Discussion

Explanation:

Let number of people who can speak
Exactly one of the three languages = a
Exactly one of the three languages = b
Exactly one of the three languages = c
None of the three languages = n

⇒ a + b + c + n = 100   …(1)

Also, a + 2b + 3c = 65 + 55 + 60
⇒ a + 2b + 3c = 180   …(2)

Now, we need to minimize b.

Let’s see if b can be 0.
From (1), if b = 0 and highest possible value of c = 55, then highest value of a = 45.
⇒ a + 2b + 3c = 210

But we want this sum to be 180 [from (2)].

So we will have to reduce the value of c and increase the value of the variable with smaller coefficient i.e., a or n.

Now, if we decrease c by 1 unit and increase n by 1 unit.
a + 2b + 3c = 207 i.e., the sum decreases by 3 units.

∴ To decrease the sum from 210 to 180, we need to decrease it by 30 units, hence we need to transfer 10 units from c to n.

∴ Value of a = 45, c = 45, b = 0 and n = 10.

Following is a possible Venn diagram for this case.

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Hence, 0.

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