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CAT 2018 Slot 1 — QA Question 31

Venn DiagramEasy

Each of 74 students in a class studies at least one of the three subjects H, E and P. Ten students study all three subjects, while twenty study H and E, but not P. Every student who studies P also studies H or E or both. If the number of students studying H equals that studying E, then the number of students studying H is

Answer & solution

Correct answer: 52

Answer: 52

Solution

Easy

Set up a three-set Venn diagram. The key clue — "every P student also studies H or E" — means there is no "P only" region. Use the total of 7474 and the equality n(H)=n(E)n(H)=n(E) to pin the H-count.

Regions of the Venn diagram (P has no exclusive region):

all three=10,H and E only (not P)=20a=H only,c=E onlyb=H and P only,d=E and P only\begin{aligned} &\text{all three}=10,\qquad \text{H and E only (not P)}=20\\ &a=\text{H only},\quad c=\text{E only}\\ &b=\text{H and P only},\quad d=\text{E and P only} \end{aligned}
H E P a c 20 b d 10
1

Use the total headcount. All regions sum to 7474; the two fixed regions (1010 and 2020) total 3030.

a+b+c+d+20+10=74⇒ a+b+c+d=44...(1)\begin{aligned} &a+b+c+d+20+10 = 74\\ &\Rightarrow\ a+b+c+d = 44 \quad\text{...(1)} \end{aligned}
2

Use n(H)=n(E)n(H)=n(E). The shared regions (2020 and 1010) cancel on both sides, leaving the exclusive H-side equal to the exclusive E-side.

n(H)=a+b+20+10,n(E)=c+d+20+10⇒ a+b=c+d...(2)\begin{aligned} &n(H)=a+b+20+10,\qquad n(E)=c+d+20+10\\ &\Rightarrow\ a+b = c+d \quad\text{...(2)} \end{aligned}
3

Combine (1) and (2).

a+b=c+d=442=22[(1) with (2)]⇒ n(H)=(a+b)+20+10=22+30=52\begin{aligned} &a+b = c+d = \tfrac{44}{2}=22 \quad\text{[(1) with (2)]}\\ &\Rightarrow\ n(H) = (a+b)+20+10 = 22+30 = 52 \end{aligned}
n(H)=52n(H) = 52

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CAT 2018 Slot 1 QA Q31: Each of 74 students in a class studies at least one of the three subjects H, E and P. Ten students study all t — Solution | TheCATExam