M38-04

Question

Mbappé has two types of sneakers: red and blue. When packing all of his sneakers for the World Cup, he puts an equal number of sneakers in each of his suitcases. The first suitcase has 27% of all his red sneakers. Does the first suitcase have less than 25% of his blue sneakers ?

(1) No suitcase has more red sneakers than the first one.

(2) The first suitcase has between 23% and 27% of his blue sneakers.

Show Answer

Official Answer

D

(1) No suitcase has more red sneakers than the first one.
 
(2) The first suitcase has between 23% and 27% of his blue sneakers.

Bunuel · Accepted Answer

Official Solution:

Mbappé has two types of sneakers: red and blue. When packing for the World Cup, he puts equal number of sneakers in each of his suitcases. The first suitcase has 27% of all his red sneakers. Does the first suitcase have less than 25% of his blue sneakers ? 
 
Say Mbappé has x suitcases. Since he put   equal number   of sneakers in each of his suitcases, then each suitcase has 100/x% of all his sneakers. For example:
  
 If Mbappé has 10 suitcases, then each will have 100/10 = 10% of all his sneakers .
  
Next, notice that, if a suitcase has LESS than  \frac{100}{x}\%  of Mbappé's   red   sneakers, it must have MORE than  \frac{100}{x}\%  of his   blue   sneakers (and vise-versa). For example:
  
 If Mbappé has 10 suitcases, each having  10\%  of all his sneakers, and one specific suitcase has  5\%  of his   red   sneakers, then it must have MORE than  10\%  of his   blue   sneakers (if a suitcase has less than  10\%  of red sneakers and less than  10\%  of blue sneakers, it cannot have  10\%  of all sneakers). 
  
(1) No suitcase has more red sneakers than the first one.
 
How many suitcases can Mbappé have? Can he have 3? No, because in this case at least one of his suitcases must have MORE than 100/3 ≈ 33% of his red sneakers (all of his 3 suitcases cannot have less than 33% of his read sneakers). So, Mbappé must have 4 or more suitcases. This means that  each of them has LESS than 25% of all his sneakers . For example:
  
If he has 4 suitcases then each has 100/4 = 25% of all his sneakers; 
 
If he has 5 suitcases then each has 100/5 = 20% of all his sneakers; 
 
...
  
From the stem we know that the first suitcase has 27% (more than 25%) of all his   red   sneakers, thus it must have less than 25% of his   blue   sneakers. Sufficient.
 
(2) The first suitcase has between 23% and 27% of his blue sneakers.
 
From, the stem we now that the first suitcase has 27% of all his   red   sneakers and from this statement that the first suitcase has more than 23% of his   blue   sneakers:
  
 0.23 \leq \frac{1}{x} \leq 0.27 
  
Only integer which satisfies the above is  x=4 . So, Mbappé has 4 suitcases and each has 25% of all his sneakers. Since the first suitcase has 27% (more than 25%) of all his   red   sneakers, then it must have less than 25% of his   blue   sneakers. Sufficient.
 
Answer: D

Krishnak13 · Answer

Fantastic explanation. Could you please share some problems related to this concept

The learning that I can take from this question are:

1. If B1, B2 are two bags containing equal number of balls and each contain Red and Blue bags
Inference-1:
Number of total balls in each bag B1, B2 = 100/2 = 50%

Now, Let's say it is given as B1 has 25% red balls
Inference-2:
Since we know total balls in B1 = 50% and 25% red balls, the number of blue balls in bag-1 must be >50% for sure

2. If 100 coins are distributed among 'n' bags
At-least one bag will have coins greater than or equal to (100/n)
At-least one bag will have coins less than or equal to (100/n)
Application:
1. If say B1 has 27% of red balls in B1 and is the highest number of red balls in any bag.
=> If there are two bags, then B2 must have 73%
If there are 3 bags, B1 is given as 27% = <33% (=100/3) => the other bag must have more than 33%. But it contradicts the given condition, which is B1 has highest red balls.
If there are 4 bags, B1 is given as 27% = >25% (=100/4) => The other bag must have <25%. And 25% < % of red balls in B1. hence, min number of bags = 4.
Now once we know if there are blue balls => at-least one bag must have (<100/4 = < 25%) and have at-least one bag that has (>100/4 =>25%)

ZZ800 · Answer

Would this be an 800 question? Is there a way to check the difficulty level of all of these questions?

Bunuel · Answer

Yes, it's a very hard question. You can check stats on it in the first post here: https://gmatclub.com/forum/gmat-club-wo ... 94961.html

pryga · Answer

Hey! 
this part " he puts an equal number of sneakers in each of his suitcases" was a little confusing for me: did it mean that it was equal number of the sneaker of each color or equal total number of sneakers ( 3:3, 5:5 ect) or (3:7 5:5 ect)

Bunuel · Answer

It means an equal  total number of sneakers  in each suitcase, not an equal number of each color.

So arrangements like 3 red + 7 blue and 5 red + 5 blue are allowed, since both suitcases contain 10 sneakers in total.

M38-04

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