Question

A survey of 100 students produced the following statics: 32 study mathematics, 20 study physics, 45 study Biology, 15 study mathematics and biology, 7 study mathematics and physics, 10 study physics and Biology, 30 do not study any of the three subjects
1, Present the information in a Venn Diagram
ii, Find the number of students who study all the three subjects
iii, Take exactly one of the three subjects

Answers

Reiny
We don't know the intersection of all three circles, so label it x
Now go to the intersection of two of the circles, say, the M and B circles
It says that sum is 15, but we have already counted x of them, so in the remaining part of the football-shaped region is 15-x
Do that for the other two intersection of pairs to have 10-x and 7-x

Now go to the whole circles, start with M
We have already counted, 15-x, x, and 7-x
or 22-x
But the whole math circle contains 32, so the open part must be 32-(22-x) = 10+x
Do the same with the other whole circles and get
20+x and 3+x

Now add them all up.
We can use a short-cut here, we know that everything in the Math circle is 32, so ..

32 + 10-x + 20+x + 3+x = 70
x = 5

ii) 5 take all three subjects
ii) add up all the parts in the outer parts of the circle which do not intersect with any other circles
that is,
Math only = 27
Biology only = 25
Physics only = 8

or

using the formula:
N(M or P or B) = N(M) + N(B) + N(P) - N(M and P) - N(M and B) - N(P and B) + N(M and P and B)
70 = 32+45+20 - 7 - 15 -10 + x
x= 5

Vaidurya Nigam
Maths only is 25 because 45+5-25
Bio only is 15 because 32+5-22
Physics only is 8 because 20+5-17
KDC
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