Of the shoe stores in City X, 30 carry Brand A

Question

Of the shoe stores in City X, 30 carry Brand A shoes, 40 carry Brand B shoes, and 25 carry Brand C shoes. If each store carries at least one of the brands, 32 of the stores carry two of the three shoe brands, and none of the stores carry all three, how many shoe stores are there in City X?

OA:

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63

Does anyone know how to solve this using a venn diagram? I'm having trouble placing the 32 in the right place.

Thanks!

Jose

OA:
 63

Does anyone know how to solve this using a venn diagram? I'm having trouble placing the 32 in the right place.

Thanks!

Jose

Bunuel · Answer

https://gmatclub.com/forum/download/file.php?id=18933 30 carry Brand A --> A=30; 40 carry Brand B --> B=40; 25 carry Brand C --> C=25; Each store carries at least one --> Neither=0. 32 of the stores carry two of the three shoe brands --> EXACTLY 2=32 (d+e+f). None of the stores carry all three --> All three=0 (g=0) Total = A + B + C - (sum \ of \ EXACTLY \ 2-group \ overlaps) - 2*(all \ three) + Neither=30+40+25-32-2*0+0=63 (explained here: advanced-overlapping-sets-problems-144260.html). Theory on Overlapping Sets: advanced-overlapping-sets-problems-144260.html (ADVANCED OVERLAPPING SETS PROBLEMS) how-to-draw-a-venn-diagram-for-problems-98036.html All DS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=45 All PS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=65

Viven · Answer

This much easier without using Venn diagrams

Total number of shoes =  30+ 40 + 25 = 95

32 stores carry exactly two of the shoe, thus the remaining shoes are =  95 -32*2 = 31

No store carries all three, so the remaining 31 shoes have to be carried by 31 stores

So the total number of stores = 32 + 31 = 63

PareshGmat · Answer

Refer Venn Diagram below:

From information given, we derive at 3 equations:

a+x+y = 30............... (i)
b+y+z=40.................(ii)
c+y+z=25..................(iii)

Adding (i), (ii) & (iii)

a+b+c+2(x+y+z) = 95

x+y+z = 32 (From the given information)
so, a+b+c = 31

So, total stores = 32+31 = 63

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Of the shoe stores in City X, 30 carry Brand A

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Of the shoe stores in City X, 30 carry Brand A

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