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CAT 2024 Slot 3 — QA Question 6

Venn DiagramEasy

In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

Answer & solution

  • A

    75 and 90, respectively

  • 72 and 80, respectively

  • C

    75 and 96, respectively

  • D

    72 and 88, respectively

Solution

Hard

For each gender, swimming% + running% exceeds 100%, so the surplus is exactly the percentage doing both (since everyone does at least one). Express "both" as a function of the number of girls, then push that count to its allowed extremes (44% and 60% of 250).

1

Both = overlap surplus. Each student opts for swimming or running or both, so for any group, swim+run−both=total\text{swim}+\text{run}-\text{both}=\text{total}, i.e. both=swim+run−total\text{both}=\text{swim}+\text{run}-\text{total}.

Boys both=(50%+70%−100%) of boys=20% of boys⇒ Girls both=(80%+60%−100%) of girls=40% of girls\begin{aligned} &\text{Boys both}=(50\%+70\%-100\%)\text{ of boys}=20\%\ \text{of boys}\\ &\Rightarrow\ \text{Girls both}=(80\%+60\%-100\%)\text{ of girls}=40\%\ \text{of girls} \end{aligned}
2

Total "both" in terms of girls. Let gg = number of girls, so boys =250−g=250-g.

Both=0.20(250−g)+0.40g⇒ Both=50−0.2g+0.4g(expand)⇒ Both=50+0.2g\begin{aligned} &\text{Both}=0.20(250-g)+0.40g\\ &\Rightarrow\ \text{Both}=50-0.2g+0.4g\quad\text{(expand)}\\ &\Rightarrow\ \text{Both}=50+0.2g \end{aligned}
3

Range of girls. Girls are at least 44%44\% and at most 60%60\% of 250250. (Both endpoints give whole numbers, so they are attainable.)

gmin⁡=0.44×250=110⇒ gmax⁡=0.60×250=150\begin{aligned} &g_{\min}=0.44\times 250=110\\ &\Rightarrow\ g_{\max}=0.60\times 250=150 \end{aligned}
4

Plug in the extremes. "Both" increases with gg, so use g=110g=110 for the minimum and g=150g=150 for the maximum.

Bothmin⁡=50+0.2(110)=72(from step 2,3)⇒ Bothmax⁡=50+0.2(150)=80\begin{aligned} &\text{Both}_{\min}=50+0.2(110)=72\quad\text{(from step 2,3)}\\ &\Rightarrow\ \text{Both}_{\max}=50+0.2(150)=80 \end{aligned}
72 and 80, respectively72\ \text{and}\ 80,\ \text{respectively}
CAT 2024 Slot 3 QA Q6: In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students — Solution | TheCATExam