Question:medium

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

Updated On: Jan 15, 2026

  • 48

  • 53

  • 47

  • 52

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The Correct Option is C

Solution and Explanation

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

Step 2: Formulate equations
Based on the provided data, we establish two equations:
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 isolate (s):
\([s+2d+3t-(n+s+d+t)=205-100]\)
Simplification yields:
\([d+2t-n=105]\)
This equation establishes a relationship between (d), (n), and (t).

Step 4: Determine the maximum and minimum values of (t)
Given the constraints \(0 \leq n,s,d,t \leq 100\) and \((n+s+d+t=100)\), we analyze extreme scenarios to find the range of (t):
a) Maximum value: Assume \((t=52)\) (maximum possible students liking all three).
Solving Equation 3 for (d):
\([d + 2(52)-n=105\) --> \((d-n=1)\)
With non-negative values for (d) and (n), the only solution is \((d=1)\) and \((n=0).\)
b) Minimum value: Assume \((t=5)\) (minimum plausible students liking all three).
Solving Equation 3 for (d):
\([d+2(5)-n=105\) --> \((d-n=95)\)
The sole valid solution is \((d=95)\) and \((n=0).\)

Step 5: Calculate the difference
The difference between the maximum and minimum possible values of (t) is calculated as:
\([47=52-5]\)
Consequently, the difference between the highest and lowest possible number of students liking all three drinks is 47.

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