Answers:
x = 7
y = 5
z = 5
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Explanation:
There are 4 distinct regions:
- The region inside circle P, but not inside circle Q. This is marked with x.
- The region in both circles P and Q. This is marked as y.
- The region inside circle Q, but not inside circle P. This is marked with z.
- The region outside both circles, but inside the rectangle. This is marked with 3
With circle P, the values of x and y must add to 12, since n(P) = 12. So x+y = 12. This solves to y = 12-x.
Similarly, y+z = 10 because n(Q) = 10, and y,z are in circle Q. This solves to y = 10-z
The values in the four regions must add to 20, as this is the total number in the universal set.
x+y+z+3 = 20
x+12-x+z = 20 ... plug in y = 12-x; solve for z
12+z+3 = 20
15+z = 20
z = 20-15
z = 5
Use this to find y
y = 10-z
y = 10-5
y = 5
Use this to find x
x+y = 12
x+5 = 12
x = 12-5
x = 7
