Question

In a class of 100 students, 73 like coffee, 80 like tea, and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

• A. 48
• B. 53
• C. 47
• D. 52

Answer 1

The Correct Option is C

Step 1: Define the variables
Let (n),(s),(d),and (t) represent the number of students who like none, exactly one, exactly two, and all three drinks respectively.

Step 2: Write the equations
We have two equations based on the information given:
Equation \(1:\)\((n+s+d+t=100)\)
Equation \(2:(s+2d+3t=205)\)

Step 3: Express (d) in terms of (n) and (t)
Subtract Equation 1 from Equation 2 to eliminate (s):
\([s+2d+3t-(n+s+d+t)=205-100]\)
Simplify:
\([d+2t-n=105]\)
This equation relates the number of students who like exactly two drinks (d) with the number of students who like none (n) and all three (t)

Step 4: Find maximum and minimum values of (t)
We know that (0 leq n,s,d,t leq 100) and \((n+s+d+t=100). \)
To find the maximum and minimum possible values of (t), we consider extreme cases:
a) Maximum: Let \((t=52)\) (all students like all three drinks). 
Solve Equation 3 for (d):
\([d + 2(52)-n=105\) implies \(d-n=1]\)
Since (d) and (n) should be non-negative, the only possible solution is \((d=1)\) and \((n=0).\)
b) Minimum: Let \((t=5)\) (only a few students like all three drinks). 
Solve Equation 3 for (d):
\([d+2(5)-n=105\) implies \(d-n=95]\)
Again, the only possible solution is \((d=95) \)and\( (n=0).\)

Step 5: Calculate the difference
The difference between the maximum and minimum possible values of (t) is:
\([47=52-5]\)
Therefore, the difference between the maximum and minimum number of students who like all three drinks is 47.

Answer 2

Let's assume the following:
n be the number of students who likes none of the drinks
s be the number of students who likes exactly one drink
d be the number of students who likes exactly two drink
t be the number of students who likes all three drink

Now, given:
n + s + d + t = 100 …. (i)
s + 2d + 3t = 73 + 80 + 52
s + 2d = 205 …. (ii)

By subtracting equation (ii) from (i), we get:
d + 2t -n = 105

Now, maximum value which can be assigned to t is 52 i.e, t = 52, d =1 and n = 0
Minimum value can be assigned to t is 5 i.e, t =5, d = 95 and n = 0

So, the difference between this value is:
= 52 - 5
= 47
So, the correct option is (C): 47.