largest groups must total 68.
Since mango + lime = 53 and cherry = 29, mango and lime cannot both be greater than 29. So cherry must be one of the two largest groups. So either mango + cherry = 68, giving mango = 39 and lime = 14, or lime + cherry = 68, giving lime = 39 and mango = 14. Not sufficient.
For example, if there are 39 mango, 29 cherry, and 14 lime candies, then to guarantee getting at least one candy of each flavor Rita must pick 69 candies, because in the worst case she could pick 39 mango and 29 cherry first. That is 68 candies and she still would not have all three flavors. The next pick must be lime, so with the 69th pick she gets all three flavors.
(2) There are fewer lime candies than mango candies.
We know mango + lime = 53 and lime is less than mango. That still allows many possible values for mango. Not sufficient.
(1)+(2) From (1) mango = 39 and lime = 14, or lime = 39 and mango = 14. From (2), lime < mango. Therefore, mango must be 39. So there are 39 mango candies. Sufficient.
Answer: C
