Alarm Count
Please provide the venn diagram solution for the following question.?
64 cars were studied to count alarm systems, leather seats, and CD players.
45 had an alarm
32 had two options
8 had three options
22 had an alarm and a CD player
3 had only a CD player
10 had an alarm and leather seats but not a CD player
37 did not have leather seats
a: How many cars have only leather seats?
b: How many cars have at least one option?
For the Venn Diagram with three circles for the three options, you obviously have 8 distinct regions. They are:
a) alarm system only
b) leather seats only
c) CD players only
d) alarm systems and leather seats, but not CD players
e) alarm systems and CD players, but not leather seats
f) leather seats and CD players, but not alarm systems
g) all 3 of alarm systems, leather seats, and CD players
h) no options
Unfortunately two of your data statements are worded slightly vaguely. Does “32 had two options” include the those with three options? And does “22 had an alarm and a CD player” include those with all three as well? For this example, I will take the answer to both of those questions as a yes, since that is the most literal interpretation of them.
So let’s evaluate the data now,
@ 8 had three options – means that the g above is 8, and that you’d put 8 into the region which is the intersection of the three circles
@ 10 had an alarm and leather seats but not a CD player – means that d is 10, and that you put 8 into the region which is the intersection of the alarm systems circle and the leather seats circle, excluding the CD players circle
@ 3 had only a CD player – means that c is 3, and you put 3 in the region that is the circle for the CD player that does not intersect the other two circles
@ 22 had an alarm and a CD player – means that e is 14 (22 alarm systems + CD players only – 8 with all three = 14), so you put 14 in the region where the alarm systems circle intersects the CD players circle excluding the part where all three intersect
@ 32 had two options – now tells you that f is 0 (32 with two or three options – 10 for d – 14 for e – 8 for g = 0), so the region where only the leather seats and CD players intersect should have 0 in it
@ 45 had an alarm – now tells you that a is 13 (45 total with alarms – 10 for d – 14 for e – 8 for g = 13 with alarm systems only), so the region of the circle for the alarm systems that does not intersect anything else has 13 in it
@ 37 did not have leather seats – now tells you that b is 9 (64 cars – 37 without leather seats = 27 with leather seats, then 27 with leather seats – 10 for d – 0 for f – 8 for g = 9 with leather seats only), so the region of the circle for leather seats that does not intersect anything should have a 9 in it
Since the a + b + c + d + e + f + g + h = 64, and a + b + c + d + e + f + g = 57, then h = 7, so the region outside the three circles has 7 in it.
This gives you the full Venn diagram, so now the answers are easy
A: How many cars have only leather seats? 9
B: How many cars have at least one option? 57
(Note that of the four possible ways to handle the two data elements that were slightly vague, one of them will not give a real solution, while the other two both give 57 as the answer to how many cars have at least one option. But for the question about only leather seats, one will give the same answer as for this one, i.e. 9, while the other gives 1 for that region.)
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