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CAT 2024 Slot 3 QA Question & Solution

ArithmeticMedium

Question

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

Options

72 and 88, respectively
75 and 96, respectively
72 and 80, respectively
75 and 90, respectively

Solution

Total number of students = 250.

Given that the percentage of girls is at least 44% and at most 60%, we have: $$ 0.44(250) \le G \le 0.60(250) $$ $$ 110 \le G \le 150 $$

Let:

  • Number of boys = $B$
  • Number of girls = $G$ So,
    $$ B + G = 250 $$

Statement 1: If 50% of boys and 80% of girls opted for swimming, then
Swimming enrolments = $0.5B + 0.8G$

Statement 2: If 70% of boys and 60% of girls opted for running, then
Running enrolments = $0.7B + 0.6G$

Total enrolments (swimming + running): $$ (0.7B + 0.6G) + (0.5B + 0.8G) = 1.2B + 1.4G $$

Using the overlapping principle:

Let:

  • $I$ = number of people enrolled in exactly one activity
  • $II$ = number of people enrolled in both activities

We know: $$ I + II = B + G = 250 $$ $$ I + 2II = 1.2B + 1.4G $$

Subtracting these: $$ II = (1.2B + 1.4G) - (B + G) $$ $$ II = 0.2B + 0.4G $$

Factorizing: $$ II = 0.2(B + 2G) $$

Since $B + G = 250$, substitute $B = 250 - G$: $$ II = 0.2(250 + G) $$

Range of $G$:

  • Minimum $G = 110$
  • Maximum $G = 150$

Maximum value of $II$: $$ II_{\max} = 0.2(250 + 150) = 80 $$

Minimum value of $II$: $$ II_{\min} = 0.2(250 + 110) = 72 $$

Final answer:

The number of students enrolled in both activities lies in the range $\boxed{72 \text{ to } 80}$