Averages , Ratio and Proportion Question 1 - CAT 2024
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:
Step-by-Step Explanation
- Find the boundaries for the student count:
Total students = 250. Let G represent the number of girls and B represent the number of boys.
0.44 × 250 ≤ G ≤ 0.60 × 250
⟹ 110 ≤ G ≤ 150
- Set up the Venn diagram enrollment system:
Total Swimming registrations = 0.5B + 0.8G
Total Running registrations = 0.7B + 0.6G
Total Registrations = (0.5B + 0.8G) + (0.7B + 0.6G)
= 1.2B + 1.4G
- Apply the set inclusion-exclusion principle:
Total Registrations = Total Students + Both
1.2B + 1.4G = 250 + Both
Substitute B = 250 − G:
1.2(250 − G) + 1.4G = 250 + Both
300 + 0.2G = 250 + Both
⟹ Both = 50 + 0.2G
- Calculate the extreme boundaries:
- Minimum: Occurs when G = 110
Bothₘᵢₙ = 50 + 0.2(110) = 72
- Maximum: Occurs when G = 150
Bothₘₐₓ = 50 + 0.2(150) = 80
Correct Option: C
💡 FundaMakers Tip:
Summing the enrollment rates (1.2B + 1.4G) and subtracting the student base (B + G = 250) quickly isolates the overlap formula Both = 0.2B + 0.4G. Substituting B = 250 − G simplifies the expression to Both = 50 + 0.2G, turning a multi-set check into a quick validation of endpoints.