CAT 2024

Slot 3

QA

Question & SolutionQues & Sol

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

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 $\boxed{72 \text{ to } 80}$