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Of the 500 patients who were checked for allergic diseases/symptoms, 40% had allergy from pollen, 50% had allergy from fur and 60% had allergy from dust. If all the patients had at least one of the aforesaid allergies and 20% of the patients had exactly two of the aforesaid allergies, how many of the patients had only one of these allergies?
  • a)
    275
  • b)
    300
  • c)
    325
  • d)
    350
Correct answer is option 'C'. Can you explain this answer?
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Of the 500 patients who were checked for allergic diseases/symptoms, 4...
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Of the 500 patients who were checked for allergic diseases/symptoms, 4...
Let a = number of patients having allergy from only one of the things, b = number of patients having allergy from exactly two of the things and c = number of patients having allergy from all three of the things.
Then, a + b + c = 500 (since each of the 500 patients had allergy from at least one of the things).
Given that b = 20% of 500 = 100,
So, a + 100 + c = 500 or, a + c = 400……… І
Also, if the number of patients who have allergy from pollen (40% of 500, i.e. 200) is added to the number of patients who have allergy from fur (50% of 500, i.e. 250), and this sum is added to the number of patients who have allergy from dust (60% of 500, i.e. 300), then each patient who has allergy from only one of the given things is counted exactly once, each patient who has allergy from exactly two of the given things is counted exactly twice and each patient who has allergy from all of the three given things is counted exactly 3 times.
Therefore, a + 2b + 3c = 200 + 250 + 300 = 750.
Using b = 100, we get:
a + 2(100) + 3c = 750 or, a + 3c = 550……. ІІ
Solving equations І and ІІ, we get,
a = 325.
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Of the 500 patients who were checked for allergic diseases/symptoms, 40% had allergy from pollen, 50% had allergy from fur and 60% had allergy from dust. If all the patients had at least one of the aforesaid allergies and 20% of the patients had exactly two of the aforesaid allergies, how many of the patients had only one of these allergies?a)275b)300c)325d)350Correct answer is option 'C'. Can you explain this answer?
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