The following diagram is based on a survey conducted among students of a class in the month of February. It shows how many students liked Tea, Coffee (hot drinks) and Coke (a cold drink), including the overlaps between the three.
Question: 1
When another survey was conducted in June among the same students, the result was different. All of them liked Coke. 12 liked Tea, but no one liked Coffee. How many students liked only Coke?
Show Hint
Find the total class strength from the February Venn diagram (35 students), then subtract the June Tea drinkers, since everyone in June already liked Coke.
Step 1: Note what stays fixed between surveys. The class does not change between February, June and December, only the drink preferences change. From the February Venn diagram, the total strength of the class is the sum of all seven regions: $4+2+11+5+3+4+6=35$ students.
Step 2: Translate the June facts into a simple picture. In June, picture just two groups inside the class of 35: those who like Coke and those who like Tea. Every student is in the Coke group, and 12 students are in the Tea group. Since no student likes Coffee at all in June, there is no third group to worry about.
Step 3: Use the fact that Tea drinkers sit inside the Coke group. Because all 35 students like Coke, the 12 Tea drinkers are automatically inside the Coke group too, so they overlap completely. The part of the Coke group not shared with Tea is what we want. $\text{Coke only} = \text{Total} - \text{Tea drinkers} = 35 - 12$
Step 4: Compute the answer. $35 - 12 = 23$ So 23 students like Coke and nothing else in the June survey. \[ \boxed{23} \]
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Question: 2
When another survey was conducted in December among the same students, the result was again different. All of them liked at least one hot drink. 16 liked Coke. No one liked all the three drinks. Which of the following conclusions is true?
Show Hint
The 16 Coke drinkers must each also like a hot drink, with no triple overlap, so the remaining 35 minus 16 equals 19 students hold the whole Tea and Coffee group, which can never reach 20.
Number of students who liked both Tea and Coffee is less than 20.
Only 10 liked both Tea and Coffee.
Show Solution
The Correct Option isC
Solution and Explanation
Step 1: Assign letters to each group. Let $t$ be students who like Tea only, $c$ be students who like Coffee only, $x$ be students who like both Tea and Coffee but not Coke, $p$ be students who like both Coke and Tea but not Coffee, and $q$ be students who like both Coke and Coffee but not Tea. Since no student likes all three drinks, there is no all three group to add.
Step 2: Use the at least one hot drink condition. Since every one of the 35 students likes Tea or Coffee, no student can sit in a Coke only group or a likes nothing group. So the five groups $t, c, x, p, q$ must add up to the full class. $t + c + x + p + q = 35$
Step 3: Use the Coke total. Every Coke drinker is counted in either $p$ (Coke and Tea) or $q$ (Coke and Coffee), since the all three group is empty. We are told 16 students like Coke, so $p + q = 16$. Substituting this into the total from Step 2 gives $t + c + x = 35 - 16 = 19$.
Step 4: Bound the Tea and Coffee group. The group liking both Tea and Coffee, without Coke, is exactly $x$. From $t + c + x = 19$ and $t, c \geq 0$, the largest $x$ can possibly be is 19, when $t = c = 0$. So $x \leq 19$, meaning $x$ is always strictly less than 20, no matter how the 19 non Coke students split between $t$, $c$ and $x$.
Step 5: Rule out the rest. No one liked Coke and Coffee would need $q = 0$, but $q$ can be anywhere from 0 to 16, so this is not always true. Some liked Coke and Tea would need $p > 0$, but $p$ could equally be 0 if $q = 16$, so this is not always true either. Only 10 liked both Tea and Coffee fixes $x = 10$ exactly, but $x$ can be anywhere from 0 to 19, so this is just one possibility, not a certainty. \[ \boxed{x < 20 \text{ always holds, so option 3 is correct}} \]
