Question 202298: In a survey of 75 resorts, it was reported that:
45 provided refrigerators in the guest rooms
37 provided laundry services
44 provided business centers
23 provided refrigerators in the guest rooms and laundry services
29 provided refrigerators in the guest rooms and business centers
21 provided laundry services and business centers
12 provided all three features
Construct a Venn diagram and use it to answer the following questions:
(a) How many of the resorts provided only refrigerators in the guest rooms?
(b) How many of the resorts provided exactly one of the features?
(c) How many of the resorts provided at least one of the features?
(d) How many of the resorts provided exactly two of the features?
(e) How many of the resorts provided none of the features?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
You need to work this from the inside out, or in the order of the data presented, from the bottom to the top.
Start by drawing 3 circles. Make the circles overlap so that you have 7 regions, 1 in the center that is part of all three circles, 3 that are overlaps of two circles only, and 3 that are regions only in one of the circles each. In these 'one-circle-only' regions, put a label for each of the circles: R for refrigerators, L for laundry, and B for business.
Start with your first (last) bit of information, namely that 12 of the resorts offer all three services. In the center region where all three circles overlap, write the number 12.
Take the next piece of information, namely that 21 resorts provide laundry and business centers. Now this 21 includes both those resorts that provide ONLY laundry and business centers plus those that provide all three services. Since there were 12 that provide all three, subtract 12 from 21 to get 9 that ONLY provide laundry and business centers. Put the number 9 in the region that is the overlap between circles L and B.
Likewise, the overlap of L and R would get an 11 and the overlap between B and R would get a 17.
Now, the number that provides refrigerators includes the number that provides ONLY refrigerators, refrigerators and business centers, refrigerators and laundry, and all three services, so 45 minus (11 plus 17 plus 12) = 5. And you put a 5 in the part of the R circle that doesn't overlap anything else.
You should be able to fill in the final two regions by yourself.
(a) the number in the part of the R circle that doesn't overlap anything else.
(b) the sum of the three numbers in the parts of the three circles that don't overlap anything
(c) the sum of all the numbers in the diagram
(d) the sum of the numbers in the three areas that represent overlaps of two circles (all of the overlap regions except the center one)
(e) 75 minus your answer to (c)
John
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